
 and i cherish more than anything else the analogies , my most trustworthy masters .
they know all the secrets of nature , and they ought least to be neglected in geometry. johannes kepler 
 figure 1artistic impression of cascading sound cones ( in the geometrical acoustics limit ) forming an acoustic black hole when supersonic flow tips the sound cones past the vertical . 

figure 2artistic impression of trapped waves ( in the physical acoustics limit ) forming an acoustic black hole when supersonic flow forces the waves to move downstream . 
 
they know all the secrets of nature , and they ought least to be neglected in geometry . artistic impression of cascading sound cones ( in the geometrical acoustics limit ) forming an acoustic black hole when supersonic flow tips the sound cones past the vertical .
artistic impression of trapped waves ( in the physical acoustics limit ) forming an acoustic black hole when supersonic flow forces the waves to move downstream .
analogies have played a very important role in physics and mathematics  they provide new ways of looking at problems that permit cross - fertilization of ideas among different branches of science .
a carefully chosen analogy can be extremely useful in focussing attention on a specific problem , and in suggesting unexpected routes to a possible solution . in this review article
, the development of analogies ( typically but not always based on condensed matter physics ) to probe aspects of the physics of curved spacetime  and in particular to probe aspects of curved space quantum field theory .
the most well - known of these analogies is the use of sound waves in a moving fluid as an analogue for light waves in a curved spacetime .
supersonic fluid flow can then generate a  dumb hole  , the acoustic analogue of a  black hole  , and the analogy can be extended all the way to mathematically demonstrating the presence of phononic hawking radiation from the acoustic horizon .
this particular provides ( at least in principle ) a concrete laboratory model for curved - space quantum field theory in a realm that is technologically accessible to experiment .
there are many other  analogue models  that may be useful for this or other reasons  some of the analogue models are interesting for experimental reasons , others are useful for the way they provide new light on perplexing theoretical questions .
the information flow is in principle bidirectional and sometimes insights developed within the context of general relativity can be used to understand aspects of the analogue model . of course analogy
is not identity , and we are in no way claiming that the analogue models we consider are completely equivalent to general relativity  merely that the analogue model ( in order to be interesting ) should capture and accurately reflect a sufficient number of important features of general relativity ( or sometimes special relativity ) .
the list of analogue models is extensive , and in this review we will seek to do justice both to the key models , and to the key features of those models . in the following chapters we shall : 
 discuss the flowing fluid analogy in some detail.summarise the history and motivation for various analogue models.discuss the many physics issues various researchers have addressed.provide a ( hopefully complete ) catalogue of extant models.discuss the main physics results obtained to date.outline the many possible directions for future research.summarise the current state of affairs . discuss the flowing fluid analogy in some detail .
discuss the many physics issues various researchers have addressed . provide a ( hopefully complete ) catalogue of extant models .
the interested reader will have had a quite thorough introduction to the ideas , techniques , and hopes of the analogue gravity programme .
apart from this present review article , and the references contained herein , there are several key items that stand out as starting points for any deeper investigation : 
 the book  artificial black holes  , edited by mario novello , matt visser , and grigori volovik .the websites for the  analogue models  workshop : 

 http://www.cbpf.br/bscg/analog/ 

 http://www.mcs.vuw.ac.nz/visser/analog/ 

 http://www.physics.wustl.edu/visser/analog/ 
 the book  the universe in a helium droplet  , by grigori volovik .the physics reports article ,  superfluid analogies of cosmological phenomena  , by grigori volovik .
the book  artificial black holes  , edited by mario novello , matt visser , and grigori volovik .
the websites for the  analogue models  workshop : 

 http://www.cbpf.br/bscg/analog/ 

 http://www.mcs.vuw.ac.nz/visser/analog/ 

 http://www.physics.wustl.edu/visser/analog/ 
 
 http://www.cbpf.br/bscg/analog/ 
 

http://www.physics.wustl.edu/visser/analog/ 
  the book  the universe in a helium droplet  , by grigori volovik .
the physics reports article ,  superfluid analogies of cosmological phenomena  , by grigori volovik .
acoustics in a moving fluid is the simplest and cleanest example of an analogue model [ 376 , 387 , 391 , 389 ] . the basic physics is simple , the conceptual framework is simple , and specific computations are often simple ( whenever , that is , they are not impossibly hard).1 the basic physics is this : a moving fluid will drag sound waves along with it , and if the speed of the fluid ever becomes supersonic , then in the supersonic sound waves will never be able to fight their way back upstream [ 376 , 387 , 391 , 389 ] .
this implies the existence of a  dumb hole  , a region from which sound can not escape.2 of course this sounds very similar , at the level of a non - mathematical verbal analogy , to the notion of a  black hole  in general relativity .
the real question is whether this verbal analogy can be turned into a precise mathematical and physical statement  it is only after we have a precise mathematical and physical connection between ( in this example ) the physics of acoustics in a fluid flow and at least some significant features of general relativity that we can claim to have an
 analogue model of ( some aspects of ) gravity  . we ( and the community at large ) often abuse language by referring to such a model as  analogue gravity  for short . 

figure 3a moving fluid will drag sound pulses along with it . a moving fluid will drag sound pulses along with it . now the features of general relativity that one typically captures in an  analogue model  are the kinematic features that have to do with how fields ( classical or quantum ) are defined on curved spacetime , and the sine qua non of any analogue model is the existence of some  effective metric  that captures the notion of the curved spacetimes that arise in general relativity .
( at the very least , one might wish to capture the notion of the minkowski geometry of special relativity . ) indeed , the verbal description above ( and its generalizations in other physical frameworks ) can be converted into a precise mathematical and physical statement , which ultimately is the reason that analogue models are of physical interest .
the advantage of geometrical acoustics is that the derivation of the precise mathematical form of the analogy is so simple as to be almost trivial , and that the derivation is extremely general .
the disadvantage is that in the geometrical acoustics limit one can deduce only the causal structure of the spacetime , and does not obtain a unique effective metric .
the advantage of physical acoustics is that while the derivation of the analogy holds in a more restricted regime , the analogy can do more for you in that it can now specify a specific effective metric and accommodate a wave equation for the sound waves . at the level of geometrical acoustics we need
only assume that : 
 the speed of sound c , relative to the fluid , is well defined.the velocity of the fluid v , relative to the laboratory , is well defined .
the velocity of the fluid v , relative to the laboratory , is well defined .
then , relative to the laboratory , the velocity of a sound ray propagating , with respect to the fluid , along the direction defined by the unit vector n is 1\documentclass[12pt]{minimal } 
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				 \begin{document}$${{{\rm{d}}{\bf{x } } } \over { { \rm{d}}t } } = c\,{\bf{n } } + { \bf{v}}.$$\end{document } this defines a sound cone in spacetime given by the condition n = 1 , i.e. , 2\documentclass[12pt]{minimal } 
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				 \begin{document}$$- { c^2}\,{\rm{d}}{t^2 } + { ( { \rm{d}}{\bf{x } } - { \bf{v}}\,{\rm{d}}t)^2 } = 0.$$\end{document } that is 3\documentclass[12pt]{minimal } 
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				 \begin{document}$$- [ { c^2 } - { v ^2}]\;{\rm{d}}{t^2 } - 2{\bf{v}}\ , \cdot \,{\rm{d}}{\bf{x}}\,{\rm{d}}t + { \rm{d}}{\bf{x}}\ , \cdot \,{\rm{d}}{\bf{x } } = 0.$$\end{document } 
 figure 4a moving fluid will tip the 
supersonic flow will tip the sound cones past the vertical . solving this quadratic equation for dx as a function of dt provides a double cone associated with each point in space and time .
this is associated with a conformal class of lorentzian metrics [ 376 , 387 , 391 , 389 , 284 ] 4 where  is an unspecified but non - vanishing function .
the virtues of the geometric approach are its extreme simplicity and the fact that the basic structure is dimension - independent . moreover
this logic rapidly ( and relatively easily ) generalises to more complicated physical situations.3 it is well known that for a static homogeneous inviscid fluid the propagation of sound waves is governed by the simple wave equation [ 219 , 221 , 264 , 353 ] 5\documentclass[12pt]{minimal } 
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				 \begin{document}$$\partial _
t^2\phi = { c^2}{\nabla^2}\phi.$$\end{document } generalizing this result to a fluid that is non - homogeneous , or to a fluid that is in motion , possibly even in non - steady motion , is more subtle than it at first would appear . to derive a wave equation in this more general situation
we shall start by adopting a few simplifying assumptions to allow us to derive the following theorem .
if a fluid is barotropic and inviscid , and the flow is irrotational ( though possibly time dependent ) then the equation of motion for the velocity potential describing an acoustic disturbance is identical to the dalembertian equation of motion for a minimally coupled massless scalar field propagating in a ( 3 + 1)-dimensional lorentzian geometry 
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				 \begin{document}$$\delta \phi \equiv { 1 \over { \sqrt { - g}}}{\partial _ \mu}(\sqrt { - g } \,{g^{\mu \nu}}\,{\partial _
\nu}\phi ) = 0.$$\end{document } 
 under these conditions , the propagation of sound is governed by an acoustic metric  g(t , x ) .
this acoustic metric describes a ( 3 + 1)-dimensional lorentzian ( pseudo - riemannian ) geometry . the metric depends algebraically on the density , velocity of flow , and local speed of sound in the fluid .
specifically 
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				 \begin{document}$${g_{\mu \nu}}(t,{\bf{x } } ) \equiv { \rho \over c}\left [ { \begin{array}{*{20}c } { - ( { c^2 } - { v ^2 } ) } & \vdots & { - { { \bf{v}}^t } } \\ { \cdots \cdots \cdots \cdots } & \cdot & { \cdots \cdots } \\ { - { \bf{v } } } & \vdots & { \bf{i } } \\ \end{array } } \right].$$\end{document } 
 ( here 
 i 
 is the 3  3 identity matrix . ) in general , when the fluid is non - homogeneous and flowing , the acoustic riemann tensor associated with this lorentzian metric will be nonzero . comment .
it is quite remarkable that even though the underlying fluid dynamics is newtonian , nonrelativistic , and takes place in flat space plus time , the fluctuations ( sound waves ) are governed by a curved ( 3 + 1)-dimensional lorentzian ( pseudo - riemannian ) spacetime geometry . for practitioners of general relativity this observation describes a very simple and concrete physical model for certain classes of lorentzian spacetimes , including ( as we shall later see ) black holes . on the other hand , this discussion is also potentially of interest to practitioners of continuum mechanics and fluid dynamics in that it provides a simple concrete introduction to lorentzian differential geometric techniques .
the fundamental equations of fluid dynamics [ 219 , 221 , 264 , 353 ] are the equation of continuity 8\documentclass[12pt]{minimal } 
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				 \begin{document}$${\partial _ t}\rho + { \bf{\nabla } } \cdot \left({\rho \;{\bf{v } } } \right ) = 0,$$\end{document } and euler s equation ( equivalent to f = ma applied to small lumps of fluid ) 9\documentclass[12pt]{minimal } 
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				 \begin{document}$$\rho { { { \rm{d}}{\bf{v } } } \over { { \rm{d}}t } } \equiv \rho \,[{\partial _ t}{\bf{v } } + ( { \bf{v } } \cdot { \bf{\nabla}}){\bf{v } } ] = { \bf{f}}.$$\end{document } start the analysis by assuming the fluid to be inviscid ( zero viscosity ) , with the only forces present being those due to pressure.4 then for the force density we have 10\documentclass[12pt]{minimal } 
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				 \begin{document}$${\bf{f } } = - { \bf{\nabla } } p.$$\end{document } via standard manipulations the euler equation can be rewritten as 11\documentclass[12pt]{minimal } 
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				 \begin{document}$${\partial _
t}{\bf{v } } = { \bf{v } } \times ( { \bf{\nabla } } \times { \bf{v } } ) - { 1 \over \rho}{\bf{\nabla}}p - { \bf{\nabla}}\left({{1 \over 2}\;{v ^2 } } \right).$$\end{document } now take the flow to be vorticity free , that is , locally irrotational . introduce the velocity potential  such that v =  , at least locally .
if one further takes the fluid to be barotropic ( this means that  is a function of p only ) , it becomes possible to define 12\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{array}{*{20}c } { h(p ) = \int\nolimits_0^p { { { { \rm{d}}p{\prime } } \over { \rho ( p\prime ) } } } ; } & { { \rm{so}}\;{\rm{that } } } & { { \bf{\nabla}}h = { 1 \over \rho}{\bf{\nabla}}p } \\
\end{array}.$$\end{document } thus the specific enthalpy , h(p ) , is a function of p only .
euler s equation now reduces to 13\documentclass[12pt]{minimal } 
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				 \begin{document}$$- { \partial _
t}\phi + h + { 1 \over 2}{({\bf{\nabla}}\phi)^2 } = 0.$$\end{document } this is a version of bernoulli s equation .
now linearise these equations of motion around some assumed background ( 0,p0 , 0 ) . set 14\documentclass[12pt]{minimal } 
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				 \begin{document}$$\rho = { \rho _ 0 } + \epsilon { \rho _ 1 } + o({\epsilon ^2}),$$\end{document } 
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				 \begin{document}$$p = { p_0 } + \epsilon { p_1 } + o({\epsilon ^2}),$$\end{document } 
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				 \begin{document}$$\phi = { \phi _ 0 } + \epsilon { \phi
_ 1 } + o({\epsilon ^2}).$$\end{document } sound is defined to be these linearised fluctuations in the dynamical quantities .
note that this is the standard definition of ( linear ) sound and more generally of acoustical disturbances . in principle , of course , a fluid mechanic might really be interested in solving the complete equations of motion for the fluid variables (  , p ,  ) . in practice ,
it is both traditional and extremely useful to separate the exact motion , described by the exact variables , (  , p ,  ) , into some average bulk motion , ( 0 , p0 , 0 ) , plus low amplitude acoustic disturbances , ( 1 , p1,1 ) .
see , for example [ 219 , 221 , 264 , 353 ] . since this is a subtle issue that we have seen cause considerable confusion in the past ,
let us be even more explicit by asking the rhetorical question :  how can we tell the difference between a wind gust and a sound wave ? 
the answer is that the difference is to some extent a matter of convention  sufficiently low - frequency long - wavelength disturbances ( wind gusts ) are conventionally lumped in with the average bulk motion .
if you wish to be hyper - technical , we can introduce a high - pass filter function to define the bulk motion by suitably averaging the exact fluid motion .
there are no deep physical principles at stake here  merely an issue of convention .
the place where we are making a specific physical assumption that restricts the validity of our analysis is in the requirement that the amplitude of the high - frequency short - wavelength disturbances be small .
this is the assumption underlying the linearization programme , and this is why sufficiently high - amplitude sound waves must be treated by direct solution of the full equations of fluid dynamics .
linearizing the continuity equation results in the pair of equations 17\documentclass[12pt]{minimal } 
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				 \begin{document}$${\partial _
t}{\rho _ 0 } + { \bf{\nabla } } \cdot ( { \rho _ 0}\;{{\bf{v}}_0 } ) = 0,$$\end{document } 
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				 \begin{document}$${\partial _
1 } + { \bf{\nabla}}\cdot({\rho _ 1}\;{{\bf{v}}_0 } + { \rho _ 0}\;{{\bf{v}}_1 } ) = 0.$$\end{document } now , the barotropic condition implies 19\documentclass[12pt]{minimal } 
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				 \begin{document}$$h(p ) = h({p_0 } + \epsilon { p_1 } + o({\epsilon ^2 } ) ) = { h_0 } + \epsilon { { { p_1 } } \over { { \rho _ 0 } } } + o({\epsilon ^2}).$$\end{document } use this result in linearizing the euler equation .
we obtain the pair 20\documentclass[12pt]{minimal } 
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				 \begin{document}$$- { \partial _ t}{\phi _ 0 } + { h_0 } + { 1 \over 2}{({\bf{\nabla}}{\phi _ 0})^2 } = 0.$$\end{document } 
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				 \begin{document}$$- { \partial _ t}{\phi _ 1 } + { { { p_0 } } \over { { \rho _ 0 } } } - { { \bf{v}}_0 } \cdot { \bf{\nabla}}{\phi _ 1 } = 0.$$\end{document } this last equation may be rearranged to yield 22\documentclass[12pt]{minimal } 
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				 \begin{document}$${p_1 } = { \rho _ 0}\;({\partial _ t}{\phi _ 1 } + { { \bf{v}}_0 } \cdot { \bf{\nabla}}{\phi _ 1}).$$\end{document } use the barotropic assumption to relate 23\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rho _ 1 } = { { \partial \rho } \over { \partial p}}\,{p_1 } = { { \partial \rho } \over { \partial p}}\,{\rho _
t}{\phi _ 1 } + { { \bf{v}}_0 } \cdot { \bf{\nabla}}{\phi _ 1}).$$\end{document } now substitute this consequence of the linearised euler equation into the linearised equation of continuity .
we finally obtain , up to an overall sign , the wave equation : 24\documentclass[12pt]{minimal } 
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				 \begin{document}$$- { \partial _
t}{\phi _ 1 } + { { \bf{v}}_0 } \cdot { \bf{\nabla}}{\phi _ 1 } ) } \right ) + { \bf{\nabla } } \cdot \left({{\rho _ 0}{\bf{\nabla}}{\phi _ 1 } - { { \partial \rho } \over { \partial p}}{\rho _
1 } + { { \bf{v}}_0 } \cdot { \phi _ 1 } ) } \right ) = 0.$$\end{document } this wave equation describes the propagation of the linearised scalar potential 1 .
once 1 is determined , equation ( 22 ) determines p1 , and equation ( 23 ) then determines 1 .
the background fields p0 , 0 and v0 = 0 , which appear as time - dependent and position - dependent coefficients in this wave equation , are constrained to solve the equations of fluid motion for a barotropic , inviscid , and irrotational flow .
apart from these constraints , they are otherwise permitted to have arbitrary temporal and spatial dependencies .
now , written in this form , the physical import of this wave equation is somewhat less than pellucid . to simplify things algebraically ,
observe that the local speed of sound is defined by 25\documentclass[12pt]{minimal } 
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				 \begin{document}$${c^{- 2 } } \equiv { { \partial \rho } \over { \partial p}}.$$\end{document } now construct the symmetric 4  4 matrix 26\documentclass[12pt]{minimal } 
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				 \begin{document}$${f^{\mu \nu}}(t,{\bf{x } } ) \equiv { { { \rho _ 0 } } \over { { c^2}}}\;\left [ { \begin{array}{*{20}c } { - 1 } & \vdots & { - v _ 0^j } \\ { \cdots \cdots } & \cdot & { \cdots \cdots \cdots \cdots } \\ { - v _ 0^i } & \vdots & { ( { c^2}\,{\delta ^{ij } } - v _ 0^iv _ 0^j ) } \\ \end{array } } \right].$$\end{document } ( greek indices run from 03 , while roman indices run from 13 . ) then , introducing ( 3 + 1)-dimensional space - time coordinates , which we write as x  ( t ; x ) , the above wave equation ( 24 ) is easily rewritten as 27\documentclass[12pt]{minimal } 
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				 \begin{document}$${\partial _
\mu}({f^{\mu \nu}}\,{\partial _ \nu}{\phi _ 1 } ) = 0.$$\end{document } this remarkably compact formulation is completely equivalent to equation ( 24 ) and is a much more promising stepping - stone for further manipulations .
the remaining steps are a straightforward application of the techniques of curved space ( 3 + 1)-dimensional lorentzian geometry .
now in any lorentzian ( i.e. , pseudo - riemannian ) manifold the curved space scalar dalembertian is given in terms of the metric g(t , x ) by ( see , for example , [ 125 , 266 , 369 , 265 , 164 , 422 ] ) 28\documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \phi \equiv { 1 \over { \sqrt { - g}}}{\partial _ \mu}(\sqrt { - g } \,{g^{\mu \nu}}\,{\partial _ \nu}\phi).$$\end{document } the inverse metric , g(t , x ) , is pointwise the matrix inverse of g(t , x ) , while g  det(g ) .
thus one can rewrite the physically derived wave equation ( 24 ) in terms of the dalembertian provided one identifies 29\documentclass[12pt]{minimal } 
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				 \begin{document}$$\sqrt { - g } \,{g^{\mu \nu } } = { f^{\mu \nu}}.$$\end{document } this implies , on the one hand 30\documentclass[12pt]{minimal } 
				
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				 \begin{document}$$\det ( { f^{\mu \nu } } ) = { ( \sqrt { - g})^4}{g^{- 1 } } = g.$$\end{document } on the other hand , from the explicit expression ( 26 ) , expanding the determinant in minors yields 31\documentclass[12pt]{minimal } 
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				 \begin{document}$$\det ( { f^{\mu \nu } } ) = { \left({{{{\rho _ 0 } } \over { { c^2 } } } } \right)^4 } \cdot [ ( - 1 ) \cdot ( { c^2 } - v _ 0 ^ 2 ) - { ( - { v _ 0})^2 } ] \cdot [ { c^2 } ] \cdot [ { c^2 } ] = - { { \rho _ 0 ^ 4 } \over { { c^2}}}.$$\end{document } thus 32\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{array}{*{20}c } { g = - { { \rho _ 0 ^ 4 } \over { { c^2 } } } ; } & { \sqrt { - g } = { { \rho _ 0 ^ 2 } \over c}. } \\ \end{array}$$\end{document } we can therefore pick off the coefficients of the inverse acoustic metric 33\documentclass[12pt]{minimal } 
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				 \begin{document}$${g^{\mu \nu}}(t,{\bf{x } } ) \equiv { 1 \over { { \rho _ 0}^c}}\;\left [ { \begin{array}{*{20}c } { - 1 } & \vdots & { - v _ 0^j } \\
{ \cdots \cdots } & \cdot & { \cdots \cdots \cdots \cdots } \\ { - v _ 0^i } & \vdots & { ( { c^2}\,{\delta ^{ij } } - v _ 0^i\,v _ 0^j ) } \\ \end{array } } \right].$$\end{document } we could now determine the metric itself simply by inverting this 4  4 matrix ( and if the reader is not a general relativist , proceeding in this direct manner is definitely the preferred option ) . on the other hand , for general relativists it is even easier to recognise that one has in front of one a specific example of the arnowitt - deser - misner split of a ( 3 + 1)-dimensional lorentzian spacetime metric into space + time , more commonly used in discussing initial value data in general relativity .
the acoustic metric is then read off by inspection 34\documentclass[12pt]{minimal } 
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				 \begin{document}$${g_{\mu \nu } } \equiv { { { \rho _ 0 } } \over c}\;\left [ { \begin{array}{*{20}c } { - ( { c^2}\ , - v _ 0 ^ 2\ , ) } & \vdots & { - v _ 0^j } \\ { \cdots \cdots \cdots \cdots } & \cdot & { \cdots \cdots } \\ { - v _ 0^i } & \vdots & { { \delta _ { ij } } } \\ \end{array } } \right].$$\end{document } equivalently , the acoustic interval can be expressed as 35\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } \equiv { g_{\mu \nu}}\;{\rm{d}}{x^\mu}\;{\rm{d}}{x^\nu } = { { { \rho _ 0 } } \over c}\;\left [ { - { c^2}\;{\rm{d}}{t^2 } + ( { \rm{d}}{x^i } - v _ 0^i\,{\rm{d}}t)\;{\delta _ { ij}}\;({\rm{d}}{x^j } - v _ 0^j\,{\rm{d}}t ) } \right].$$\end{document } this completes the proof of the theorem .
 we have presented the theorem and proof , which closely follows the discussion in , in considerable detail because it is a standard template that can be readily generalised in many ways .
this discussion can then be used as a starting point to initiate the analysis of numerous and diverse physical models .
a few brief comments should be made before proceeding further : 
 observe that the signature of this effective metric is indeed (  , + , + , + ) , as it should be to be regarded as lorentzian.observe that in physical acoustics it is the inverse metric density , 36\documentclass[12pt]{minimal } 
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				 \begin{document}$${f^{\mu \nu } } = \sqrt { - g } \,{g^{\mu \nu}}$$\end{document } that is of more fundamental significance for deriving the wave equation than is the metric g itself .
( this observation continues to hold in more general situations where it is often significantly easier to calculate the tensor density f than it is to calculate the effective metric g.)it should be emphasised that there are two distinct metrics relevant to the current discussion : 
 the physical spacetime metric is in this case just the usual flat metric of minkowski space : 37\documentclass[12pt]{minimal } 
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				 \begin{document}$${\eta _ { \mu \nu } } \equiv { ( { \rm{diag } } [ - c_{{\rm{light}}}^2,1,1,1])_{\mu \nu}}.$$\end{document } ( here clight is the speed of light in vacuum . ) the fluid particles couple only to the physical metric . in fact the fluid motion is completely non - relativistic , so that v0  clight , and it is quite sufficient to consider galilean relativity for the underlying fluid mechanics.sound waves on the other hand , do not  see  the physical metric at all . acoustic perturbations couple only to the effective acoustic metric g.it is quite remarkable that ( to the best of our knowledge ) the acoustic metric was first derived and used in moncrief s studies of the relativistic hydrodynamics of accretion flows surrounding black holes .
we shall come back to this work later on , in our historical section.the geometry determined by the acoustic metric does however inherit some key properties from the existence of the underlying flat physical metric .
for instance , the topology of the manifold does not depend on the particular metric considered .
the acoustic geometry inherits the underlying topology of the physical metric  ordinary   with possibly a few regions excised ( due to whatever hard - wall boundary conditions one might wish to impose on the fluid ) . in systems constrained to have effectively less than 3 spacelike dimensions one can reproduce more complicated topologies ( consider for example an effectively one - dimensional flow in a tubular ring).furthermore , the acoustic geometry automatically inherits from the underlying newtonian time parameter , the important property of  stable causality  [ 164 , 422 ] .
note that 38\documentclass[12pt]{minimal } 
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				 \begin{document}$${g^{\mu \nu}}({\nabla _
\nu}t ) = - { 1 \over { { \rho _ 0}\,c } } < 0.$$\end{document } this precludes some of the more entertaining causality - related pathologies that sometimes arise in general relativity .
( for a general discussion of causal pathologies in general relativity , see for example [ 164 , 161 , 162 , 72 , 163 , 396]).other concepts that translate immediately are those of  ergo - region 
,  trapped surface  ,  apparent horizon  , and  event horizon  .
these notions will be developed more fully in the following subsection.the properly normalised four - velocity of the fluid is 39\documentclass[12pt]{minimal } 
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				 \begin{document}$${v^\mu } = { { ( 1;\;v _ 0^i ) } \over { \sqrt { { \rho _ 0}\,c}}},$$\end{document } so that 40\documentclass[12pt]{minimal } 
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				 \begin{document}$${g_{\mu \nu}}{v^\mu}{v^\nu } = g(v , v ) = - 1.$$\end{document } this four - velocity is related to the gradient of the natural time parameter by 41\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{array}{*{20}c } { { \nabla _ \mu}t = ( 1,0,0,0 ) ; } & { { \nabla ^\mu}t = - { { ( 1;\;v _ 0^i ) } \over { { \rho _ 0}\;c } } = - { { { v^\mu } } \over { \sqrt { { \rho _ 0}\;c}}}. } \\ \end{array}$$\end{document } thus the integral curves of the fluid velocity field are orthogonal ( in the lorentzian metric ) to the constant time surfaces .
the acoustic proper time along the fluid flow lines ( streamlines ) is 42\documentclass[12pt]{minimal } 
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				 \begin{document}$$\tau = \int { \sqrt { { \rho _ 0}\;c } \,{\rm{d}}t},$$\end{document } and the integral curves are geodesics of the acoustic metric if and only if 0 
 c is position independent.observe that in a completely general ( 3 + 1)-dimensional lorentzian geometry the metric has 6 degrees of freedom per point in spacetime .
( 4  4 symmetric matrix  10 independent components ; then subtract 4 coordinate conditions).in contrast , the acoustic metric is more constrained .
being specified completely by the three scalars 0 ( t , x ) , 0(t , x ) , and c(t , x ) , the acoustic metric has at most 3 degrees of freedom per point in spacetime .
the equation of continuity actually reduces this to 2 degrees of freedom , which can be taken to be 0(t , x ) and c(t , x).thus the simple acoustic metric of this section can at best reproduce some subset of the generic metrics of interest in general relativity.a point of notation : where the general relativist uses the word  stationary 
static  translates to a rather messy constraint on the fluid flow ( to be discussed more fully below).finally , we should emphasise that in einstein gravity the spacetime metric is related to the distribution of matter by the non - linear einstein - hilbert differential equations . in contrast , in the present context , the acoustic metric is related to the distribution of matter in a simple algebraic fashion .
observe that the signature of this effective metric is indeed (  , + , + , + ) , as it should be to be regarded as lorentzian .
observe that in physical acoustics it is the inverse metric density , 36\documentclass[12pt]{minimal } 
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				 \begin{document}$${f^{\mu \nu } } = \sqrt { - g } \,{g^{\mu \nu}}$$\end{document } that is of more fundamental significance for deriving the wave equation than is the metric g itself .
( this observation continues to hold in more general situations where it is often significantly easier to calculate the tensor density f than it is to calculate the effective metric g. ) it should be emphasised that there are two distinct metrics relevant to the current discussion : 
 the physical spacetime metric is in this case just the usual flat metric of minkowski space : 37\documentclass[12pt]{minimal } 
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				 \begin{document}$${\eta _ { \mu \nu } } \equiv { ( { \rm{diag } } [ - c_{{\rm{light}}}^2,1,1,1])_{\mu \nu}}.$$\end{document } ( here clight is the speed of light in vacuum . ) the fluid particles couple only to the physical metric . in fact the fluid motion is completely non - relativistic , so that v0  clight , and it is quite sufficient to consider galilean relativity for the underlying fluid mechanics.sound waves on the other hand , do not  see  the physical metric at all .
acoustic perturbations couple only to the effective acoustic metric g. the physical spacetime metric is in this case just the usual flat metric of minkowski space : 37\documentclass[12pt]{minimal } 
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				 \begin{document}$${\eta _ { \mu \nu } } \equiv { ( { \rm{diag } } [ - c_{{\rm{light}}}^2,1,1,1])_{\mu \nu}}.$$\end{document } ( here clight is the speed of light in vacuum . ) the fluid particles couple only to the physical metric . in fact the fluid motion is completely non - relativistic , so that v0  clight , and it is quite sufficient to consider galilean relativity for the underlying fluid mechanics .
sound waves on the other hand , do not  see  the physical metric at all .
acoustic perturbations couple only to the effective acoustic metric g. it is quite remarkable that ( to the best of our knowledge ) the acoustic metric was first derived and used in moncrief s studies of the relativistic hydrodynamics of accretion flows surrounding black holes .
the geometry determined by the acoustic metric does however inherit some key properties from the existence of the underlying flat physical metric .
for instance , the topology of the manifold does not depend on the particular metric considered .
the acoustic geometry inherits the underlying topology of the physical metric  ordinary   with possibly a few regions excised ( due to whatever hard - wall boundary conditions one might wish to impose on the fluid ) . in systems constrained to have effectively less than 3 spacelike dimensions one can reproduce more complicated topologies
( consider for example an effectively one - dimensional flow in a tubular ring ) .
furthermore , the acoustic geometry automatically inherits from the underlying newtonian time parameter , the important property of  stable causality  [ 164 , 422 ] .
note that 38\documentclass[12pt]{minimal } 
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				 \begin{document}$${g^{\mu \nu}}({\nabla _
\nu}t ) = - { 1 \over { { \rho _ 0}\,c } } < 0.$$\end{document } this precludes some of the more entertaining causality - related pathologies that sometimes arise in general relativity .
( for a general discussion of causal pathologies in general relativity , see for example [ 164 , 161 , 162 , 72 , 163 , 396 ] ) .
 trapped surface  ,  apparent horizon  , and  event horizon  .
the properly normalised four - velocity of the fluid is 39\documentclass[12pt]{minimal } 
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				 \begin{document}$${v^\mu } = { { ( 1;\;v _ 0^i ) } \over { \sqrt { { \rho _ 0}\,c}}},$$\end{document } so that 40\documentclass[12pt]{minimal } 
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				 \begin{document}$${g_{\mu \nu}}{v^\mu}{v^\nu } = g(v , v ) = - 1.$$\end{document } this four - velocity is related to the gradient of the natural time parameter by 41\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{array}{*{20}c } { { \nabla _ \mu}t = ( 1,0,0,0 ) ; } & { { \nabla ^\mu}t = - { { ( 1;\;v _ 0^i ) } \over { { \rho _ 0}\;c } } = - { { { v^\mu } } \over { \sqrt { { \rho _ 0}\;c}}}. } \\ \end{array}$$\end{document } thus the integral curves of the fluid velocity field are orthogonal ( in the lorentzian metric ) to the constant time surfaces . the acoustic proper time along the fluid flow lines ( streamlines ) is 42\documentclass[12pt]{minimal } 
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				 \begin{document}$$\tau = \int { \sqrt { { \rho _ 0}\;c } \,{\rm{d}}t},$$\end{document } and the integral curves are geodesics of the acoustic metric if and only if 0 
 c is position independent . observe that in a completely general ( 3 + 1)-dimensional lorentzian geometry the metric has 6 degrees of freedom per point in spacetime .
( 4  4 symmetric matrix  10 independent components ; then subtract 4 coordinate conditions ) .
being specified completely by the three scalars 0 ( t , x ) , 0(t , x ) , and c(t , x ) , the acoustic metric has at most 3 degrees of freedom per point in spacetime .
the equation of continuity actually reduces this to 2 degrees of freedom , which can be taken to be 0(t , x ) and c(t , x ) .
thus the simple acoustic metric of this section can at best reproduce some subset of the generic metrics of interest in general relativity .
static  translates to a rather messy constraint on the fluid flow ( to be discussed more fully below ) .
finally , we should emphasise that in einstein gravity the spacetime metric is related to the distribution of matter by the non - linear einstein - hilbert differential equations . in contrast , in the present context , the acoustic metric is related to the distribution of matter in a simple algebraic fashion .
let us start with the notion of an ergo - region : consider integral curves of the vector 43\documentclass[12pt]{minimal } 
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				 \begin{document}$${k^\mu } \equiv { ( \partial/\partial t)^\mu } = { ( 1,0,0,0)^\mu}.$$\end{document } if the flow is steady then this is the time translation killing vector . even if the flow is not steady the background minkowski metric provides us with a natural definition of  at rest  .
then5 
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				 \begin{document}$${g_{\mu \nu}}{(\partial/\partial t)^\mu}{(\partial/\partial t)^\nu } = { g_{tt } } = - [ { c^2 } - { v ^2}].$$\end{document } this quantity changes sign when v > c. thus any region of supersonic flow is an ergo - region .
( and the boundary of the ergo - region may be deemed to be the ergo - surface . )
the analogue of this behaviour in general relativity is the ergosphere surrounding any spinning black hole  it is a region where space  moves  with superluminal velocity relative to the fixed stars [ 265 , 164 , 422 ] .
a trapped surface in acoustics is defined as follows : take any closed two - surface . if the fluid velocity is everywhere inward - pointing and the normal component of the fluid velocity is everywhere greater than the local speed of sound , then no matter what direction a sound wave propagates , it will be swept inward by the fluid flow and be trapped inside the surface .
( for comparison with the usual situation in general relativity see [ 164 , pages 319323 ] or [ 422 , pages 310311 ] . )
inner - trapped surfaces ( anti - trapped surfaces ) can be defined by demanding that the fluid flow is everywhere outward - pointing with supersonic normal component .
it is only because of the fact that the background minkowski metric provides a natural definition of  at rest  that we can adopt such a simple and straightforward definition . in ordinary general relativity
we need to develop considerable additional technical machinery , such as the notion of the  expansion  of bundles of ingoing and outgoing null geodesics , before defining trapped surfaces . that the above definition for acoustic geometries is a specialization of the usual one can be seen from the discussion on pages 262263 of hawking and ellis .
the acoustic trapped region is now defined as the region containing outer trapped surfaces , and the acoustic ( future ) apparent horizon as the boundary of the trapped region .
that is , the acoustic apparent horizon is the two - surface for which the normal component of the fluid velocity is everywhere equal to the local speed of sound .
( we can also define anti - trapped regions and past apparent horizons but these notions are of limited utility in general relativity.)6 the event horizon ( absolute horizon ) is defined , as in general relativity , by demanding that it be the boundary of the region from which null geodesics ( phonons ) can not escape .
a past event horizon can be defined in terms of the boundary of the region that can not be reached by incoming phonons  strictly speaking this requires us to define notions of past and future null infinities , but we will simply take all relevant incantations as understood . in particular the event horizon is a null surface , the generators of which are null geodesics . in all stationary geometries
the apparent and event horizons coincide , and the distinction is immaterial . in time - dependent geometries
the distinction is often important . when computing the surface gravity we shall restrict attention to stationary geometries ( steady flow ) . in fluid flows of high symmetry ( spherical symmetry , plane symmetry ) , the ergosphere may coincide with the acoustic apparent horizon , or even the acoustic event horizon
this is the analogue of the result in general relativity that for static ( as opposed to stationary ) black holes the ergosphere and event horizon coincide . for many more details , including appropriate null coordinates and carter - penrose diagrams , both in stationary and time - dependent situations , see . 

figure 5a moving fluid can form  trapped surfaces  when supersonic flow tips the sound cones past the vertical .
a moving fluid can form  trapped surfaces  when supersonic flow tips the sound cones past the vertical . because of the definition of event horizon in terms of phonons ( null geodesics ) that can not escape the acoustic black hole , the event horizon is automatically a null surface , and the generators of the event horizon are automatically null geodesics . in the case of acoustics
there is one particular parameterization of these null geodesics that is  most natural  , which is the parameterization in terms of the newtonian time coordinate of the underlying physical metric .
this allows us to unambiguously define a  surface gravity  even for non - stationary ( time - dependent ) acoustic event horizons , by calculating the extent to which this natural time parameter fails to be an affine parameter for the null generators of the horizon .
( this part of the construction fails in general relativity where there is no universal natural time - coordinate unless there is a timelike killing vector  this is why extending the notion of surface gravity to non - stationary geometries in general relativity is so difficult . )
when it comes to explicitly calculating the surface gravity in terms of suitable gradients of the fluid flow , it is nevertheless very useful to limit attention to situations of steady flow ( so that the acoustic metric is stationary ) .
this has the added bonus that for stationary geometries the notion of  acoustic surface gravity  in acoustics is unambiguously equivalent to the general relativity definition .
it is also useful to take cognizance of the fact that the situation simplifies considerably for static ( as opposed to merely stationary ) acoustic metrics . to set up the appropriate framework , write the general stationary acoustic metric in the form 45\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } = { \rho \over c}\left [ { - { c^2}\,{\rm{d}}{t^2 } + { { ( { \rm{d}}{\bf{x } } - { \bf{v}}\,{\rm{d}}t)}^2 } } \right].$$\end{document }
the time translation killing vector is simply \documentclass[12pt]{minimal } 
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				 \begin{document}${k^\mu } = ( 1;\vec 0)$\end{document } with 46\documentclass[12pt]{minimal } 
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				 \begin{document}$${k^2 } \equiv { g_{\mu \nu}}{k^\mu}{k^\nu } \equiv - { \left\vert { \bf{k } } \right\vert ^2 } = - { \rho \over c}[{c^2 } - { v ^2}].$$\end{document } the metric can also be written as 47\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } = { \rho \over c}\left [ { - ( { c^2 } - { v ^2})\;{\rm{d}}{t^2 } - 2{\bf{v } } \cdot { \rm{d}}{\bf{x}}\;{\rm{d}}t + { { ( { \rm{d}}{\bf{x}})}^2 } } \right].$$\end{document } now suppose that the vector v/(c  v ) is integrable , then we can define a new time coordinate by 48\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}\tau = { \rm{d}}t + { { { \bf{v } } \cdot { \rm{d}}{\bf{x } } } \over { { c^2 } - { v ^2}}}.$$\end{document } substituting this back into the acoustic line element gives 49\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } = { \rho \over c}\left [ { - ( { c^2 } - { v ^2})\;{\rm{d}}{\tau ^2 } + \left\{{{\delta _ { ij } } + { { { v ^i}{v ^j } } \over { { c^2 } - { v ^2 } } } } \right\}\;{\rm{d}}{x^i}\,{\rm{d}}{x^j } } \right].$$\end{document } in this coordinate system the absence of the time - space cross - terms makes manifest that the acoustic geometry is in fact static ( there exists a family of spacelike hypersurfaces orthogonal to the timelike killing vector ) . the condition that an acoustic geometry be static , rather than merely stationary ,
is thus seen to be 50\documentclass[12pt]{minimal } 
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				 \begin{document}$${\bf{\nabla } } \times \left\{{{{\bf{v } } \over { ( { c^2 } - { v ^2 } ) } } } \right\ } = 0,$$\end{document } that is , ( since in deriving the existence of the effective metric we have already assumed the fluid to be irrotational ) , 51\documentclass[12pt]{minimal } 
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				 \begin{document}$${\bf{v } } \times { \bf{\nabla}}({c^2 } - { v ^2 } ) = 0.$$\end{document } this requires the fluid flow to be parallel to another vector that is not quite the acceleration but is closely related to it .
( note that , because of the vorticity free assumption , v is just the three - acceleration of the fluid , it is the occurrence of a possibly position dependent speed of sound that complicates the above . ) once we have a static geometry , we can of course directly apply all of the standard tricks   for calculating the surface gravity developed in general relativity .
we set up a system of fiducial observers ( fidos ) by properly normalizing the time - translation killing vector 52\documentclass[12pt]{minimal } 
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				 \begin{document}$${{\bf{v}}_{fido } } \equiv { { \bf{k } } \over { \left\vert { \bf{k } } \right\vert } } = { { \bf{k } } \over { \sqrt { ( \rho / c)\;[{c^2 } - { v ^2}]}}}.$$\end{document } the four - acceleration of the fidos is defined as 53\documentclass[12pt]{minimal } 
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				 \begin{document}$${{\bf{a}}_{{\rm{fido } } } } \equiv ( { { \bf{v}}_{{\rm{fido } } } } \cdot \nabla){{\bf{v}}_{{\rm{fido}}}},$$\end{document } and using the fact that k is a killing vector , it may be computed in the standard manner 54\documentclass[12pt]{minimal } 
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				 \begin{document}$${{\bf{a}}_{{\rm{fido } } } } = + { 1 \over 2}{{{\bf{\nabla}}{{\left\vert { \bf{k } } \right\vert}^2 } } \over { { { \left\vert { \bf{k } } \right\vert}^2}}}.$$\end{document } that is 55\documentclass[12pt]{minimal } 
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				 \begin{document}$${{\bf{a}}_{{\rm{fido } } } } = { 1 \over 2}\left [ { { { { \bf{\nabla}}({c^2 } - { v ^2 } ) } \over { ( { c^2 } - { v ^2 } ) } } + { { { \bf{\nabla}}(\rho / c ) } \over { ( \rho / c ) } } } \right].$$\end{document } the surface gravity is now defined by taking the norm afido  , multiplying by the lapse function , \documentclass[12pt]{minimal } 
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				 \begin{document}$\left\| { \rm{k } } \right\| = \sqrt { ( \rho / c)[{c^2 } - { v ^2}]}$\end{document } , and taking the limit as one approaches the horizon : v  c ( remember that we are currently dealing with the static case ) .
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				 \begin{document}$$\left\vert { { { \bf{a}}_{{\rm{fido } } } } } \right\vert \;\left\vert { \bf{k } } \right\vert = { 1 \over 2}{\bf{v } } \cdot { \bf{\nabla}}({c^2 } - { v ^2 } ) + o({c^2 } - { v ^2}),$$\end{document } so that the surface gravity is given in terms of a normal derivative by7 
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				 \begin{document}$${g_h } = { 1 \over 2}{{\partial ( { c^2 } - { v ^2 } ) } \over { \partial
n } } = c{{\partial ( c - v ) } \over { \partial n}}.$$\end{document } this is not quite unruh s result [ 376 , 377 , 378 ] since he implicitly took the speed of sound to be a position - independent constant . the fact that prefactor /c drops out of the final result for the surface gravity
though derived in a totally different manner , this result is also compatible with the expression for  surface - gravity  obtained in the solid - state black holes of reznik , wherein a position dependent ( and singular ) refractive index plays a role analogous to the acoustic metric . as a further consistency check , one can go to the spherically symmetric case and check that this reproduces the results for  dirty black holes  enunciated in .
since this is a static geometry , the relationship between the hawking temperature and surface gravity may be verified in the usual fast - track manner  using the wick rotation trick to analytically continue to euclidean space .
if you do nt like euclidean signature techniques ( which are in any case only applicable to equilibrium situations ) you should go back to the original hawking derivations [ 159 , 160].8 one final comment to wrap up this section : the coordinate transform we used to put the acoustic metric into the explicitly static form is perfectly good mathematics , and from the general relativity point of view is even a simplification .
however , from the point of view of the underlying newtonian physics of the fluid , this is a rather bizarre way of deliberately de - synchronizing your clocks to take a perfectly reasonable region  the boundary of the region of supersonic flow  and push it out to  time  plus infinity . from the fluid dynamics point of view this coordinate transformation is correct but perverse , and it is easier to keep a good grasp on the physics by staying with the original newtonian time coordinate .
if the fluid flow does not satisfy the integrability condition which allows us to introduce an explicitly static coordinate system , then defining the surface gravity is a little trickier . recall that by construction the acoustic apparent horizon is in general defined to be a two - surface
for which the normal component of the fluid velocity is everywhere equal to the local speed of sound , whereas the acoustic event horizon ( absolute horizon ) is characterised by the boundary of those null geodesics ( phonons ) that do not escape to infinity . in the stationary case
these notions coincide , and it is still true that the horizon is a null surface , and that the horizon can be ruled by an appropriate set of null curves .
suppose we have somehow isolated the location of the acoustic horizon , then in the vicinity of the horizon we can split up the fluid flow into normal and tangential components 58\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{array}{*{20}c } { { \bf{v } } = { { \bf{v } } _ \bot } + { { \bf{v}}_\parallel } ; } & { { \rm{where } } } & { { { \bf{v } } _ \bot } = { v _ \bot}\,{\bf{\hat n}}. } \\ \end{array}$$\end{document } here ( and for the rest of this particular section ) it is essential that we use the natural newtonian time coordinate inherited from the background newtonian physics of the fluid .
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				 \begin{document}${{\rm{\hat n}}}$\end{document } is a unit vector field that at the horizon is perpendicular to it , and away from the horizon is some suitable smooth extension .
( for example , take the geodesic distance to the horizon and consider its gradient . )
we only need this decomposition to hold in some open set encompassing the horizon and do not need to have a global decomposition of this type available .
furthermore , by definition we know that v = c at the horizon . now consider the vector field 59\documentclass[12pt]{minimal } 
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				 \begin{document}$${l^\mu } = ( 1;\;v _ \parallel ^i).$$\end{document } since the spatial components of this vector field are by definition tangent to the horizon , the integral curves of this vector field will be generators for the horizon .
furthermore the norm of this vector ( in the acoustic metric ) is 60\documentclass[12pt]{minimal } 
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				 \begin{document}$${\left\vert { \bf{l } } \right\vert ^2 } = - { \rho \over c}\left [ { - ( { c^2 } - { v ^2 } ) - 2{{\bf{v}}_\parallel } \cdot { \bf{v } } + { { \bf{v}}_\parallel } \cdot { { \bf{v}}_\parallel } } \right ] = { \rho \over c}({c^2 } - v _ \bot ^2).$$\end{document } in particular , on the acoustic horizon
l defines a null vector field , the integral curves of which are generators for the acoustic horizon .
we shall now verify that these generators are geodesics , though the vector field l is not normalised with an affine parameter , and in this way shall calculate the surface gravity .
consider the quantity ( l  )l and calculate 61\documentclass[12pt]{minimal } 
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				 \begin{document}$${l^\alpha}{\nabla _ \alpha}{l^\mu } = { l^\alpha}({\nabla _ \alpha}{l_\beta } - { \nabla _ \beta}{l_\alpha}){g^{\beta \mu } } + { 1 \over 2}{\nabla _ \beta}({l^2}){g^{\beta \mu}}.$$\end{document } to calculate the first term note that 62\documentclass[12pt]{minimal } 
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				 \begin{document}$${l_\mu } = { \rho \over c}(- [ { c^2 } - v _
^2];{{\bf{v } } _ \bot}).$$\end{document } thus 63\documentclass[12pt]{minimal } 
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				 \begin{document}$${l_{[\alpha , \beta ] } } = - \left [ { \begin{array}{*{20}c } 0 & \vdots & { - { \nabla _ i}\left [ { { \rho \over c}({c^2 } - v _ \bot ^2 ) } \right ] } \\ { \cdots \cdots \cdots \cdots } & \cdot & { \cdots \cdots } \\ { + { \nabla _ j}\left [ { { \rho \over c}({c^2 } - v _ \bot ^2 ) } \right ] } & \vdots & { { { \left({{\rho \over c}\;{v ^ \bot } } \right)}_{[i , j ] } } } \\
so : 64\documentclass[12pt]{minimal } 
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				 \begin{document}$${l^\alpha}{l_{[\beta , \alpha ] } } = \left({{{\bf{v}}_\parallel } \cdot { \bf{\nabla}}\left [ { { \rho \over c}({c^2 } - v _
\bot ^2 ) } \right];{\nabla _ j}\left [ { { \rho \over c}({c^2 } - v _ \bot ^2 ) } \right ] + v _ \parallel ^i{{\left({{\rho \over
c}\;{v ^ \bot } } \right)}_{[j , i ] } } } \right).$$\end{document } on the horizon , where c = v , this simplifies tremendously 65\documentclass[12pt]{minimal } 
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				 \begin{document}$$({l^\alpha}{l_{[\beta , \alpha ] } } ) { \vert _ { { \rm{horizon } } } } = - { \rho \over c}\;(0;{\nabla _ j}({c^2 } - v _ \bot ^2 ) ) = - { \rho \over c}{{\partial ( { c^2 } - v _ \bot ^2 ) } \over { \partial n}}\;(0;{\hat n_j}).$$\end{document } similarly , for the second term we have 66\documentclass[12pt]{minimal } 
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				 \begin{document}$${\nabla _ \beta}({l^2 } ) = \left({0;{\nabla _ j}\left [ { { \rho \over c}({c^2 } - v _
\bot ^2 ) } \right ] } \right).$$\end{document } on the horizon this again simplifies 67\documentclass[12pt]{minimal } 
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				 \begin{document}$${\nabla _ \beta}({l^2}){\vert _ { { \rm{horizon } } } } = + { \rho \over c}\;\;(0;{\nabla _ j}({c^2 } - v _
\bot ^2 ) ) = + { \rho \over c}{{\partial ( { c^2 } - v _ \bot ^2 ) } \over { \partial n}}\;(0;{\hat n_j}).$$\end{document } there is partial cancellation between the two terms , and so 68\documentclass[12pt]{minimal } 
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				 \begin{document}$${({l^\alpha}{\nabla _ \alpha}{l_\mu})_{{\rm{horizon } } } } = + { 1 \over 2}\;{\rho \over c}{{\partial ( { c^2 } - v _ \bot ^2 ) } \over { \partial n}}\;(0;{\hat n_j}),$$\end{document } while 69\documentclass[12pt]{minimal } 
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				 \begin{document}$${({l_\mu})_{{\rm{horizon } } } } = { \rho \over c}\;(0;c\,{\hat n_j}).$$\end{document } comparing this with the standard definition of surface gravity   9 
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				 \begin{document}$${({l^\alpha}{\nabla _ \alpha}{l_\mu})_{{\rm{horizon } } } } = + { { { g_h } } \over c}\;{({l_\mu})_{{\rm{horizon}}}},$$\end{document } we finally have 71\documentclass[12pt]{minimal } 
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				 \begin{document}$${g_h } = { 1 \over 2}{{\partial ( { c^2 } - v _ \bot ^2 ) } \over { \partial
n } } = c{{\partial ( c - { v _ \bot } ) } \over { \partial n}}.$$\end{document } this is in agreement with the previous calculation for static acoustic black holes , and insofar as there is overlap , is also consistent with results of unruh [ 376 , 377 , 378 ] , reznik , and the results for  dirty black holes  . from the construction
it is clear that the surface gravity is a measure of the extent to which the newtonian time parameter inherited from the underlying fluid dynamics fails to be an affine parameter for the null geodesics on the horizon.10 again , the justification for going into so much detail on this specific model is that this style of argument can be viewed as a template  it will ( with suitable modifications ) easily generalise to more complicated analogue models . as an example of a fluid flow where the distinction between ergosphere and acoustic event horizon is critical
we shall model a draining bathtub by a ( 3 + 1 ) dimensional flow with a linear sink along the z - axis .
let us start with the simplifying assumption that the background density  is a position - independent constant throughout the flow ( which automatically implies that the background pressure p and speed of sound c are also constant throughout the fluid flow ) .
the equation of continuity then implies that for the radial component of the fluid velocity we must have 72\documentclass[12pt]{minimal } 
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				 \begin{document}$${v ^{\hat r } } \propto { 1 \over r}.$$\end{document } in the tangential direction , the requirement that the flow be vorticity free ( apart from a possible delta - function contribution at the vortex core ) implies , via stokes theorem , that 73\documentclass[12pt]{minimal } 
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				 \begin{document}$${v ^{\hat t } } \propto { 1 \over r}.$$\end{document } ( if these flow velocities are nonzero , then following the discussion of   there must be some external force present to set up and maintain the background flow . fortunately it is easy to see that this external force affects only the background flow and does not influence the linearised fluctuations we are interested in . ) for the background velocity potential we must then have 74\documentclass[12pt]{minimal } 
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				 \begin{document}$$\phi ( r,\theta ) = - a\;{\rm{ln}}(r / a ) - b\;\theta.$$\end{document } note that , as we have previously hinted , the velocity potential is not a true function ( because it has a discontinuity on going through 2 radians ) .
the velocity potential must be interpreted as being defined patch - wise on overlapping regions surrounding the vortex core at r = 0 .
the velocity of the fluid flow is 75\documentclass[12pt]{minimal } 
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				 \begin{document}$${\bf{v } } = - { \bf{\nabla}}\phi = { { ( a\;\hat r + b\;\hat \theta ) } \over r}.$$\end{document } dropping a position - independent prefactor , the acoustic metric for a draining bathtub is explicitly given by 76\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } = - { c^2}\,{\rm{d}}{t^2 } + { \left({{\rm{d}}r - { a \over r}{\rm{d}}t } \right)^2 } + { \left({r\,{\rm{d}}\theta - { b \over r}{\rm{d}}t } \right)^2 } + { \rm{d}}{z^2}.$$\end{document } equivalently 77\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } = - \left({{c^2 } - { { { a^2 } + { b^2 } } \over { { r^2 } } } } \right)\,{\rm{d}}{t^2 } - 2{a \over r}{\rm{d}}r\,{\rm{d}}t - 2b\,{\rm{d}}\theta \,{\rm{d}}t + { \rm{d}}{r^2 } + { r^2}{\rm{d}}{\theta ^2 } + { \rm{d}}{z^2}.$$\end{document } a similar metric , restricted to a=0 ( no radial flow ) , and generalised to an anisotropic speed of sound , has been exhibited by volovik , that metric being a model for the acoustic geometry surrounding physical vortices in superfluid he .
( for a survey of the many analogies and similarities between the physics of superfluid he and the standard electroweak model see , this reference is also useful as background to understanding the lorentzian geometric aspects of he fluid flow . ) note that the metric given above is not identical to the metric of a spinning cosmic string , which would instead take the form   78\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } = - { c^2}\,{({\rm{d}}t - \tilde a\,{\rm{d}}\theta)^2 } + { \rm{d}}{r^2 } + ( 1 - \tilde b){r^2}{\rm{d}}{\theta ^2 } + { \rm{d}}{z^2}.$$\end{document } in conformity with previous comments , the vortex fluid flow is seen to possess an acoustic metric that is stably causal and which does not involve closed timelike curves .
( at large distances it is possible to approximate the vortex geometry by a spinning cosmic string , but this approximation becomes progressively worse as the core is approached . ) the ergosphere forms at 79\documentclass[12pt]{minimal } 
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				 \begin{document}$${r_{{\rm{ergosphere } } } } = { { \sqrt { { a^2 } + { b^2 } } } \over c}.$$\end{document } note that the sign of a is irrelevant in defining the ergosphere and ergo - region : it does not matter if the vortex core is a source or a sink . 

figure 6a collapsing vortex geometry ( draining bathtub ) : the green spirals denote streamlines of the fluid flow .
the outer circle represents the ergo - surface while the inner circle represents the [ outer ] event horizon . a collapsing vortex geometry ( draining bathtub ) : the green spirals denote streamlines of the fluid flow .
the outer circle represents the ergo - surface while the inner circle represents the [ outer ] event horizon .
the acoustic event horizon forms once the radial component of the fluid velocity exceeds the speed of sound , that is at 80\documentclass[12pt]{minimal } 
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				 \begin{document}$${r_{{\rm{horizon } } } } = { { \vert a\vert } \over c}.$$\end{document } the sign of a now makes a difference . for a <
0 we are dealing with a future acoustic horizon ( acoustic black hole ) , while for a > 0 we are dealing with a past event horizon ( acoustic white hole ) .
a popular model for the investigation of event horizons in the acoustic analogy is the one - dimensional slab geometry where the velocity is always along the z direction and the velocity profile depends only on z. the continuity equation then implies that (z ) v(z ) is a constant , and the acoustic metric becomes 81\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } \propto { 1 \over { v ( z)\,c(z)}}\left [ { - c{{(z)}^2}{\rm{d}}{t^2 } + { { \{{\rm{d}}z - v ( z)\,{\rm{d}}t\}}^2 } + { \rm{d}}{x^2 } + { \rm{d}}{y^2}}\right].$$\end{document } that is 82\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } \propto { 1 \over { v ( z)\,c(z)}}\left [ { - \{c{{(z)}^2 } - v { { ( z)}^2}{\rm{\ } d}}{t^2 } - 2v ( z)\,{\rm{d}}z\,{\rm{d}}t + { \rm{d}}{x^2 } + { \rm{d}}{y^2 } + { \rm{d}}{z^2 } } \right].$$\end{document } if we set c = 1 and ignore the conformal factor we have the toy model acoustic geometry discussed by unruh [ 378 , page 2828 , equation ( 8) ] , jacobson [ 188 , page 7085 , equation ( 4 ) ] , corley and jacobson , and corley .
( in this situation one must again invoke an external force to set up and maintain the fluid flow . since the conformal factor is regular at the event horizon , we know that the surface gravity and hawking temperature are independent of this conformal factor . ) in the general case it is important to realise that the flow can go supersonic for either of two reasons : the fluid could speed up , or the speed of sound could decrease . when it comes to calculating the  surface gravity  both of these effects will have to be taken into account . to see how close the acoustic metric can get to reproducing the schwarzschild geometry
it is first useful to introduce one of the more exotic representations of the schwarzschild geometry : the painlev - gullstrand line element , which is simply an unusual choice of coordinates on the schwarzschild spacetime.11 in modern notation the schwarzschild geometry in ingoing ( + ) and outgoing (  ) painlev - gullstrand coordinates may be written as : 83\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } = - { \rm{d}}{t^2 } + { \left({{\rm{d}}r \pm \sqrt { { { 2gm } \over r } } { \rm{d}}t } \right)^2 } + { r^2}({\rm{d}}{\theta ^2 } + { \sin ^2}\theta \,{\rm{d}}{\phi ^2}).$$\end{document } equivalently 84\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } = - \left({1 - { { 2gm } \over r } } \right)\,{\rm{d}}{t^2 } \pm \sqrt { { { 2gm } \over r } } { \rm{d}}r\,{\rm{d}}t + { \rm{d}}{r^2 } + { r^2}({\rm{d}}{\theta ^2 } + { \sin ^2}\theta \,{\rm{d}}{\phi ^2}).$$\end{document } this representation of the schwarzschild geometry was not ( until the advent of the analogue models ) particularly well - known , and it has been independently rediscovered several times during the 20th century . see for instance painlev , gullstrand , lematre , the related discussion by israel , and more recently , the paper by kraus and wilczek .
the painlev - gullstrand coordinates are related to the more usual schwarzschild coordinates by 85\documentclass[12pt]{minimal } 
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				 \begin{document}$${t_{{\rm{pg } } } } = { t_{\rm{s } } } \pm \left [ { 4m\;\,{\rm{arctanh}}\left({\sqrt { { { 2gm } \over r } } } \right ) - 2\sqrt { 2gmr } } \right].$$\end{document } or equivalently 86\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{t_{{\rm{pg } } } } = { \rm{d}}{t_{\rm{s } } } \pm { { \sqrt { 2gm / r } } \over { 1 - 2gm / r}}{\rm{d}}r.$$\end{document } with these explicit forms in hand , it becomes an easy exercise to check the equivalence between the painlev - gullstrand line element and the more usual schwarzschild form of the line element .
it should be noted that the + sign corresponds to a coordinate patch that covers the usual asymptotic region plus the region containing the future singularity of the maximally extended schwarzschild spacetime .
it thus covers the future horizon and the black hole singularity . on the other hand
the  sign corresponds to a coordinate patch that covers the usual asymptotic region plus the region containing the past singularity .
as emphasised by kraus and wilczek , the painlev - gullstrand line element exhibits a number of features of pedagogical interest .
in particular the constant time spatial slices are completely flat  the curvature of space is zero , and all the spacetime curvature of the schwarzschild geometry has been pushed into the time - time and time - space components of the metric . given the painlev - gullstrand line element , it might seem trivial to force the acoustic metric into this form : simply take  and c to be constants , and set \documentclass[12pt]{minimal } 
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				 \begin{document}$v = \sqrt { 2gm / r}$\end{document } while this certainly forces the acoustic metric into the painlev - gullstrand form the problem with this is that this assignment is incompatible with the continuity equation   ( v )  0 that was used in deriving the acoustic equations .
the best we can actually do is this : pick the speed of sound c to be a position independent constant , which we normalise to unity ( c = 1 ) .
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				 \begin{document}$v = \sqrt { 2gm / r}$\end{document } , and use the continuity equation   ( v ) = 0 to deduce v
 1/r so that   r. since the speed of sound is taken to be constant we can integrate the relation c = dp / d to deduce the equation of state must be p = p + c and that the background pressure satisfies p  p  cr .
overall the acoustic metric is now 87\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } \propto { r^{- 3/2}}\left [ { - { \rm{d}}{t^2 } + { { \left({{\rm{d}}r \pm \sqrt { { { 2gm } \over r } } { \rm{d}}t } \right)}^2 } + { r^2}({\rm{d}}{\theta ^2 } + { { \sin}^2}\theta \;{\rm{d}}{\phi ^2 } ) } \right].$$\end{document } so we see that the net result is conformal to the painlev - gullstrand form of the schwarzschild geometry but not identical to it . for many purposes
this is quite good enough : we have an event horizon , we can define surface gravity , we can analyse hawking radiation.12 since surface gravity and hawking temperature are conformal invariants   this is sufficient for analysing basic features of the hawking radiation process . the only way in which the conformal factor can influence the hawking radiation is through backscattering off the acoustic metric . (
the phonons are minimally coupled scalars , not conformally coupled scalars so there will in general be effects on the frequency - dependent greybody factors . )
if we focus attention on the region near the event horizon , the conformal factor can simply be taken to be a constant , and we can ignore all these complications . up to now ,
this can be justified either with a view to using fluid mechanics to teach us more about general relativity , or to using the techniques of lorentzian geometry to teach us more about fluid mechanics .
for example , given the machinery developed so far , taking the short wavelength / high frequency limit to obtain geometrical acoustics is now easy .
compare this to general relativity where in the geometrical optics approximation light rays ( photons ) follow null geodesics of the physical spacetime metric .
since null geodesics are insensitive to any overall conformal factor in the metric [ 265 , 164 , 422 ] one might as well simplify life by considering a modified conformally related metric 88\documentclass[12pt]{minimal } 
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				 \begin{document}$${h_{\mu \nu } } \equiv \left [ { \begin{array}{*{20}c } { - ( { c^2 } - v _ 0 ^ 2 ) } & \vdots & { - v _ 0^j } \\ { \cdots \cdots \cdots \cdots } & \cdot & { \cdots \cdots } \\ { - v _ 0^i } & \vdots & { { \delta ^{ij } } } \\ \end{array } } \right].$$\end{document } this immediately implies that , in the geometric acoustics limit , sound propagation is insensitive to the density of the fluid . in this limit
, acoustic propagation depends only on the local speed of sound and the velocity of the fluid .
it is only for specifically wave related properties that the density of the medium becomes important .
we can rephrase this in a language more familiar to the acoustics community by invoking the eikonal approximation .
express the linearised velocity potential , 1 , in terms of an amplitude , a , and phase ,  , by 1  ae then , neglecting variations in the amplitude a , the wave equation reduces to the eikonal equation 
 89\documentclass[12pt]{minimal } 
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				 \begin{document}$${h^{\mu \nu}}\,{\partial _
\nu}\varphi = 0.$$\end{document } this eikonal equation is blatantly insensitive to any overall multiplicative prefactor ( conformal factor ) . as a sanity check on the formalism , it is useful to re - derive some standard results .
for example , let the null geodesic be parameterised by x(t )  ( t ; x(t ) ) .
then the null condition implies 90\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{array}{*{20}c } { { h_{\mu \nu}}{{{\rm{d}}{x^\mu } } \over { { \rm{d}}t}}{{{\rm{d}}{x^\nu } } \over { { \rm{d}}t } } = 0\quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ { \leftrightarrow - ( { c^2 } - v _ 0 ^ 2 ) - 2v _ 0^i{{{\rm{d}}{x^i } } \over { { \rm{d}}t } } + { { { \rm{d}}{x^i } } \over { { \rm{d}}t}}{{{\rm{d}}{x^i } } \over { { \rm{d}}t } } = 0 } \\ { \leftrightarrow \left\vert { { { { \rm{d}}{\bf{x } } } \over { { \rm{d}}t } } - { { \bf{v}}_0 } } \right\vert = c.\quad \quad \quad \quad \quad \quad \quad \quad } \\ \end{array}$$\end{document } here the norm is taken in the flat physical metric .
this has the obvious interpretation that the ray travels at the speed of sound , c , relative to the moving medium .
let x(s )  ( t(s ) ; x(s ) ) be some null path from x1 to x2 , parameterised in terms of physical arc length ( i.e. ,  dx / ds  1 ) .
then the tangent vector to the path is 91\documentclass[12pt]{minimal } 
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				 \begin{document}$${{{\rm{d}}{x^\mu } } \over { { \rm{d}}s } } = \left({{{{\rm{d}}t } \over { { \rm{d}}s}};{{{\rm{d}}{x^i } } \over { { \rm{d}}s } } } \right).$$\end{document } the condition for the path to be null ( though not yet necessarily a null geodesic ) is 92\documentclass[12pt]{minimal } 
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				 \begin{document}$${g_{\mu \nu}}{{{\rm{d}}{x^\mu } } \over { { \rm{d}}s}}{{{\rm{d}}{x^\nu } } \over { { \rm{d}}s } } = 0.$$\end{document } using the explicit algebraic form for the metric , this can be expanded to show 93\documentclass[12pt]{minimal } 
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				 \begin{document}$$- ( { c^2 } - v _ 0 ^ 2){\left({{{{\rm{d}}t } \over { { \rm{d}}s } } } \right)^2 } - 2v _ 0^i\left({{{{\rm{d}}{x^i } } \over { { \rm{d}}s } } } \right)\left({{{{\rm{d}}t } \over { { \rm{d}}s } } } \right ) + 1 = 0.$$\end{document } solving this quadratic 94\documentclass[12pt]{minimal } 
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				 \begin{document}$$\left({{{{\rm{d}}t } \over { { \rm{d}}s } } } \right ) = { { - v _ 0^i\left({{{{\rm{d}}{x^i } } \over { { \rm{d}}s } } } \right ) + \sqrt { { c^2 } - v _ 0 ^ 2 + { { \left({v _ 0^i{{{\rm{d}}{x^i } } \over { { \rm{d}}s } } } \right)}^2 } } } \over { { c^2 } - v _ 0 ^ 2}}.$$\end{document } therefore , the total time taken to traverse the path is 95\documentclass[12pt]{minimal } 
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				 \begin{document}$$t\,[\gamma ] = \int\nolimits_{{{\bf{x}}_1}}^{{{\bf{x}}_2 } } { ( { \rm{d}}t/{\rm{d}}s)\,{\rm{d}}s } = \int\nolimits_\gamma { { 1 \over { { c^2 } - v _ 0 ^ 2 } } } \left\{{\sqrt { ( { c^2 } - v _ 0 ^ 2){\rm{d}}{s^2 } + { { ( v _ 0^i\,{\rm{d}}{x^i})}^2 } } - v _ 0^i\,{\rm{d}}{x^i } } \right\}.$$\end{document } if we now recall that extremising the total time taken is fermat s principle for sound rays , we see that we have checked the formalism for stationary geometries ( steady flow ) by reproducing the discussion on page 262 of landau and lifshitz .
there are a large number of ways in which the present particularly simple analogue model can be generalised .
obvious issues within the current physical framework are : 
 adding external forces.working in ( 1 + 1 ) or ( 2 + 1 ) dimensions.adding vorticity , to go beyond the irrotational constraint . 
 beyond these immediate questions , we could also seek similar effects in other physical or mathematical frameworks . adding external forces . working in ( 1 + 1 ) or ( 2 + 1 ) dimensions .
adding vorticity , to go beyond the irrotational constraint . adding external forces is easy , an early discussion can be found in   and more details are available in .
the key point is that with an external force one can to some extent shape the background flow ( see for example the discussion on ) . upon linearization ,
the role of spacetime dimension in these acoustic geometries is sometimes a bit surprising and potentially confusing .
this is important because there is a real physical distinction , for instance , between truly ( 2 + 1)-dimensional systems and effectively ( 2 + 1)-dimensional systems in the form of ( 3 + 1)-dimensional systems with cylindrical symmetry .
similarly there is a real physical distinction between a truly ( 1 + 1)-dimensional system and a ( 3 + 1)-dimensional system with transverse symmetry .
we emphasise that in cartesian coordinates the wave equation 96\documentclass[12pt]{minimal } 
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				 \begin{document}$${\partial \over { \partial { x^\mu}}}\left({{f^{\mu \nu}}{\partial \over { \partial { x^\nu}}}\;\phi } \right ) = 0,$$\end{document } where 97 holds independent of the dimensionality of spacetime .
it depends only on the euler equation , the continuity equation , a barotropic equation of state , and the assumption of irrotational flow [ 376 , 387 , 391 , 389 ] .
introducing the inverse acoustic metric g , defined by 98\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{array}{*{20}c } { { f^{\mu \nu } } = \sqrt { - g } \,{g^{\mu \nu } } ; } & { g = { 1 \over { \det ( { g^{\mu \nu } } ) } } } \\ \end{array}$$\end{document }
the wave equation ( 96 ) corresponds to the dalembertian wave equation in a curved space - time with contravariant metric tensor : 99 where d is the dimension of space ( not spacetime ) .
the covariant acoustic metric is then 100 
 d = 3 : the acoustic line element for three space and one time dimension reads 101 
 d = 2 : the acoustic line element for two space and one time dimension reads 102 this situation would be appropriate , for instance , when dealing with surface waves or excitations confined to a particular substrate .
d = 1 : the naive form of the acoustic metric in ( 1 + 1 ) dimensions is ill - defined , because the conformal factor is raised to a formally infinite power .
this is a side effect of the well - known conformal invariance of the laplacian in 2 dimensions .
the wave equation in terms of the densitised inverse metric f continues to make good sense ; it is only the step from f to the effective metric that breaks down .
note that this issue only presents a difficulty for physical systems that are intrinsically one - dimensional .
a three - dimensional system with plane symmetry , or a two - dimensional system with line symmetry , provides a perfectly well behaved model for ( 1 + 1 ) dimensions , as in the cases d = 3 and d = 2 above . for the preceding analysis to hold
it is necessary and sufficient that the flow locally be vorticity free ,   v = 0 , so that velocity potentials exist on an atlas of open patches .
note that the irrotational condition is automatically satisfied for the super - fluid component of physical superfluids .
( this point has been emphasised by comer , who has also pointed out that in superfluids there will be multiple acoustic metrics  and multiple acoustic horizons  corresponding to first and second sound . ) even for normal fluids , vorticity free flows are common , especially in situations of high symmetry . furthermore
, the previous condition enables us to handle vortex filaments , where the vorticity is concentrated into a thin vortex core , provided we do not attempt to probe the vortex core itself .
it is not necessary for the velocity potential  to be globally defined . though physically important , dealing with situations of distributed vorticity is much more difficult , and the relevant wave equation is more complicated in that the velocity scalar is now insufficient to completely characterise the fluid flow.13 an approach similar to the spirit of the present discussion , but in terms of clebsch potentials , can be found in .
the eikonal approximation ( geometrical acoustics ) leads to the same conformal class of metrics previously discussed , but in the realm of physical acoustics the wave equation is considerably more complicated than a simple dalembertian .
( roughly speaking , the vorticity becomes a source for the dalembertian , while the vorticity evolves in response to gradients in a generalised scalar potential .
this seems to take us outside the realm of models of direct interest to the general relativity community.)14 as a first ( and rather broad ) example of the very abstract ways in which the notion of an acoustic metric can be generalised , we start from the simple observation that irrotational barotropic fluid mechanics can be described by a lagrangian , and ask if we can extend the notion of an acoustic metric to all ( or at least some wide class of ) lagrangian systems ? indeed , suppose we have a single scalar field  whose dynamics is governed by some generic lagrangian \documentclass[12pt]{minimal } 
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				 \begin{document}${\mathcal l}({\partial _ \mu}\phi , \phi)$\end{document } , which is some arbitrary function of the field and its first derivatives ( here we will follow the notation and ideas of ) . in the general analysis that follows the previous irrotational and inviscid fluid system is included as a particular case ; the dynamics of the scalar field  is now much more general .
we want to consider linearised fluctuations around some background solution 0(t , x ) of the equations of motion , and to this end we write 103\documentclass[12pt]{minimal } 
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				 \begin{document}$$\phi ( t,{\rm{x } } ) = { \phi _ 0}(t,{\rm{x } } ) + \epsilon { \phi _ 1}(t,{\rm{x } } ) + { { { \epsilon ^2 } } \over 2}{\phi _ 2}(t,{\rm{x } } ) + o({\epsilon ^3}).$$\end{document } now use this to expand the lagrangian around the classical solution 0(t , x ) : 104\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{array}{*{20}c } { { \mathcal l}({\partial _ \mu}\phi , \phi ) = { \mathcal l}({\partial _
\mu}{\phi _ 0},{\phi _ 0 } ) + \epsilon \left [ { { { \partial { \mathcal l } } \over { \partial ( { \partial _ \mu}\phi)}}{\partial _ \mu}{\phi _ 1 } + { { \partial { \mathcal l } } \over { \partial \phi}}{\phi _ 1 } } \right]\quad \quad \quad \quad \quad \quad \quad } \\ { + { { { \epsilon ^2 } } \over 2}\left [ { { { \partial { \mathcal l } } \over { \partial ( { \partial _ \mu}\phi)}}{\partial _ \mu}{\phi _ 2 } + { { \partial { \mathcal l } } \over { \partial \phi}}{\phi _ 2 } } \right]\quad \quad \quad \quad \quad \quad \quad \quad } \\
{ + { { { \epsilon ^2 } } \over 2}\left [ { { { { \partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}\phi)\,\partial ( { \partial _ \nu}\phi)}}{\partial _ \mu}{\phi _ 1}\,{\partial _ \nu}{\phi _ 1 } } \right .
+ 2{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}\phi)}}{\partial _ \mu}{\phi _ 1}\,{\phi _ 1 } }
{ + { { { \partial ^2}{\mathcal l } } \over { \partial \phi \,\partial \phi}}{\phi _ 1}\,{\phi _ 1 } } \right ] + o({\epsilon ^3})\,.\quad \quad \quad \quad } \\
\end{array}$$\end{document } it is particularly useful to consider the action 105\documentclass[12pt]{minimal } 
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				 \begin{document}$${s_{\left [ \phi \right ] } } = \int { { { \rm{d}}^{d + 1}}x } \,{\mathcal l}({\partial _ \mu}\phi , \phi),$$\end{document } since doing so allows us to integrate by parts .
( note that the lagrangian \documentclass[12pt]{minimal } 
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				 \begin{document}${\mathcal l}$\end{document } is taken to be a scalar density , not a true scalar . )
we can now use the euler - lagrange equations for the background field 106\documentclass[12pt]{minimal } 
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				 \begin{document}$${\partial _ \mu}\left({{{\partial { \mathcal l } } \over { \partial ( { \partial _ \mu}\phi ) } } } \right ) - { { \partial { \mathcal l } } \over { \partial \phi } } = 0,$$\end{document } to discard the linear terms ( remember we are linearizing around a solution of the equations of motion ) and so we get 107\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{array}{*{20}c } { s\left [ \phi \right ] = s\left [ { { \phi _ 0 } } \right ] + { { { \epsilon ^2 } } \over 2}\int { { { \rm{d}}^{d + 1}}x\left [ { \left\{{{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}\phi)\,\partial ( { \partial _ \nu}\phi ) } } } \right\ } } \right.\,{\partial _ \mu}{\phi _ 1}\,{\partial _ \nu}{\phi _ 1}\quad \quad \quad \quad \quad } } \\ { \left .
{ + \left({{{{\partial ^2}{\mathcal l } } \over { \partial \phi \,\partial \phi } } - { \partial _
\mu}\left\{{{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}\phi)\,\partial \phi } } } \right\ } } \right)\,{\phi _ 1}\,{\phi _ 1 } } \right ] + o({\epsilon ^3 } ) } \\ \end{array}$$\end{document } having set things up this way , the equation of motion for the linearised fluctuation is now easily read off as 108\documentclass[12pt]{minimal } 
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				 \begin{document}$${\partial _
\mu}\left({\left\{{{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}\phi)\,\partial ( { \partial _ \nu}\phi ) } } } \right\}\,{\partial _ \nu}{\phi _ 1 } } \right ) - \left({{{{\partial ^2}{\mathcal l } } \over { \partial \phi \,\partial \phi } } - { \partial _
\mu}\left\{{{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}\phi)\,\partial \phi } } } \right\ } } \right)\,{\phi _ 1 } = 0.$$\end{document } this is a second - order differential equation with position - dependent coefficients ( these coefficients all being implicit functions of the background field 0 ) .
this can be given a nice clean geometrical interpretation in terms of a dalembertian wave equation  provided we define the effective spacetime metric by 109\documentclass[12pt]{minimal } 
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				 \begin{document}$$\sqrt { - g } { g^{\mu \nu } } \equiv { f^{\mu \nu } } \equiv { \left . {
\left\{{{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}\phi)\,\partial ( { \partial _ \nu}\phi ) } } } \right\ } } \right\vert _ { { \phi _ 0}}}.$$\end{document } note that this is another example of a situation in which calculating the inverse metric density is easier than calculating the metric itself . suppressing the 0 except when necessary for clarity , this implies [ in ( d+1 ) dimensions , d space dimensions plus 1 time dimension ] 110\documentclass[12pt]{minimal } 
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				 \begin{document}$${(- g)^{(d - 1)/2 } } = - \det \left\{{{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}\phi)\,\partial ( { \partial _ \nu}\phi ) } } } \right\}.$$\end{document } therefore 111\documentclass[12pt]{minimal } 
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				 \begin{document}$${\left .
{ { g^{\mu \nu}}({\phi _ 0 } ) = { { \left({- \det \left\{{{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}\phi)\,\partial ( { \partial _ \nu}\phi ) } } } \right\ } } \right)}^{- 1/(d - 1 ) } } } \right\vert _ { { \phi _ 0}}}{\left .
{ \left\{{{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}\phi)\,\partial ( { \partial _ \nu}\phi ) } } } \right\ } } \right\vert _ { { \phi _ 0}}}.$$\end{document } and , taking the inverse 112\documentclass[12pt]{minimal } 
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				 \begin{document}$${\left .
{ { g_{\mu \nu}}({\phi _ 0 } ) = { { \left({- \det \left\{{{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}\phi)\,\partial ( { \partial _ \nu}\phi ) } } } \right\ } } \right)}^{- 1/(d - 1 ) } } } \right\vert _ { { \phi _ 0}}}{\left .
{ \left\{{{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}\phi)\,\partial ( { \partial _ \nu}\phi ) } } } \right\ } } \right\vert _ { { \phi _ 0}}}.$$\end{document } we can now write the equation of motion for the linearised fluctuations in the geometrical form 113\documentclass[12pt]{minimal } 
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				 \begin{document}$$\left [ { \delta ( g({\phi _ 0 } ) ) - v({\phi _ 0 } ) } \right]{\phi _ 1 } = 0,$$\end{document } where  is the dalembertian operator associated with the effective metric g(0 ) , and v(0 ) is the background - field - dependent ( and so in general position - dependent )  mass term  : 114\documentclass[12pt]{minimal } 
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				 \begin{document}$$v({\phi _ 0 } ) = { 1 \over { \sqrt { - g}}}\left({{{{\partial ^2}{\mathcal l } } \over { \partial \phi \,\partial \phi } } - { \partial _
\mu}\left\{{{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}\phi)\,\partial \phi } } } \right\ } } \right)$$\end{document } 
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				 \begin{document}$$= { \left({- \det \left\{{{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}\phi)\,\partial ( { \partial _ \nu}\phi ) } } } \right\ } } \right)^{- 1/(d - 1 ) } } \times \,\,\left({{{{\partial ^2}{\mathcal l } } \over { \partial \phi \,\partial \phi } } - { \partial _
\mu}\left\{{{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}\phi)\,\partial \phi } } } \right\ } } \right).$$\end{document } thus v(0 ) is a true scalar ( not a density ) .
note that the differential equation ( 113 ) is automatically formally self - adjoint ( with respect to the measure \documentclass[12pt]{minimal } 
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				 \begin{document}$v = \sqrt { - g } { { \rm{d}}^{d + 1}}x$\end{document}. it is important to realise just how general the result is ( and where the limitations are ) : it works for any lagrangian depending only on a single scalar field and its first derivatives .
the linearised pde will be hyperbolic ( and so the linearised equations will have wave - like solutions ) if and only if the effective metric g has lorentzian signature [,+ ] observe that if the lagrangian contains nontrivial second derivatives you should not be too surprised to see terms beyond the dalembertian showing up in the linearised equations of motion . as a specific example of the appearance of effective metrics due to lagrangian dynamics we reiterate the fact that inviscid irrotational barotropic hydrodynamics naturally falls into this scheme ( which is why , with hindsight , the derivation of the acoustic metric presented earlier in this review was so relatively straightforward ) . in inviscid irrotational barotropic hydrodynamics
the lack of viscosity ( dissipation ) guarantees the existence of a lagrangian ; which a priori could depend on several fields .
since the flow is irrotational v =  is a function only of the velocity potential , and the lagrangian is a function only of this potential and the density .
finally the equation of state can be used to eliminate the density leading to a lagrangian that is a function only of the single field  and its derivatives .  
the class of analogue models based on fluid mechanics is now quite large and the literature is extensive .
most of the relevant discussion will be deferred until subsequent sections , so for the time being we shall just mention reasonably immediate generalizations such as : 
 working with specific fluids . 

we next turn to a brief historical discussion , seeking to place the work of the last decade into its proper historical perspective . working with specific fluids . 

from the point of view of the general relativity community the history of analogue models can reasonably neatly ( but superficially ) be divided into a  historical  period ( essentially pre-1981 ) and a  modern  period ( essentially post-1981 ) .
the key event in the  modern  period ( though largely unrecognised at the time ) was the 1981 publication of unruh s paper  experimental black hole evaporation  , which implemented an analogue model based on fluid flow , and then used the power of that analogy to probe fundamental issues regarding hawking radiation from  real  general relativity black holes .
we believe that unruh s 1981 article represents the first observation of the now widely established fact that hawking radiation has nothing to do with general relativity per se , but that hawking radiation is instead a fundamental curved - space quantum field theory phenomenon that occurs whenever a horizon is present in an effective geometry.15 though unruh s 1981 paper was seminal in this regard , it lay largely unnoticed for many years .
some 10 years later jacobson s article  black - hole evaporation and ultrashort distances    used unruh s analogy to build a physical model for the  trans - planckian modes  believed to be relevant to the hawking radiation process .
progress then sped up with the relatively rapid appearance of   and [ 377 , 378 ] .
( this period also saw the independent rediscovery of the fluid analogue model by one of the present authors , and the first explicit consideration of superfluids in this regard . )
the later 1990 s then saw continued work by jacobson and his group [ 187 , 188 , 88 , 90 , 198 ] , with new and rather different contributions coming in the form of the solid state models considered by reznik [ 319 , 318].16 this period also saw the introduction of the more general class of superfluid models considered by volovik and his collaborators [ 402 , 403 , 213 , 110 , 407 , 405 , 406 , 199 , 409 , 410 ] , more precise formulations of the notions of horizon , ergosphere , and surface gravity in analogue models [ 389 , 391 ] , and discussions of the implications of analogue models regarding bekenstein - hawking entropy [ 390 , 391 ] .
finally , analogue spacetimes based on special relativistic acoustics were considered in . by the year 2000 , articles on one or another aspect of analogue gravity were appearing at the rate of over 20 per year , and it becomes impractical to summarise more than a selection of them .
key developments in 2000 were the introduction , by garay and collaborators , of the use of bose - einstein condensates as a working fluid [ 136 , 137 ] , and the extension of those ideas by the present authors .
further afield , the trans - planckian problem also reared its head in the context of cosmological inflation , and analogue model ideas previously applied to hawking radiation were reused in that context [ 205 , 273 ] .
that year also marked the appearance of a review article on superfluid analogues , more work on  near - horizon 
theories into cosmology where they can be used as the basis for a precise definition of what is meant by a vsl (  variable speed of light  ) cosmology .
models based on nonlinear electrodynamics were investigated in , he  a based models were reconsidered in [ 193 , 411 ] , and 
the most radical proposal to appear in 2000 was that of laughlin et al . .
based on taking a superfluid analogy rather literally they mooted an actual physical breakdown of general relativity at the horizon of a black hole . additionally , the workshop on  analogue models of general relativity  , held at cbpf ( rio de janeiro ) gathered some 20 international participants and greatly stimulated the field , leading ultimately to the publication of the book   in 2002 .
this year saw more applications of analogue - inspired ideas to cosmological inflation [ 107 , 263 , 262 , 207 , 275 ] , to neutron star cores , and to the cosmological constant [ 414 , 416 ] .
normal mode  analysis in [ 15 , 16 , 398 ] , the development of dielectric analogues in , speculations regarding the possibly emergent nature of einstein gravity [ 20 , 398 ] , and further developments regarding the use of he  a   as an analogue for electromagnetism .
vorticity was discussed in , and the use of becs as a model for the breakdown of lorentz invariance in .
the excitation spectrum in superfluids , specifically the fermion zero modes , were investigated in [ 412 , 182 ] , while the relationship between rotational friction in superfluids and super - radiance in rotating spacetimes was discussed in .
 was the title and theme of parentani s article in 2002 , while schutzhold and unruh developed a rather different fluid - based analogy based on gravity waves in shallow water [ 344 , 345 ] .
super - radiance was investigated in , while the propagation of phonons and quasiparticles was discussed in [ 122 , 121 ] .
the stability of an acoustic white hole was investigated in , while further developments regarding analogue models based on nonlinear electrodynamics were presented by novello and collaborators in [ 102 , 103 , 282 , 278 , 126 ] .
though analogue models lead naturally to the idea of high - energy violations of lorentz invariance , it must be stressed that definite observational evidence for violations of lorentz invariance is lacking  in fact there are rather strong constraints on how strong any possible lorentz violating effect might be [ 195 , 194 ] .
that year saw further discussion of analogue - inspired models for black hole entropy and the cosmological constant [ 419 , 421 ] , and the development of analogue models for frw geometries [ 115 , 114 , 17 , 105 , 242 ] .
there were several further developments regarding the foundations of bec - based models in [ 18 , 116 ] , while analogue spacetimes in superfluid neutron stars were further investigated in .
effective geometry was the theme in , while applications of nonlinear electrodynamics ( and its effective metric ) to cosmology were presented in .
super - radiance was further investigated in [ 26 , 24 ] , while the limitations of the  slow light 
the review article by burgess , emphasised the role of general relativity as an effective field theory  the sine qua non for any attempt at interpreting general relativity as an emergent theory .
the lecture notes by jacobson   give a nice introduction to hawking radiation and its connection to analogue spacetimes .
the year 2004 saw the appearance of some 30 articles on ( or closely related to ) analogue models .
effective geometries in astrophysics were discussed by perez bergliaffa , while the physical realizability of acoustic hawking radiation was addressed in [ 95 , 382 ] .
more cosmological issues were raised in [ 382 , 424 ] , while a specifically astrophysical use of the acoustic analogy was invoked in [ 96 , 97 , 98 ] .
bec - based horizons were again considered in [ 149 , 148 ] , while backreaction effects were the focus of attention in [ 10 , 9 , 208 ] .
more issues relating to the simulation of frw cosmologies were raised in [ 118 , 119 ] .
unruh and schtzhold discussed the universality of the hawking effect , and a new proposal for possibly detecting hawking radiation in a electromagnetic wave guide .
the causal structure of analogue spacetimes was considered in , while quasinormal modes attracted attention in [ 31 , 237 , 64 , 269 ] .
spacetimes , a proposal for using analogue models to generate massive phonon modes in becs , and an extension of the usual formalism for representing weak - field gravitational lensing in terms of an analogue refractive index .
finally we mention the development of yet more strong observational bounds on possible ultra high energy lorentz violation [ 196 , 197 ] .
the first few months of 2005 have seen continued and vigourous activity on the analogue model front .
more studies of the super - resonance phenomenon have appeared [ 25 , 113 , 209 , 354 ] , and a mini - survey was presented in .
quasinormal modes have again received attention in , while the magnus force is reanalysed in terms of the acoustic geometry in .
singularities in the acoustic geometry are considered in , while back - reaction has received more attention in .
we shall in the next subsection focus more precisely on the early history of analogue models , and specifically those that seem to us to have had a direct historical connection with the sustained burst of work carried out in the last 15 years .
of course the division into pre-1981 and post-1981 articles is at a deeper level somewhat deceptive .
there have been several analogue models investigated over the years , with different aims , different levels of sophistication , and ultimately different levels of development .
armed with a good library and some hindsight it is possible to find interesting analogues in a number of places.17 perhaps the first paper to seriously discuss analogue models and effective metric techniques was that of gordon ( yes , he of the klein - gordon equation ) .
that is : gordon wanted to use a gravitational field to mimic a dielectric medium .
what is now often referred to as the gordon metric is the expression 116\documentclass[12pt]{minimal } 
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				 \begin{document}$${\left [ { { g_{{\rm{effective } } } } } \right]_{\mu \nu } } = { \eta _ { \mu \nu } } + \left [ { 1 - { n^{- 2 } } } \right]{v_\mu}{v_\nu},$$\end{document } where  is the flat minkowski metric , n is the refractive index , and v is the 4-velocity of the medium .
one historically important contribution was one of the problems in the well - known book  the classical theory of fields  by landau and lifshitz .
see the end of chapter 10 , paragraph 90 , and the problem immediately thereafter :  equations of electrodynamics in the presence of a gravitational field  .
note that in contrast to gordon , here the interest is in using dielectric media to mimic a gravitational field . in france
the idea was taken up by pham mau quan , who showed that ( under certain conditions ) maxwell s equations can be expressed directly in terms of the effective metric specified by the coefficients 117\documentclass[12pt]{minimal } 
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				 \begin{document}$${\left [ { { g_{{\rm{effective } } } } } \right]_{\mu \nu } } = { g_{\mu \nu } } + \left [ { 1 - { 1 \over { \epsilon \mu } } } \right]{v_\mu}{v_\nu},$$\end{document } where g is the ordinary spacetime metric ,  and  are the permeability and permittivity , and v is the 4-velocity of the medium .
the trajectories of the electromagnetic rays are interpreted in this case as geodesics of null length of this new effective metric .
three articles that directly used the dielectric analogy to analyse specific physics problems are those of skrotskii , balazs , and winterberg .
the general formalism was more fully developed in articles such as those by peblanski [ 304 , 303 ] , and good summary of this classical period can be found in the article by de felice . in summary and with the benefit of hindsight : an arbitrary gravitational field can always be represented as an equivalent optical medium , but subject to the somewhat unphysical restriction that 118\documentclass[12pt]{minimal } 
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				 \begin{document}$$\left [ { { \rm{magnetic}}\,{\rm{permitivity } } } \right]\infty \left [ { { \rm{electric}}\,{\rm{permeability } } } \right].$$\end{document } if an optical medium does not satisfy this constraint ( with a position independent proportionality constant ) then it is not completely equivalent to a gravitational field . for a position dependent proportionality constant complete equivalence can be established in the geometric optics limit , but for wave optics the equivalence is not complete . there were several papers in the 1980 s using an acoustic analogy to investigate the propagation of shockwaves in astrophysical situations , most notably those of moncrief   and matarrese [ 259 , 260 , 258 ] .
in particular in moncrief s work   the linear perturbations of a relativistic perfect fluid on an arbitrary general relativistic metric were studied , and it was shown that the wave equation for such perturbations can be expressed as a relativistic wave equation on some effective ( acoustic ) metric ( which can admit acoustic horizons ) . in this sense   can be seen as a precursor to the later works on acoustic geometries and acoustic horizons.18 the so - called  electro - mechanical analogy  has also had a long history within the engineering community .
electro - mechanical - acoustic  analogy , or even an  electro - thermal  analogy .
unfortunately the issues of interest to the engineering community rarely resonate within the relativity community , and these engineering analogies ( though powerful in their own right ) have no immediate impact for our purposes.19 the motivation for these investigations ( both historical and current ) is rather mixed . in modern language
the reasons to investigate analogue models are : 
 partly to use condensed matter to gain insight into classical general relativity.partly to use condensed matter to gain insight into curved - space quantum field theory.partly to develop an observational window on curved - space quantum field theory.partly to use classical general relativity to gain insight into condensed matter physics.partly ( much more tenuous ) to gain insight into new and radically different ways of dealing with  quantum gravity  . partly to use condensed matter to gain insight into classical general relativity . partly to use condensed matter to gain insight into curved - space quantum field theory . partly to develop an observational window on curved - space quantum field theory . partly to use classical general relativity to gain insight into condensed matter physics .
partly ( much more tenuous ) to gain insight into new and radically different ways of dealing with  quantum gravity  . to further complicate the history
, there is large body of work for which analogue spacetime ideas provide part of the background gestalt , even if the specific connection may sometimes be somewhat tenuous . among such articles we mention : 
 analogue - based  geometrical  interpretations of pseudo - momentum , iordanskii forces , magnus forces , and the acoustic aharanov - bohm effect [ 133 , 365 , 366 , 367 , 368 , 408].an analogue - inspired interpretation of the kerr spacetime .the use of analogies to clarify the newtonian limit of general relativity , to provide heuristics for motivating interest in specific spacetimes [ 320 , 395 ] , and to discuss a simple interpretation of the notion of a horizon .discrete   and non - commutative   spacetimes partially influenced and flavoured by analogue ideas.analogue-based hints on how to implement 
 ( dsr ) [ 215 , 216 , 217 , 370 ] , and a cautionary analysis of why this might be difficult .possible black - hole phase transitions placed in an analogue context .cosmological structure formation viewed as noise amplification .modified inflationary scenarios [ 81 , 83].discussions of unusual topology ,  acoustic wormholes  , and unusual temporal structure [ 270 , 272 , 313 , 357 , 358 , 433].analogue models based on plasmon physics [ 357 , 358].abstract quantum field theoretic considerations of the unruh effect .numerous suggestions regarding possible trans - planckian physics [ 7 , 29 , 69 , 74 , 75 , 171 , 262 , 322 , 371].numerous suggestions regarding a minimum length in quantum gravity [ 32 , 35 , 49 , 85 , 135 , 175 , 176 , 215 , 216 , 217 , 243 , 245 , 244 , 355].standard quantum field theory physics reformulated in the light of analogue models [ 4 , 5 , 117 , 127 , 238 , 239 , 240 , 241 , 248 , 285 , 286 , 295 , 294 , 296 , 301 , 314 , 428].standard general relativity supplemented with analogue viewpoints and insights [ 212 , 225 , 248].the discussion of , and argument for , a possible reassessment of fundamental features of quantum physics and general relativity [ 6 , 152 , 206 , 226 , 241 , 297 , 328 , 335].non - standard viewpoints on quantum physics and general relativity [ 93 , 174 , 290 , 324 , 323 , 336 , 337 , 338 , 339].soliton physics , defect physics , and the fizeau effect , presented with an analogue flavour.analogue-inspired models of black hole accretion [ 315 , 316].cosmological horizons from an analogue spacetime perspective .analogue - inspired insights into renormalization group flow .an analysis of  wave catastrophes  inspired by analogue models .improved numerical techniques for handling wave equations , and analytic techniques for handling wave tails , partially based on analogue ideas . analogue - based  geometrical  interpretations of pseudo - momentum , iordanskii forces , magnus forces , and the acoustic aharanov - bohm effect [ 133 , 365 , 366 , 367 , 368 , 408 ] .
an analogue - inspired interpretation of the kerr spacetime . the use of analogies to clarify the newtonian limit of general relativity , to provide heuristics for motivating interest in specific spacetimes [ 320 , 395 ] , and to discuss a simple interpretation of the notion of a horizon .
discrete   and non - commutative   spacetimes partially influenced and flavoured by analogue ideas .
[ 215 , 216 , 217 , 370 ] , and a cautionary analysis of why this might be difficult .
discussions of unusual topology ,  acoustic wormholes  , and unusual temporal structure [ 270 , 272 , 313 , 357 , 358 , 433 ] .
numerous suggestions regarding possible trans - planckian physics [ 7 , 29 , 69 , 74 , 75 , 171 , 262 , 322 , 371 ] .
numerous suggestions regarding a minimum length in quantum gravity [ 32 , 35 , 49 , 85 , 135 , 175 , 176 , 215 , 216 , 217 , 243 , 245 , 244 , 355 ] .
standard quantum field theory physics reformulated in the light of analogue models [ 4 , 5 , 117 , 127 , 238 , 239 , 240 , 241 , 248 , 285 , 286 , 295 , 294 , 296 , 301 , 314 , 428 ] .
standard general relativity supplemented with analogue viewpoints and insights [ 212 , 225 , 248 ] .
the discussion of , and argument for , a possible reassessment of fundamental features of quantum physics and general relativity [ 6 , 152 , 206 , 226 , 241 , 297 , 328 , 335 ] .
non - standard viewpoints on quantum physics and general relativity [ 93 , 174 , 290 , 324 , 323 , 336 , 337 , 338 , 339 ] .
soliton physics , defect physics , and the fizeau effect , presented with an analogue flavour .
improved numerical techniques for handling wave equations , and analytic techniques for handling wave tails , partially based on analogue ideas . from the above the reader
can easily appreciate the broad interest in , and wide applicability of , analogue spacetime models .
we have doubtless missed some articles of historical importance , but with a good library or a fast internet connection the reader will be in as good a position as we are to find any additional historical articles .
in this chapter , we will attempt to categorise the very many analogue models researchers have investigated . perhaps the most basic subdivision is into classical models and quantum models , but even then many other levels of refinement are possible .
consider for instance the following list : 
 classical models : 
 classical sound.water waves ( gravity waves).classical refractive index.normal modes.quantum models : 
 bose - einstein condensates ( becs).the heliocentric universe.(helium as an exemplar for just about anything.)slow light . 

quantum models : 
 bose - einstein condensates ( becs).the heliocentric universe.(helium as an exemplar for just about anything.)slow light .
sound in a moving fluid has already been extensively discussed in section 2 , and we will not repeat such discussion here . in contrast , sound in a solid exhibits its own distinct and interesting features , notably in the existence of a generalization of the normal notion of birefringence  longitudinal modes travel at a different speed ( typically faster ) than do transverse modes . this may be viewed as an example of an analogue model which breaks the  light cone  into two at the classical level ; as such this model is not particularly useful if one is trying to simulate special relativistic kinematics with its universal speed of light , though it may be used to gain insight into yet another way of  breaking  lorentz invariance .
a wonderful example of the occurrence of an effective metric in nature is that provided by gravity waves in a shallow basin filled with liquid   ( see figure 7).20 if one neglects the viscosity and considers an irrotational flow , v =  , one can write bernoulli s equation in the presence of earth s gravity as 119\documentclass[12pt]{minimal } 
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				 \begin{document}$${\partial _ t}\phi + { 1 \over 2}{(\nabla \phi)^2 } = - { p \over \rho } - gz - { v_\parallel}.$$\end{document } 
 figure 7gravity waves in a shallow fluid basin with a background horizontal flow .
 is the density of the fluid ,  its pressure , g the gravitational acceleration and v a potential associated with some external force necessary to establish an horizontal flow in the fluid .
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				 \begin{document}${\bf{v}}_{\rm{b}}^{||}$\end{document}. we must also impose the boundary conditions that the pressure at the surface , and the vertical velocity at the bottom , both vanish : p(z = hb ) = 0 , v(z = 0 ) = 0 . once a horizontal background flow is established , one can see that the perturbations of the velocity potential satisfy 120\documentclass[12pt]{minimal } 
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				 \begin{document}$${\partial _
t}\,\delta \phi + { \bf{v}}_{\rm{b}}^\parallel \cdot { \nabla _ \parallel}\delta \phi = - { p \over \rho}.$$\end{document } if we now expand this perturbation potential in a taylor series 121\documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \phi = \sum\limits_{n = 0}^\infty { { { { z^n } } \over { n ! } } } \,\delta { \phi _ n}(x , y),$$\end{document } it is not difficult to prove   that surface waves with long wavelengths ( long compared with the depth of the basin ,   hb ) , can be described to a good approximation by 0 ( x , y ) and that this field  sees
 an effective metric of the form 122\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } = { 1 \over { { c^2}}}\left [ { - \,({c^2 } - \nu _ { \rm{b}}^{\parallel 2})\,{\rm{d}}{t^2 } - 2{\rm{v}}_{\rm{b}}^\parallel \,\cdot\,{\bf{dx}}\,{\rm{d}}t + { \bf{dx}}\,\cdot\,{\bf{dx } } } \right],$$\end{document } where \documentclass[12pt]{minimal } 
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				 \begin{document}$c \equiv \sqrt { g{h_{\rm{b}}}}$\end{document}. the link between small variations of the potential field and small variations of the position of the surface is provided by the following equation 123\documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta { \nu _ \bot } = - { h_{\rm{b}}}\,\nabla _ \parallel ^2\delta { \phi _ 0 } = { \partial _ t}\,\delta h + { \rm{v}}_{\rm{b}}^\parallel \cdot { \nabla _
\parallel}\delta h = { { \rm{d } } \over { { \rm{d}}t}}\delta h.$$\end{document } the entire previous analysis can be generalised to the case in which the bottom of the basin is not flat , and the background flow not purely horizontal .
therefore , one can create different effective metrics for gravity waves in a shallow fluid basin by changing ( from point to point ) the background flow velocity and the depth , hb(x , y ) .
the main advantage of this model is that the velocity of the surface waves can very easily be modified by changing the depth of the basin .
this velocity can be made very slow , and consequently , the creation of ergoregions should be relatively easier than in other models .
as described here , this model is completely classical given that the analogy requires long wavelengths and slow propagation speeds for the gravity waves .
although the latter feature is convenient for the practical realization of analogue horizons , it is a disadvantage in trying to detect analogue hawking radiation as the relative temperature will necessarily be very low .
( this is why , in order to have a possibility of experimentally observing hawking evaporation and other quantum phenomena , one would need to use ultra cold quantum fluids . )
however , the gravity wave analogue can certainly serve to investigate the classical phenomena of mode mixing that underlies the quantum processes .
the macroscopic maxwell equations inside a dielectric take the well - known form 124\documentclass[12pt]{minimal } 
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				 \begin{document}$$\nabla \ , \cdot \,{\bf{b } } = 0,\quad \,\,\nabla \ , \times \,\,{\bf{e}}\,{\rm{+}}\,{\partial _ t}{\bf{b}}\,{\rm{=}}\,{\rm{0,}}$$\end{document } 
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				 \begin{document}$$\nabla \,\cdot\,{\bf{d } } = 0,\quad \,\,\nabla \ , \times \,\,{\bf{h}}\ , - \,{\partial _
t}{\bf{d}}\,{\rm{=}}\,{\rm{0}},$$\end{document } with the constitutive relations h =   b and d =  # x00b7 ; e. here ,  is the 3  3 permittivity tensor and  the 3  3 permeability tensor of the medium .
these equations can be written in a condensed way as 126\documentclass[12pt]{minimal } 
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				 \begin{document}$${\partial _
\alpha \nu \beta}}\,{f_{\nu \beta } } } \right ) = 0$$\end{document } where f = a[, ] is the electromagnetic tensor , 127\documentclass[12pt]{minimal } 
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				 \begin{document}$${f_{0i } } = - { f_{i0 } } = - { e_{i,}}\quad { f_{ij } } = { \epsilon _ { ijk}}{b^k},$$\end{document } and ( assuming the medium is at rest ) the non - vanishing components of the 4th rank tensor z are given by 128\documentclass[12pt]{minimal } 
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				 \begin{document}$${z^{0i0j } } = - { z^{0ij0 } } = { z^{i0j0 } } = - { z^{i00j } } = - { 1 \over 2}{\epsilon ^{ij}};$$\end{document } 
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				 \begin{document}$${z^{ijkl } } = { 1 \over 2}{\epsilon ^{ijm}}\,{\epsilon ^{kln}}\,\mu _ { mn}^{- 1};$$\end{document } supplemented by the conditions that z is antisymmetric on its first pair of indices and antisymmetric on its second pair of indices . without significant loss of generality we can ask that z also be symmetric under pairwise interchange of the first pair of indices with the second pair  thus z exhibits most of the algebraic symmetries of the riemann tensor , though this appears to merely be accidental , and not fundamental in any way .
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				 \begin{document}$${\mathcal
l } = \sqrt { - g } \,{g^{\mu \alpha}}\,{g^{\nu \beta}}{f_{\mu \nu}}\,{f_{\alpha \beta}}$$\end{document } we see that in curved spacetime we can also write the electromagnetic equations of motion in the form ( 126 ) where now ( for some constant k ) : 131\documentclass[12pt]{minimal } 
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				 \begin{document}$${z^{\mu \nu \alpha \beta } } = k\sqrt { - g } \left\{{{g^{\mu \alpha}}{g^{\nu \beta } } - { g^{\mu \beta}}{g^{\nu \alpha } } } \right\}$$\end{document } if we consider a static gravitational field we can always re - write it as a conformal factor multiplying an ultra - static metric 132\documentclass[12pt]{minimal } 
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				 \begin{document}$${g_{\mu \nu } } = { \omega ^2}\,\left\{{- 1 \oplus { g_{ij } } } \right\}$$\end{document } then 133\documentclass[12pt]{minimal } 
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				 \begin{document}$${z^{0i0j } } = - { z^{0ij0 } } = { z^{i0j0 } } = - { z^{i00j } } = - k\,\sqrt { - g } \,{g^{ij}};$$\end{document } 
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				 \begin{document}$${z^{ijkl } } = k\,\sqrt { - g } \,\left\{{{g^{ik}}\,{g^{jl } } - { g^{il}}\,{g^{jk } } } \right\}$$\end{document } the fact that z is independent of the conformal factor  is simply the reflection of the well - known fact that the maxwell equations are conformally invariant in ( 3 + 1 ) dimensions . thus if we wish to have the analogy ( between a static gravitational field and a dielectric medium at rest ) hold at the level of the wave equation ( physical optics ) we must satisfy the two stringent constraints 135\documentclass[12pt]{minimal } 
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				 \begin{document}$$k\,\sqrt { - g } \,{g^{ij } } = { 1 \over 2}\,{\epsilon ^{ij}};$$\end{document } 
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				 \begin{document}$$k\,\sqrt { - g } \left\{{{g^{ik}}\,{g^{jl } } - { g^{il}}\,{g^{jk } } } \right\ } = { 1 \over 2}\,{\epsilon ^{ijm}}\,{\epsilon ^{kln}}\,\mu _
{ mn}^{- 1}.$$\end{document } the second of these constraints can be written as 137\documentclass[12pt]{minimal } 
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				 \begin{document}$$k\,\sqrt { - g } \,{\epsilon _ { ijm}}\,{\epsilon _ { kln}}\,\left\{{{g^{ik}}\,{g^{jl } } } \right\ } = \mu _ { mn}^{- 1}.$$\end{document } in view of the standard formula for 3  3 determinants 138\documentclass[12pt]{minimal } 
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				 \begin{document}$${\epsilon _ { ijm}}\,{\epsilon _ { kln}}\,\left\{{{x^{ik}}{x^{jl } } } \right\ } = 2\,\det \,x\,x_{mn}^{- 1},$$\end{document } this now implies 139\documentclass[12pt]{minimal } 
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				 \begin{document}$$2k{{{g_{ij } } } \over { \sqrt { - g } } } = \mu _ { ij}^{- 1},$$\end{document } whence 140\documentclass[12pt]{minimal } 
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				 \begin{document}$${1 \over { 2k}}\,\sqrt { - g } \,{g^{ij } } = { \mu ^{ij}}.$$\end{document } comparing this with 141\documentclass[12pt]{minimal } 
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				 \begin{document}$$2k\,\sqrt { - g } \,{g^{ij } } = { \epsilon ^{ij}},$$\end{document } we now have : 142\documentclass[12pt]{minimal } 
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				 \begin{document}$${\epsilon ^{ij } } = 4\,{k^2}\,{\mu ^{ij}};$$\end{document } 
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				 \begin{document}$${g^{ij } } = { { 4\,{k^2 } } \over { \det \,\epsilon}}{\epsilon ^{ij}};$$\end{document } 
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				 \begin{document}$${g^{ij } } = { 1 \over { 4\,{k^2}\,\det \,\mu}}{\mu ^{ij}}.$$\end{document } to rearrange this , introduce the matrix square root [  ] , which always exists because  is real positive definite and symmetric .
then 145\documentclass[12pt]{minimal } 
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				 \begin{document}$${g^{ij } } = { \left [ { { { \left\{{{{{\mu ^{1/2}}\,\epsilon \,{\mu ^{1/2 } } } \over { \det ( \mu \,\epsilon ) } } } \right\}}^{1/2 } } } \right]^{ij}}.$$\end{document } note that if you are given the static gravitational field ( in the form  , gij ) one can always solve to find an equivalent analogue in terms of permittivity / permeability ( albeit an analogue that satisfies the mildly unphysical constraint   ).21 on the other hand , if you are given permeability and permittivity tensors  and  , then it is only for that subclass of media that satisfy 
  that one can perfectly mimic all of the electromagnetic effects by an equivalent gravitational field .
of course this can be done provided one only considers wavelengths that are sufficiently long for the macroscopic description of the medium to be valid . in this respect
it is interesting to note that the behaviour of the refractive medium at high frequencies has been used to introduce an effective cutoff for the modes involved in hawking radiation .
we shall encounter this model later on when we shall consider the trans - planckian problem for hawking radiation . with a bit more work this discussion
can be extended to a medium in motion , leading to an extension of the gordon metric .
alternatively , one can agree to ask more limited questions by working at the level of geometrical optics ( adopting the eikonal approximation ) , in which case there is no longer any restriction on the permeability and permittivity tensors . to see this
construct the matrix 146\documentclass[12pt]{minimal } 
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				 \begin{document}$${c^{\mu \nu } } = { z^{\mu \alpha \nu \beta}}\,{k_\alpha}{k_\beta}.$$\end{document } the dispersion relations for the propagation of photons ( and therefore the sought for geometrical properties ) can be obtained from the reduced determinant of c ( notice that the [ full ] determinant of c is identically zero as c 
 k = 0 ; the reduced determinant is that associated with the three directions orthogonal to k = 0 ) . by choosing the gauge a0 =
0 one can see that this reduced determinant can be obtained from the determinant of the 3  3 sub - matrix c. this determinant is 147\documentclass[12pt]{minimal } 
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				 \begin{document}$$\det ( { c^{ij } } ) = { 1 \over 8}\det \left({- { \omega ^2}{\epsilon ^{ij } } + { \epsilon ^{ikm}}\,{\epsilon ^{jln}}\,\mu _ { mn}^{- 1}{k_k}{k_l } } \right).$$\end{document } or , after making some manipulations , 148\documentclass[12pt]{minimal } 
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				 \begin{document}$$\det ( { c^{ij } } ) = { 1 \over 8}\det \,\left [ { - { \omega ^2}{\epsilon ^{ij } } + \,{{(\det \,\mu)}^{- 1}}({\mu ^{ij}}{\mu ^{kl}}{k_k}{k_l } - { \mu ^{im}}{k_m}{\mu ^{jl}}{k_l } ) } \right].$$\end{document } to simplify this , again introduce the matrix square roots [  ] and [ ]ij , which always exist because the relevant matrices are real positive definite and symmetric
. then define 149\documentclass[12pt]{minimal } 
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				 \begin{document}$${\tilde k^i } = { \left [ { { \mu ^{1/2 } } } \right]^{ij}}\,{k_j}$$\end{document } and 150\documentclass[12pt]{minimal } 
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				 \begin{document}$${\left [ { \tilde \epsilon } \right]^{ij } } = \det ( \mu)\,{\left [ { { \mu ^{- 1/2}}\,\epsilon \,{\mu ^{- 1/2 } } } \right]_{ij}}$$\end{document } so that 151\documentclass[12pt]{minimal } 
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				 \begin{document}$$\det ( { c^{ij } } ) \propto \det \left\{{- { \omega ^2}{{[\tilde \epsilon]}^{ij } } + { \delta ^{ij}}[{\delta _ { mn}}\,{{\tilde k}^m}\,{{\tilde k}^n } ] - { { \tilde k}^i}\,{{\tilde k}^j } } \right\}.$$\end{document } the behaviour of this dispersion relation now depends critically on the way that the eigenvalues of \documentclass[12pt]{minimal } 
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				 \begin{document}${\bf{\tilde \epsilon}}$\end{document } are distributed .
if all eigenvalues are degenerate then \documentclass[12pt]{minimal } 
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				 \begin{document}${\bf{\tilde \epsilon } } = \tilde \epsilon \,{\bf{i}}$\end{document } , implying    but now with the possibility of a position dependent proportionality factor ( in the case of physical optics the proportionality factor was constrained to be a position - independent constant ) . in this case
we now easily evaluate 152\documentclass[12pt]{minimal } 
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				 \begin{document}$$\epsilon = { { { \rm{tr}}(\epsilon ) } \over { { \rm{tr}}(\mu)}}\mu \quad { \rm{and}}\quad \tilde \epsilon = \det \,\mu { { { \rm{tr}}(\epsilon ) } \over { { \rm{tr}}(\mu)}},$$\end{document } while 153\documentclass[12pt]{minimal } 
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				 \begin{document}$$\det \,({c^{ij } } ) \propto { \omega ^2}{\left\{{{\omega ^2 } - [ { { \tilde \epsilon}^{- 1}}{\delta _ { mn}}{{\tilde k}^m}\,{{\tilde k}^n } ] } \right\}^2}.$$\end{document } that is 154\documentclass[12pt]{minimal } 
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				 \begin{document}$$\det ( { c^{ij } } ) \propto { \omega ^2}{\left\{{{\omega ^2 } - [ { g^{ij}}\,{k_i}\,{k_j } ] } \right\}^2},$$\end{document } with 155\documentclass[12pt]{minimal } 
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				 \begin{document}$${g^{ij } } = { 1 \over { \tilde \epsilon}}{[\mu ] ^{ij } } = { { { \rm{tr}}(\mu)\,{{[\mu ] } ^{ij } } } \over { { \rm{tr}}(\epsilon)\,\det \,\mu } } = { { { \rm{tr}}(\epsilon)\,{{[\epsilon ] } ^{ij } } } \over { { \rm{tr}}(\mu)\,\det \,\epsilon}}.$$\end{document } this last result is compatible with but more general than the result obtained under the more restrictive conditions of physical optics . in the situation where both permittivity and permeability are isotropic , (  =   and  =   ) this reduces to the perhaps more expected result 156\documentclass[12pt]{minimal } 
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				 \begin{document}$${g^{ij } } = { { { \delta ^{ij } } } \over { \epsilon \,\mu}}.$$\end{document } if \documentclass[12pt]{minimal } 
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				 \begin{document}${\bf{\tilde \epsilon}}$\end{document } has two distinct eigenvalues then the determinant det(c ) factorises into a trivial factor of  and two quadratics .
this is the physical situation encountered in uni - axial crystals , where the ordinary and extraordinary rays each obey distinct quadratic dispersion relations . from the point of view of analogue models
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				 \begin{document}${\bf{\tilde \epsilon}}$\end{document } has three distinct eigenvalues then the determinant det(c ) is the product of a trivial factor of  and a non - factorizable quartic .
this is the physical situation encountered in bi - axial crystals [ 39 , 399 ] , and it seems that no meaningful notion of effective riemannian metric can be assigned to this case .
( the use of finsler geometries in this situation is an avenue that may be worth pursuing . )
hehl and co - workers have championed the idea of using the linear constitutive relations of electrodynamics as the primary quantities , and then treating the spacetime metric ( even for flat space ) as a derived concept .
in general , the permittivity and permeability tensors can be modified by applying strong electromagnetic fields ( this produces an effectively non - linear electrodynamics ) .
the entire previous discussion still applies if one considers the photon as the linear perturbation of the electromagnetic field over a background configuration 157\documentclass[12pt]{minimal } 
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				 \begin{document}$${f_{\mu \nu } } = f_{\mu \nu}^{{\rm{bg } } } + f_{\mu \nu}^{{\rm{ph}}}.$$\end{document } the background field \documentclass[12pt]{minimal } 
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				 \begin{document}$f_{\mu v}^{{\rm{bg}}}$\end{document } sets the value of (f ) , and (f ) .
equation ( 126 ) then becomes an equation for \documentclass[12pt]{minimal } 
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				 \begin{document}$f_{\mu v}^{{\rm{ph}}}$\end{document}. this approach has been extensively investigated by novello and co - workers [ 279 , 283 , 102 , 282 , 280 , 281 , 278 , 126 ] .
the propagation of photons in a dielectric medium characterised by 3  3 permeability and permittivity tensors constrained by    is equivalent ( at the level of geometric optics ) to the propagation of photons in a curved spacetime manifold characterised by the ultrastatic metric ( 155 ) , provided one only considers wavelengths that are sufficiently long for the macroscopic description of the medium to be valid .
if in addition one takes a fluid dielectric , by controlling its flow one can generalise the gordon metric and again reproduce metrics of the painleve - gullstrand type , and therefore geometries with ergo - regions .
if the proportionality constant relating    is position independent , one can make the stronger statement ( 144 ) which holds true at the level of physical optics .
we have already seen how linearizing the euler - lagrange equations for a single scalar field naturally leads to the notion of an effective spacetime metric .
if more than one field is involved the situation becomes more complicated , in a manner similar to that of geometrical optics in uni - axial and bi - axial crystals .
( this should , with hindsight , not be too surprising since electromagnetism , even in the presence of a medium , is definitely a lagrangian system and definitely involves more than one single scalar field . ) a normal mode analysis based on a general lagrangian ( many fields but still first order in derivatives of those fields ) leads to a concept of refringence , or more specifically multi - refringence , a generalization of the birefringence of geometrical optics . to see how this comes about ,
we want to consider linearised fluctuations around some background solution of the equations of motion . as in the single - field case
we write ( here we will follow the notation and conventions of ) 158\documentclass[12pt]{minimal } 
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				 \begin{document}$${\phi ^a}(t,\vec x ) = \phi _ 0^a(t,\vec x ) + \epsilon \,\phi _ 1^a(t,\vec x ) + { { { \epsilon ^2 } } \over 2}\phi _ 2^a(t,\vec x ) + o({\epsilon ^3}).$$\end{document } now use this to expand the lagrangian 159\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{array}{*{20}c}{{\mathcal l}({\partial _
\mu}\phi _ 0^a,\,\phi _ 0^a ) + \epsilon \left [ { { { \partial { \mathcal l } } \over { \partial ( { \partial _
\mu}\phi _ 1^a + { { \partial { \mathcal l } } \over { \partial { \phi ^a}}}\phi _ 1^a } \right ] } \\ { + { { { \epsilon ^2 } } \over 2}\left [ { { { \partial { \mathcal l } } \over { \partial ( { \partial
\mu}\phi _ 2^a + { { \partial { \mathcal l } } \over { \partial { \phi ^a}}}\phi _ 2^a } \right ] } \\
{ \quad + { { { \epsilon ^2 } } \over 2}\left [ { { { { \partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}{\phi ^a})\,\partial ( { \partial
\\ { \left . { \quad \quad \quad \quad \quad \quad + 2{{{\partial ^2}{\mathcal l } } \over { \partial
( { \partial _ \mu}{\phi ^a})\,\partial { \phi ^b}}}\,{\partial _ \mu}\phi _ 1^a\,\phi _ 1^b + { { { \partial ^2}{\mathcal l } } \over { \partial { \phi ^a}\,\partial { \phi ^b}}}\phi _ 1^a\,\phi _ 1^b } \right ] } \\ { + o({\epsilon ^3})\quad \quad \quad \quad \quad \quad \quad \quad } \\ \end{array}$$\end{document } consider the action 160\documentclass[12pt]{minimal } 
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				 \begin{document}$$s[{\phi ^a } ] = \int { { d^{d + 1}}x\,{\mathcal l}({\partial _
\mu}{\phi ^a},\,{\phi ^a}).}$$\end{document } doing so allows us to integrate by parts . as in the single - field case
we can use the euler - lagrange equations to discard the linear terms ( since we are linearizing around a solution of the equations of motion ) and so get 161\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{array}{*{20}c } { s[{\phi ^a } ] = s[\phi _ 0^a]\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad
\quad } \\ { + { { { \epsilon ^2 } } \over 2}\int { { d^{d + 1}}x\left [ { \left\{{{{{\partial ^2}{\mathcal l } } \over { \partial
\nu}{\phi ^b } ) } } } \right\}{\partial _ \mu}\phi _ 1^a\,{\partial _ \nu}\phi _ 1^b } \right . } } \\ { \left .
{ \quad \quad \quad \quad \quad \quad + 2\left\{{{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _
\mu}{\phi ^a})\,\partial { \phi ^b } } } } \right\}\,{\partial _ \mu}\phi _ 1^a\,\phi _ 1^b + \left\{{{{{\partial
^2}{\mathcal l } } \over { \partial { \phi ^a}\,\partial { \phi ^b } } } } \right\}\phi _ 1^a\,\phi _ 1^b } \right ] } \\ { + o({\epsilon ^3}).\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ \end{array}$$\end{document } because the fields now carry indices ( ab ) we can not cast the action into quite as simple a form as was possible in the single - field case .
the equation of motion for the linearised fluctuations are now read off as 162\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{array}{*{20}c } { { \partial _
\mu}\left({\left\{{{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}{\phi ^a})\,\partial ( { \partial _ \nu}{\phi ^b } ) } } } \right\}{\partial _ \nu}\phi _ 1^b } \right ) + \,{\partial _
\mu}\left({{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}{\phi ^a})\,\partial { \phi ^b}}}\phi _ 1^b } \right ) } \\ { - { \partial _ \mu}\phi _
\mu}{\phi ^b})\,\partial { \phi ^a } } } - \left({{{{\partial ^2}{\mathcal l } } \over { \partial { \phi ^a}\,\partial { \phi ^b } } } } \right)\phi _ 1^b = 0 . } \\
\end{array}$$\end{document } this is a linear second - order system of partial differential equations with position - dependent coefficients .
this system of pdes is automatically self - adjoint ( with respect to the trivial  flat 
first 163\documentclass[12pt]{minimal } 
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				 \begin{document}$${f^{\mu \nu}}_{ab } \equiv { 1 \over 2}\left({{{{\partial ^2}{\mathcal
l } } \over { \partial ( { \partial _ \mu}{\phi ^a})\,\partial ( { \partial _ \nu}{\phi ^b } ) } } + { { { \partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \nu}{\phi ^a})\,\partial ( { \partial _ \mu}{\phi ^b } ) } } } \right).$$\end{document } this quantity is independently symmetric under interchange of  ,  and a , b. we will want to interpret this as a generalization of the  densitised metric 
, f , but the interpretation is not as straightforward as for the single - field case .
next , define 164\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{array}{*{20}c } { { \gamma ^\mu}_{ab } \equiv + { { { \partial ^2}{\mathcal l } } \over { \partial ( { \partial
_ \mu}{\phi ^a})\,\partial { \phi ^b } } } - { { { \partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}{\phi ^b})\,\partial { \phi ^a}}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \,\ , } \\ { + { 1 \over 2}{\partial _
\nu}\left({{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \nu}{\phi ^a})\,\partial ( { \partial _ \mu}{\phi ^b } ) } } - { { { \partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}{\phi ^a})\,\partial ( { \partial _ \nu}{\phi ^b } ) } } } \right ) . } \\ \end{array}$$\end{document } this quantity is anti - symmetric in a , b. one might want to interpret this as some sort of  spin connexion  , or possibly as some generalization of the notion of  dirac matrices  .
finally , define 165\documentclass[12pt]{minimal } 
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				 \begin{document}$${k_{ab } } = - { { { \partial ^2}{\mathcal l } } \over { \partial { \phi ^a}\,\partial { \phi ^b } } } + { 1 \over 2}{\partial _
\mu}\left({{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}{\phi ^a})\,\partial { \phi ^b } } } } \right ) + { 1 \over 2}{\partial _
\mu}\left({{{{\partial ^2}{\mathcal l } } \over { \partial ( { \partial _ \mu}{\phi ^b})\,\partial { \phi ^a } } } } \right).$$\end{document } this quantity is by construction symmetric in ( ab ) .
we will want to interpret this as some sort of  potential  or  mass matrix  .
then the crucial point for the following discussion is to realise that equation ( 162 ) can be written in the compact form 166\documentclass[12pt]{minimal } 
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				 \begin{document}$${\partial _
\nu}\phi _ 1^b ) + { 1 \over 2}\left [ { \gamma _ { ab}^\mu \,{\partial _ \mu}\phi _ 1^b
1^b ) } \right ] + { k_{ab}}\,\phi _ 1^b = 0.$$\end{document } now it is more transparent that this is a formally self - adjoint second - order linear system of pdes .
similar considerations can be applied to the linearization of any hyperbolic system of second - order pdes .
consider an eikonal approximation for an arbitrary direction in field space , that is , take 167\documentclass[12pt]{minimal } 
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				 \begin{document}$${\phi ^a}(x ) = { \epsilon ^a}(x)\,\,\exp [ - i\varphi ( x)],$$\end{document } with (x ) a slowly varying amplitude , and (x ) a rapidly varying phase . in this eikonal approximation ( where we neglect gradients in the amplitude , and gradients in the coefficients of the pdes ,
retaining only the gradients of the phase ) the linearised system of pdes ( 166 ) becomes 168\documentclass[12pt]{minimal } 
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				 \begin{document}$$\{{f^{\mu \nu}}_{ab}\,{\partial _
\mu}\varphi ( x)\,{\partial _ \nu}\varphi ( x ) + { \gamma ^\mu}_{ab}\,{\partial _ \mu}\varphi ( x ) + { k_{ab}}\ } \,\,{\epsilon ^b}(x ) = 0.$$\end{document } this has a nontrivial solution if and only if (x ) is a null eigenvector of the matrix 169\documentclass[12pt]{minimal } 
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				 \begin{document}$${f^{\mu \nu}}_{ab}\,{k_\mu}\,{k_\nu } + { \gamma ^\mu}_{ab}\,{k_\mu } + { k_{ab}},$$\end{document } where k = (x ) . now
, the condition for such a null eigenvector to exist is that 170\documentclass[12pt]{minimal } 
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				 \begin{document}$$f(x,\,k ) \equiv \det \,\{{f^{\mu \nu}}_{ab}\,{k_\mu}\,{k_\nu } + { \gamma ^\mu}_{ab}\,{k_\mu } + { k_{ab}}\ } = 0,$$\end{document } with the determinant to be taken on the field space indices ab .
this is the natural generalization to the current situation of the fresnel equation of bi - refringent optics [ 39 , 224 ] . following the analogy with the situation in electrodynamics ( either nonlinear electrodynamics , or more prosaically propagation in a bi - refringent crystal )
the fresnel equation then describes how different polarizations can propagate at different velocities ( or in more geometrical language , can see different metric structures ) . in particle physics language
this determinant condition f(x , k ) = 0 is the natural generalization of the  mass shell  constraint .
indeed it is useful to define the mass shell as a subset of the cotangent space by 171\documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal f}(x ) \equiv \left\{{{k_\mu}\left\vert { f(x,\,k ) = 0 } \right . }
\right\}.$$\end{document } in more mathematical language we are looking at the null space of the determinant of the  symbol  of the system of pdes . by investigating f(x ,
k ) one can recover part ( not all ) of the information encoded in the matrices \documentclass[12pt]{minimal } 
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				 \begin{document}${f^{\mu v}}_{ab}$\end{document } , \documentclass[12pt]{minimal } 
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				 \begin{document}${\gamma ^\mu}_{ab}$\end{document } , and kab , or equivalently in the  generalised fresnel equation  ( 170 ) .
( note that for the determinant equation to be useful it should be non - vacuous ; in particular one should carefully eliminate all gauge and spurious degrees of freedom before constructing this  generalised fresnel equation 
, since otherwise the determinant will be identically zero . ) we now want to make this analogy with optics more precise , by carefully considering the notion of characteristics and characteristic surfaces .
we will see how to extract from the the high - frequency high - momentum regime described by the eikonal approximation all the information concerning the causal structure of the theory .
one of the key structures that a lorentzian spacetime metric provides is the notion of causal relationships .
this suggests that it may be profitable to try to work backwards from the causal structure to determine a lorentzian metric .
now the causal structure implicit in the system of second - order pdes given in equation ( 166 ) is described in terms of the characteristic surfaces , and it is for this reason that we now focus on characteristics as a way of encoding causal structure , and as a surrogate for some notion of lorentzian metric .
note that via the hadamard theory of surfaces of discontinuity the characteristics can be identified with the infinite - momentum limit of the eikonal approximation .
that is , when extracting the characteristic surfaces we neglect subdominant terms in the generalised fresnel equation and focus only on the leading term in the symbol ( \documentclass[12pt]{minimal } 
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				 \begin{document}${f^{\mu v}}_{ab}$\end{document } ) . in particle physics language going to the infinite - momentum limit
puts us on the light cone instead of the mass shell ; and it is the light cone that is more useful in determining causal structure . the  normal cone  at some specified point x , consisting of the locus of normals to the characteristic surfaces , is defined by 172\documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal n}(x ) \equiv \left\{{{k_\mu}\left\vert { \,\det \,({f^{\mu \nu}}_{ab}\,{k_\mu}\,{k_\mu } ) = 0 } \right . }
\right\}.$$\end{document } as was the case for the fresnel equation ( 170 ) , the determinant is to be taken on the field indices ab .
( remember to eliminate spurious and gauge degrees of freedom so that this determinant is not identically zero . )
we emphasise that the algebraic equation defining the normal cone is the leading term in the fresnel equation encountered in discussing the eikonal approximation .
if there are n fields in total then this  normal cone  will generically consist of n nested sheets each with the topology ( not necessarily the geometry ) of a cone .
often several of these cones will coincide , which is not particularly troublesome , but unfortunately it is also common for some of these cones to be degenerate , which is more problematic .
it is convenient to define a function q(x , k ) on the co - tangent bundle 173\documentclass[12pt]{minimal } 
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				 \begin{document}$$q(x,\,k ) \equiv \det \,({f^{\mu \nu}}_{ab}(x)\,{k_\mu}\,{k_\mu}).$$\end{document } the function q(x , k ) defines a completely - symmetric spacetime tensor ( actually , a tensor density ) with 2n indices 174\documentclass[12pt]{minimal } 
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				 \begin{document}$$q(x,\,k ) = { q^{{\mu _ 1}{\nu _ 1}{\mu _ 2}{\nu _ 2 } \cdots { \mu _ n}{\nu _ n}}}(x)\,\,{k_{{\mu _ 1}}}\,{k_{{\nu _ 1}}}\,{k_{{\mu _ 2}}}\,{k_{{\nu _ 2 } } } \cdots { k_{{\mu _
n}}}\,{k_{{\nu _ n}}}.$$\end{document } ( remember that \documentclass[12pt]{minimal } 
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				 \begin{document}${f^{\mu v}}_{ab}$\end{document } is symmetric in both  and ab independently . ) explicitly , using the expansion of the determinant in terms of completely antisymmetric field - space levi - civita tensors 175\documentclass[12pt]{minimal } 
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				 \begin{document}$${q^{{\mu _ 1}{\nu _ 1}{\mu _ 2}{\nu _ 2 } \cdots { \mu _ n}{\nu _ n } } } = { 1 \over { n!}}{\epsilon ^{{a_1}{a_2 } \ldots { a_n}}}\,{\epsilon ^{{b_1}{b_2 } \ldots { b_n}}}\,{f^{{\mu _ 1}{\nu _ 1}}}_{{a_1}{b_1}}\,{f^{{\mu _ 2}{\nu _ 2}}}_{{a_2}{b_2 } } \cdots { f^{{\mu _ n}{\nu _ n}}}_{{a_n}{b_n}}.$$\end{document } in terms of this q(x , k ) function the normal cone is 176\documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal n}(x ) \equiv \left\{{{k_\mu}\left\vert { q(x,\,k ) = 0 } \right . }
characteristic cone  , aka  null cone  ) is the envelope of the set of characteristic surfaces through the point x. thus the  monge cone  is dual to the  normal cone  , its explicit construction is given by ( courant and hilbert [ 92 , volume 2 , page 583 ] ) : 177\documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal m}(x ) = \left\{{{t^\mu } = \left .
{ { { \partial q(x,\,k ) } \over { \partial { k_\mu } } } } \right\vert { k_\mu } \in { \mathcal n}(x ) } \right\}.$$\end{document } the structure of the normal and monge cones encode all the information related with the causal propagation of signals associated with the system of pdes .
we will now see how to relate this causal structure with the existence of effective spacetime metrics , from the experimentally favoured single - metric theory compatible with the einstein equivalence principle to the most complicated case of pseudo - finsler geometries . 

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				 \begin{document}${f^{\mu v}}_{ab}$\end{document } factorises 178\documentclass[12pt]{minimal } 
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				 \begin{document}$${f^{\mu \nu}}_{ab } = { h_{ab}}\,{f^{\mu \nu}}.$$\end{document } then 179\documentclass[12pt]{minimal } 
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				 \begin{document}$$q(x,\,k ) = \det ( { h_{ab}})\,{\left [ { { f^{\mu \nu}}\,{k_\mu}\,{k_\nu } } \right]^n}$$\end{document } the monge cones and normal cones are then true geometrical cones ( with the n sheets lying directly on top of one another ) .
the normal modes all see the same spacetime metric , defined up to an unspecified conformal factor by g  f. this situation is the most interesting from the point of view of general relativity .
physically it corresponds to a single - metric theory , and mathematically it corresponds to a strict algebraic condition on the \documentclass[12pt]{minimal } 
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				 \begin{document}${f^{\mu v}}_{ab}$\end{document}.the next most useful situation corresponds to the commutativity condition : 180\documentclass[12pt]{minimal } 
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				 \begin{document}$${f^{\mu \nu}}_{ab}\,{f^{\alpha \beta}}_{bc } = { f^{\alpha \beta}}_{ab}\,{f^{\mu \nu}}_{bc};\quad { \rm{that}}\,{\rm{is}}\quad \left [ { { f^{\mu \nu}},\,{f^{\alpha \beta } } } \right ] = 0.$$\end{document } if this algebraic condition is satisfied , then for all spacetime indices  and  the \documentclass[12pt]{minimal } 
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				 \begin{document}${f^{\mu v}}_{ab}$\end{document } can be simultaneously diagonalised in field space leading to 181\documentclass[12pt]{minimal } 
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				 \begin{document}$${\bar f^{\mu \nu}}{\,_{ab } } = { \rm{diag}}\{\bar f_1^{\mu \nu},\,\bar f_2^{\mu \nu},\,\bar f_3^{\mu \nu } , \ldots , \bar f_n^{\mu \nu}\}$$\end{document } and then 182\documentclass[12pt]{minimal } 
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\begin{document}$$q(x,\,k ) = \prod\limits_{a = 1}^n { [ \bar f_a^{\mu \nu}\,{k_\mu}\,{k_\nu}].}$$\end{document } this case corresponds to an n - metric theory , where up to an unspecified conformal factor \documentclass[12pt]{minimal } 
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				 \begin{document}$g_a^{\mu v } \propto
\bar f_a^{\mu v}$\end{document}. this is the natural generalization of the two metric situation in bi - axial crystals.if \documentclass[12pt]{minimal } 
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				 \begin{document}${f^{\mu v}}_{ab}$\end{document } is completely general , satisfying no special algebraic condition , then q(x , k ) does not factorise and is in general a polynomial of degree 2n in the wave vector k. this is the natural generalization of the situation in bi - axial crystals . ( and for any deeper analysis of this situation one will almost certainly need to adopt pseudo - finsler techniques . )
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				 \begin{document}${f^{\mu v}}_{ab}$\end{document } factorises 178\documentclass[12pt]{minimal } 
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				 \begin{document}$${f^{\mu \nu}}_{ab } = { h_{ab}}\,{f^{\mu \nu}}.$$\end{document } then 179\documentclass[12pt]{minimal } 
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				 \begin{document}$$q(x,\,k ) = \det ( { h_{ab}})\,{\left [ { { f^{\mu \nu}}\,{k_\mu}\,{k_\nu } } \right]^n}$$\end{document } the monge cones and normal cones are then true geometrical cones ( with the n sheets lying directly on top of one another ) .
the normal modes all see the same spacetime metric , defined up to an unspecified conformal factor by g  f. this situation is the most interesting from the point of view of general relativity .
physically it corresponds to a single - metric theory , and mathematically it corresponds to a strict algebraic condition on the \documentclass[12pt]{minimal } 
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				 \begin{document}${f^{\mu v}}_{ab}$\end{document}. the next most useful situation corresponds to the commutativity condition : 180\documentclass[12pt]{minimal } 
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				 \begin{document}$${f^{\mu \nu}}_{ab}\,{f^{\alpha \beta}}_{bc } = { f^{\alpha \beta}}_{ab}\,{f^{\mu \nu}}_{bc};\quad { \rm{that}}\,{\rm{is}}\quad \left [ { { f^{\mu \nu}},\,{f^{\alpha \beta } } } \right ] = 0.$$\end{document } if this algebraic condition is satisfied , then for all spacetime indices  and  the \documentclass[12pt]{minimal } 
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				 \begin{document}${f^{\mu v}}_{ab}$\end{document } can be simultaneously diagonalised in field space leading to 181\documentclass[12pt]{minimal } 
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				 \begin{document}$${\bar f^{\mu \nu}}{\,_{ab } } = { \rm{diag}}\{\bar f_1^{\mu \nu},\,\bar f_2^{\mu \nu},\,\bar f_3^{\mu \nu } , \ldots , \bar f_n^{\mu \nu}\}$$\end{document } and then 182\documentclass[12pt]{minimal } 
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\begin{document}$$q(x,\,k ) = \prod\limits_{a = 1}^n { [ \bar f_a^{\mu \nu}\,{k_\mu}\,{k_\nu}].}$$\end{document } this case corresponds to an n - metric theory , where up to an unspecified conformal factor \documentclass[12pt]{minimal } 
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				 \begin{document}$g_a^{\mu v } \propto
\bar f_a^{\mu v}$\end{document}. this is the natural generalization of the two metric situation in bi - axial crystals . if \documentclass[12pt]{minimal } 
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				 \begin{document}${f^{\mu v}}_{ab}$\end{document } is completely general , satisfying no special algebraic condition , then q(x , k ) does not factorise and is in general a polynomial of degree 2n in the wave vector k. this is the natural generalization of the situation in bi - axial crystals . ( and for any deeper analysis of this situation one will almost certainly need to adopt pseudo - finsler techniques . )
the message to be extracted from this rather formal discussion is that effective metrics are rather general and mathematically robust objects that can arise in quite abstract settings  in the abstract setting discussed here it is the algebraic properties of the object \documentclass[12pt]{minimal } 
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				 \begin{document}${f^{\mu v}}_{ab}$\end{document } that eventually leads to mono - metricity , multi - metricity , or worse .
the current abstract discussion also serves to illustrate , yet again , 
 that there is a significant difference between the levels of physical normal modes ( wave equations ) , and geometrical normal modes ( dispersion relations ) , andthat the densitised inverse metric is in many ways more fundamental than the metric itself . that there is a significant difference between the levels of physical normal modes ( wave equations ) , and geometrical normal modes ( dispersion relations ) , and that the densitised inverse metric is in many ways more fundamental than the metric itself .
we have seen that one of the main aims of research in analogue models of gravity is the possibility of simulating semiclassical gravity phenomena , such as the hawking radiation effect or cosmological particle production . in this
sense systems characterised by a high degree of quantum coherence , very cold temperatures , and low speeds of sound offer the best test field .
hence it is not surprising that in recent years bose - einstein condensates ( becs ) have become the subject of extensive study as possible analogue models of general relativity [ 136 , 137 , 16 , 19 , 18 , 115 , 114 ] .
let us start by very briefly reviewing the derivation of the acoustic metric for a bec system , and show that the equations for the phonons of the condensate closely mimic the dynamics of a scalar field in a curved spacetime . in the dilute gas approximation , one can describe a bose gas through a quantum field \documentclass[12pt]{minimal } 
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				 \begin{document}$\hat \psi$\end{document } satisfying 183\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm i}\hbar { \partial \over { \partial t}}\hat\psi = \left({- { { { \hbar ^2 } } \over { 2m}}{\nabla ^2 } + { v_{{\rm{ext}}}}({\rm{x } } ) + \kappa ( a)\hat\psi { } ^\dagger \hat\psi } \right)\hat\psi.$$\end{document } here  parameterises the strength of the interactions between the different bosons in the gas
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				 \begin{document}$$\kappa ( a ) = { { 4\pi a{\hbar ^2 } } \over m}.$$\end{document } as usual , the quantum field can be separated into a macroscopic ( classical ) condensate and a fluctuation : \documentclass[12pt]{minimal } 
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				 \begin{document}$\hat\psi = \psi + \hat\varphi$\end{document } , with \documentclass[12pt]{minimal } 
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				 \begin{document}$\left\langle { \hat\psi } \right\rangle = \psi$\end{document}. then , by adopting the self - consistent mean field approximation ( see for example ) 185\documentclass[12pt]{minimal } 
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				 \begin{document}$$\hat\varphi { } ^\dagger \hat\varphi \hat\varphi \simeq 2\langle \hat\varphi { } ^\dagger \hat\varphi \rangle \hat\varphi + \langle \hat\varphi \hat\varphi \rangle { \hat\varphi{}^\dagger},$$\end{document } one can arrive at the set of coupled equations : 186\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{array}{*{20}c } { { \rm{i}}\hbar { \partial \over { \partial t}}\psi ( t,\,{\rm x } ) = \left({- { { { \hbar ^2 } } \over { 2m}}{\nabla ^2 } + { v_{{\rm{ext}}}}({\rm{x } } ) + \kappa \,{n_c } } \right)\psi ( t,\,{\rm x } ) } \\ { \quad \quad \quad \quad \quad \quad + \kappa \{2\tilde n\psi ( t,\,{\rm x } ) + \tilde m{\psi ^{\ast}}(t,\,{\rm x})\ } ; } \\ \end{array}$$\end{document } 
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				 \begin{document}$$\begin{array}{*{20}c } { { \rm{i}}\hbar { \partial \over { \partial t}}\hat\varphi ( t,\,{\rm{x } } ) = \left({- { { { \hbar ^2 } } \over { 2m}}{\nabla ^2 } + { v_{{\rm{ext}}}}({\rm{x } } ) + \kappa \ , 2{n_t } } \right)\hat\varphi ( t,\,{\rm{x } } ) } \\ { + \kappa \,{m_t}{{\hat\varphi}{}^\dagger}(t,\,{\rm{x } } ) . } \\
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				 \begin{document}$${n_c } \equiv \vert \psi ( t,\,{\rm{x}}){\vert ^2};\quad { m_c } \equiv { \psi ^2}(t,\,{\rm{x}});$$\end{document } 
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				 \begin{document}$$\tilde n \equiv \langle \hat\varphi { } ^\dagger \,\hat\varphi \rangle ; \quad \quad \tilde m \equiv
\langle \hat\varphi \,\hat\varphi \rangle;$$\end{document } 
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				 \begin{document}$${n_t } \equiv { n_c } + \tilde n;\quad \quad { m_t } \equiv { m_c } + \tilde m.$$\end{document } the equation for the classical wave function of the condensate is closed only when the back - reaction effect due to the fluctuations are neglected .
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				 \begin{document}${\tilde m}$\end{document } and  . )
this is the approximation contemplated by the gross - pitaevskii equation . in general one will have to solve both equations simultaneously .
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				 \begin{document}$$\psi ( t,\,{\rm{x } } ) = \sqrt { { n_c}(t,\,{\rm{x } } ) } \,\exp [ - { \rm{i}}\theta ( t,\,{\rm{x}})/\hbar ] , $ $ \end{document } and defining an irrotational  velocity field  by v  /m , the gross - pitaevskii equation can be rewritten as a continuity equation plus an euler equation : 192\documentclass[12pt]{minimal } 
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				 \begin{document}$${\partial \over { \partial t}}{n_c } + \nabla \cdot ( { n_c}{\rm{v } } ) = 0,$$\end{document } 
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				 \begin{document}$$m{\partial \over { \partial t}}{\rm{v } } + \nabla \left({{{m{v ^2 } } \over 2 } + { v_{{\rm{ext}}}}(t,\,{\rm{x } } ) + \kappa { n_c } - { { { \hbar ^2 } } \over { 2m}}{{{\nabla ^2}\sqrt { { n_c } } } \over { \sqrt { { n_c } } } } } \right ) = 0.$$\end{document } these equations are completely equivalent to those of an irrotational and inviscid fluid apart from the existence of the so - called quantum potential 194\documentclass[12pt]{minimal } 
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				 \begin{document}$${v_{{\rm{quantum } } } } = - { \hbar ^2}{\nabla ^2}\sqrt { { n_c}}/(2m\sqrt { { n_c}}),$$\end{document } which has the dimensions of an energy . note that 195\documentclass[12pt]{minimal } 
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				 \begin{document}$${n_c}\,{\nabla _ i}{v_{{\rm{quantum } } } } = { n_c}{\nabla _ i}\left [ { - { { { \hbar ^2 } } \over { 2m}}{{{\nabla ^2}\sqrt { { n_c } } } \over { \sqrt { { n_c } } } } } \right ] = { \nabla _ j}\left [ { - { { { \hbar ^2 } } \over { 4m}}{n_c}\,{\nabla _ i}{\nabla _ j}\ln \,{n_c } } \right],$$\end{document } which justifies the introduction of the so - called quantum stress tensor 196\documentclass[12pt]{minimal } 
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				 \begin{document}$$\sigma _ { ij}^{{\rm{quantum } } } = - { \hbar \over { 4m}}{n_c}{\nabla _ i}{\nabla _
j}\ln \,{n_c}.$$\end{document } this tensor has the dimensions of pressure , and may be viewed as an intrinsically quantum anisotropic pressure contributing to the euler equation .
if we write the mass density of the madelung fluid as  = m nc , and use the fact that the flow is irrotational then the euler equation takes the form 197\documentclass[12pt]{minimal } 
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				 \begin{document}$$\rho \left [ { { \partial \over { \partial t}}{\rm{v } } + ( { \rm{v } } \cdot \nabla){\rm{v } } } \right ] + \rho \,\nabla \left [ { { { { v_{{\rm{ext}}}}(t,\,{\rm{x } } ) } \over m } } \right ] + \nabla \left [ { { { \kappa { \rho ^2 } } \over { 2{m^2 } } } } \right ] + \nabla \cdot { \sigma ^{{\rm{quantum } } } } = 0.$$\end{document } note that the term vext / m has the dimensions of specific enthalpy , while /(2 m ) represents a bulk pressure .
when the gradients in the density of the condensate are small one can neglect the quantum stress term leading to the standard hydrodynamic approximation . because the flow is irrotational ,
the euler equation is often more conveniently written in hamilton - jacobi form : 198\documentclass[12pt]{minimal } 
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				 \begin{document}$$m{\partial \over { \partial t}}\theta + \left({{{{{[\nabla \theta ] } ^2 } } \over { 2 m } } + { v_{{\rm{ext}}}}(t,\,{\rm{x ) + } } \kappa { n_c } - { { { \hbar ^2 } } \over { 2m}}{{\nabla \sqrt { { n_c } } } \over { \sqrt { { n_c } } } } } \right ) = 0.$$\end{document } apart from the wave function of the condensate itself , we also have to account for the ( typically small ) quantum perturbations of the system ( 187 ) .
these quantum perturbations can be described in several different ways , here we are interested in the  quantum acoustic representation 
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				 \begin{document}$$\hat \varphi ( t,\,{\rm{x ) } } = { e^{- i\theta/\hbar}}\left({{1 \over { 2\sqrt { { n_c}}}}{{\hat n}_1 } - { \rm{i}}{{\sqrt { { n_c } } } \over \hbar}{{\hat \theta}_1 } } \right),$$\end{document } where \documentclass[12pt]{minimal } 
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				 \begin{document}${{\hat n}_1}$\end{document } , \documentclass[12pt]{minimal } 
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				 \begin{document}${\hat\theta _ 1}$\end{document } are real quantum fields . by using this representation equation ( 187 ) can be rewritten as 200\documentclass[12pt]{minimal } 
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				 \begin{document}$${\partial _
t}{\hat n_1 } + { 1 \over m}\nabla \cdot \left({{n_1}\,\nabla \theta + { n_c}\,\nabla { { \hat \theta}_1 } } \right ) = 0,$$\end{document } 
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				 \begin{document}$${\partial _ t}\hat{{\theta _ 1 } } + { 1 \over m}\nabla \theta \cdot \nabla \hat{{\theta _ 1 } } + \kappa ( a)\;{n_1 } - { { { \hbar ^2 } } \over { 2m}}\;{d_2}{\hat{n}_1 } = 0.$$\end{document } here d2 represents a second - order differential operator obtained from linearizing the quantum potential . explicitly : 202\documentclass[12pt]{minimal } 
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				 \begin{document}$${d_2}{\hat n_1 } \equiv - { 1 \over 2}n_c^{- 3/2}[{\nabla ^2}(n_c^{+ 1/2})]\;{\hat n_1 } + { 1 \over 2}n_c^{- 1/2}{\nabla ^2}(n_c^{- 1/2}{\hat n_1}).$$\end{document } the equations we have just written can be obtained easily by linearizing the gross - pitaevskii equation around a classical solution : \documentclass[12pt]{minimal } 
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				 \begin{document}${n_c } \to { n_c } + { { \hat n}_1}$\end{document } , \documentclass[12pt]{minimal } 
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				 \begin{document}$\phi \to \phi + { \hat\phi _ 1}$\end{document}. it is important to realise that in those equations the back - reaction of the quantum fluctuations on the background solution has been assumed negligible .
we also see in equations ( 200 , 201 ) , that time variations of vext and time variations of the scattering length a appear to act in very different ways . whereas the external potential only influences the background equation ( 198 ) ( and hence the acoustic metric in the analogue description ) , the scattering length directly influences both the perturbation and background equations . from the previous equations for the linearised perturbations it is possible to derive a wave equation for \documentclass[12pt]{minimal } 
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				 \begin{document}${\hat\theta _ 1}$\end{document } ( or alternatively , for \documentclass[12pt]{minimal } 
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				 \begin{document}${{\hat n}_1}$\end{document } ) .
all we need is to substitute in equation ( 200 ) the \documentclass[12pt]{minimal } 
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				 \begin{document}${{\hat n}_1}$\end{document } obtained from equation ( 201 ) .
this leads to a pde that is second - order in time derivatives but infinite order in space derivatives  to simplify things we can construct the symmetric 4  4 matrix 203\documentclass[12pt]{minimal } 
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				 \begin{document}$${f^{\mu \nu}}(t,{\bf{x } } ) \equiv \left
[ { \begin{array}{*{20}c } { { f^{00 } } } & \vdots & { { f^{0j}}}\\ { \cdots \cdots } & \cdot & { \cdots \cdots \cdots \cdots}\\ { { f^{i0 } } } & \vdots & { { f^{ij}}}\\ \end{array } } \right].$$\end{document } ( greek indices run from 03 , while roman indices run from 13 . ) then , introducing ( 3 + 1)-dimensional space - time coordinates 204\documentclass[12pt]{minimal } 
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				 \begin{document}$${x^\mu } \equiv ( t;{x^i})$$\end{document } the wave equation for 1 is easily rewritten as 205\documentclass[12pt]{minimal } 
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				 \begin{document}$${\partial _
\nu}{\hat\theta _ 1 } ) = 0{.}$$\end{document } where the f are differential operators acting on space only : 206\documentclass[12pt]{minimal } 
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				 \begin{document}$${f^{00 } } = - { \left [ { \kappa ( a ) - { { { \hbar ^2 } } \over { 2m}}{d_2 } } \right]^{- 1}}$$\end{document } 
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				 \begin{document}$${f^{0j } } = - { \left [ { \kappa ( a ) - { { { \hbar ^2 } } \over { 2m}}{d_2 } } \right]^{- 1}}{{{\nabla ^j}{\theta _ 0 } } \over m}$$\end{document } 
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				 \begin{document}$${f^{i0 } } = - { { { \nabla ^i}{\theta _ 0 } } \over m}{\left [ { \kappa ( a ) - { { { \hbar ^2 } } \over { 2m}}{d_2 } } \right]^{- 1}}$$\end{document } 
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				 \begin{document}$${f^{ij } } = { { { n_c}{\delta ^{ij } } } \over m } - { { { \nabla ^i}{\theta _ 0 } } \over m}{\left [ { \kappa ( a ) - { { { \hbar ^2 } } \over { 2m}}{d_2 } } \right]^{- 1}}{{{\nabla ^j}{\theta _ 0 } } \over m}.$$\end{document } now , if we make a spectral decomposition of the field \documentclass[12pt]{minimal } 
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				 \begin{document}${\hat\theta _ 1}$\end{document } we can see that for wavelengths larger than  / mcs ( this corresponds to the  healing length 
, as we will explain below ) , the terms coming from the linearization of the quantum potential ( the d2 ) can be neglected in the previous expressions , in which case the f can be approximated by numbers , instead of differential operators .
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				 \begin{document}$$\sqrt { - g } \;{g^{\mu \nu } } = { f^{\mu \nu}},$$\end{document } the equation for the field \documentclass[12pt]{minimal } 
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				 \begin{document}${\hat\theta _ 1}$\end{document } becomes that of a ( massless minimally coupled ) quantum scalar field over a curved background 211\documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta { \theta _ 1 } \equiv { 1 \over { \sqrt { - g}}}{\partial _ \mu}(\sqrt { - g } { g^{\mu \nu}}{\partial _
\nu})\;{\hat\theta _ 1 } = 0,$$\end{document } with an effective metric of the form 212\documentclass[12pt]{minimal } 
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				 \begin{document}$${g_{\mu \nu}}(t,{\bf{x } } ) \equiv { { { n_c } } \over { m\;{c_{\rm{s}}}(a,{n_c})}}\left [ { \begin{array}{*{20}c } { - \{{c_{\rm{s}}}{{(a,{n_c})}^2 } - { v^2}\ } } & \vdots & { - { v_j } } \\
{ \cdots \cdots \cdots \cdots } & \vdots & { \cdots \cdots } \\ { - { v_i } } & \vdots & { { \delta _ { ij } } } \\ \end{array } } \right].$$\end{document } here the magnitude cs(nc , a ) represents the speed of the phonons in the medium : 213\documentclass[12pt]{minimal } 
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				 \begin{document}$${c_{\rm{s}}}{(a,{n_c})^2 } = { { \kappa ( a){n_c } } \over m}.$$\end{document } with this effective metric now in hand , the analogy is fully established , and one is now in a position to start asking more specific physics questions . it is interesting to consider the case in which the above  hydrodynamical  approximation for becs does not hold . in order to explore a regime where the contribution of the quantum potential can not be neglected we can use the so called eikonal approximation , a high - momentum approximation where the phase fluctuation \documentclass[12pt]{minimal } 
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				 \begin{document}${\hat\theta _ 1}$\end{document } is itself treated as a slowly - varying amplitude times a rapidly varying phase .
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				 \begin{document}${{\hat n}_1}$\end{document } and \documentclass[12pt]{minimal } 
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				 \begin{document}${\hat\theta _ 1}$\end{document } fluctuations .
in fact , if one discards the unphysical possibility that the respective phases differ by a time varying quantity , any time - constant difference can be safely reabsorbed in the definition of the ( complex ) amplitudes .
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				 \begin{document}$$\hat{{\theta _ 1}}(t,{\bf{x } } ) = { \rm re } \{{{\mathcal a}_\theta}\;\exp ( - i\phi)\},$$\end{document } 
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				 \begin{document}$$\hat{{n_1}}(t,{\bf{x } } ) = { \rm{re}}\{{{\mathcal a}_\rho}\;\exp ( - i\phi)\}.$$\end{document } as a consequence of our starting assumptions , gradients of the amplitude , and gradients of the background fields , are systematically ignored relative to gradients of . ( warning : what we are doing here is not quite a  standard  eikonal approximation , in the sense that it is not applied directly on the fluctuations of the field (t , x ) but separately on their amplitudes and phases 1 and 1 . )
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				 \begin{document}$$\omega = { { \partial \phi } \over { \partial t}};\quad { k_i } = { \nabla _ i}\phi.$$\end{document } then the operator d2 can be approximated as 217\documentclass[12pt]{minimal } 
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				 \begin{document}$${d_2}{\hat n_1 } \equiv - { 1 \over 2}n_c^{- 3/2}[\delta ( n_c^{+ 1/2})]\;{\hat n_1 } + { 1 \over 2}n_c^{- 1/2}\delta ( n_c^{- 1/2}{\hat n_1})$$\end{document } 
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				 \begin{document}$$\approx + { 1 \over 2}n_c^{- 1}[\delta \;{\hat n_1}]$$\end{document } 
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				 \begin{document}$$= - { 1 \over 2}n_c^{- 1}{k^2}\;{\hat n_1}.$$\end{document } a similar result holds for d2 acting on \documentclass[12pt]{minimal } 
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				 \begin{document}${\hat\theta _ 1}$\end{document}. that is , under the eikonal approximation
we effectively replace the operator d2 by the function 
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				 \begin{document}$${d_2 } \rightarrow - { 1 \over 2}n_c^{- 1}{k^2}.$$\end{document } for the matrix f this effectively results in the replacement 221\documentclass[12pt]{minimal } 
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				 \begin{document}$${f^{00 } } \rightarrow - { \left [ { \kappa ( a ) + { { { \hbar ^2}\;{k^2 } } \over { 4m\;{n_c } } } } \right]^{- 1}}$$\end{document } 
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				 \begin{document}$${f^{0j } } \rightarrow - { \left [ { \kappa ( a ) + { { { \hbar ^2}\;{k^2 } } \over { 4m\;{n_c } } } } \right]^{- 1}}{{{\nabla ^j}{\theta _ 0 } } \over m}$$\end{document } 
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				 \begin{document}$${f^{i0 } } \rightarrow - { { { \nabla ^i}{\theta _ 0 } } \over m}{\left [ { \kappa ( a ) + { { { \hbar ^2}\;{k^2 } } \over { 4m\;{n_c } } } } \right]^{- 1}}$$\end{document } 
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				 \begin{document}$${f^{ij } } \rightarrow { { { n_c}\;{\delta ^{ij } } } \over m } - { { { \nabla ^i}{\theta _ 0 } } \over m}{\left [ { \kappa ( a ) + { { { \hbar ^2}\;{k^2 } } \over { 4m\;{n_c } } } } \right]^{- 1}}{{{\nabla ^j}{\theta _ 0 } } \over m}.$$\end{document } ( as desired , this has the net effect of making f a matrix of numbers , not operators . ) the physical wave equation ( 205 ) now becomes a nonlinear dispersion relation 225\documentclass[12pt]{minimal } 
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				 \begin{document}$${f^{00}}{\omega ^2 } + ( { f^{0i } } + { f^{i0}})\omega \;{k_i } + { f^{ij}}\;{k_i}\;{k_j } = 0.$$\end{document } after substituting the approximate d2 into this dispersion relation and rearranging , we see ( remember : k = k =  
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				 \begin{document}$$- { \omega ^2 } + 2\;v_0^i\;\omega { k_i } + { { { n_c}{k^2 } } \over m}\left [ { \kappa ( a ) + { { { \hbar ^2 } } \over { 4m{n_c}}}\;{k^2 } } \right ] - { ( v_0^i\;{k_i})^2 } = 0.$$\end{document } 
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				 \begin{document}$${(\omega - v_0^i\;{k_i})^2 } = { { { n_c}{k^2 } } \over m}\left [ { \kappa ( a ) + { { { \hbar ^2 } } \over { 4m{n_c}}}\;{k^2 } } \right].$$\end{document } introducing the speed of sound cs this takes the form : 228\documentclass[12pt]{minimal } 
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				 \begin{document}$$\omega = v_0^i\;{k_i } \pm \sqrt { c_{\rm{s}}^2{k^2 } + { { \left({{\hbar \over { 2m}}{k^2 } } \right)}^2}}.$$\end{document } at this stage some observations are in order : 
 it is interesting to recognise that the dispersion relation ( 228 ) is exactly in agreement with that found in 1947 by bogoliubov   ( reprinted in ; see also ) for the collective excitations of a homogeneous bose gas in the limit t  0 ( almost complete condensation ) . in his derivation bogoliubov applied a diagonalization procedure for the hamiltonian describing the system of bosons.it is easy to see that ( 228 ) actually interpolates between two different regimes depending on the value of the wavelength  = 2/k with respect to the  acoustic compton wavelength 
( remember that cs is the speed of sound ; this is not a standard particle physics compton wavelength . ) in particular , if we assume v0 = 0 ( no background velocity ) , then for large wavelengths
  c one gets a standard phonon dispersion relation   ck. for wavelengths   c the quasi - particle energy tends to the kinetic energy of an individual gas particle and in fact   k/(2m).we would also like to highlight that in relative terms , the approximation by which one neglects the quartic terms in the dispersion relation gets worse as one moves closer to a horizon where v0 = cs . the non - dimensional parameter that provides this information
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				 \begin{document}$$\delta \equiv { { \sqrt { 1 + { { \lambda _ c^2 } \over { 4{\lambda ^2 } } } } - 1 } \over { ( 1 - { v_0}/{c_{\rm{s } } } ) } } \simeq { 1 \over { ( 1 - { v_0}/{c_{\rm{s}}})}}{{\lambda _ c^2 } \over { 8{\lambda ^2}}}.$$\end{document } as we will discuss in section 5.1.3 , this is the reason why sonic horizons in a bec can exhibit different features from those in standard general relativity.the dispersion relation ( 228 ) exhibits a contribution due to the background flow \documentclass[12pt]{minimal } 
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				 \begin{document}$v _ 0^i{k_i}$\end{document } , plus a quartic dispersion at high momenta .
the group velocity is 230\documentclass[12pt]{minimal } 
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				 \begin{document}$$v_g^i = { { \partial \omega } \over { \partial { k_i } } } = v_0^i \pm { { \left({{c^2 } + { { { \hbar ^2 } } \over { 2{m^2}}}{k^2 } } \right ) } \over { \sqrt { { c^2}{k^2 } + { { \left({{\hbar \over { 2m}}{k^2 } } \right)}^2}}}}{k^i}.$$\end{document } dispersion relations of this type ( but in most cases with the sign of the quartic term reversed ) have been used by corley and jacobson in analysing the issue of trans - planckian modes in the hawking radiation from general relativistic black holes [ 185 , 186 , 88 ] .
the existence of modified dispersion relations ( mdr ) , that is , dispersion relations that break lorentz invariance , can be taken as a manifestation of new physics showing up at high energies / short wavelengths . in their analysis
( unruh s analysis of this problem used a slightly different toy model in which the dispersion relation saturated at high momentum . ) in our case , however , the group velocity grows without bound allowing high - momentum modes to escape from behind the horizon .
thus the acoustic horizon is not absolute in these models , but is instead frequency dependent , a phenomenon that is common once non - trivial dispersions are included.indeed , with hindsight the fact that the group velocity goes to infinity for large k was preordained : after all , we started from the generalised nonlinear schrdinger equation , and we know what its characteristic curves are . like the diffusion equation the characteristic curves of the schrdinger equation ( linear or nonlinear ) move at infinite speed . if we then approximate this generalised nonlinear schrdinger equation in any manner , for instance by linearization , we can not change the characteristic curves : for any well behaved approximation technique , at high frequency and momentum we should recover the characteristic curves of the system we started with .
however , what we certainly do see in this analysis is a suitably large region of momentum space for which the concept of the effective metric both makes sense , and leads to finite propagation speed for medium - frequency oscillations.this type of superluminal dispersion relation has also been analysed by corley and jacobson .
they found that this escape of modes from behind the horizon often leads to self - amplified instabilities in systems possessing both an inner horizon as well as an outer horizon , possibly causing them to disappear in an explosion of phonons .
this is also in partial agreement with the stability analysis performed by garay et al .
let us however leave further discussion regarding these developments to the section 5.1.3 on horizon stability .
it is interesting to recognise that the dispersion relation ( 228 ) is exactly in agreement with that found in 1947 by bogoliubov   ( reprinted in ; see also ) for the collective excitations of a homogeneous bose gas in the limit t  0 ( almost complete condensation ) . in his derivation bogoliubov
it is easy to see that ( 228 ) actually interpolates between two different regimes depending on the value of the wavelength  = 2/k with respect to the  acoustic compton wavelength 
( remember that cs is the speed of sound ; this is not a standard particle physics compton wavelength . ) in particular , if we assume v0 = 0 ( no background velocity ) , then for large wavelengths
  c one gets a standard phonon dispersion relation   ck. for wavelengths   c the quasi - particle energy tends to the kinetic energy of an individual gas particle and in fact   k/(2 m ) .
we would also like to highlight that in relative terms , the approximation by which one neglects the quartic terms in the dispersion relation gets worse as one moves closer to a horizon where v0 = cs .
the non - dimensional parameter that provides this information is defined by 229\documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \equiv { { \sqrt { 1 + { { \lambda _ c^2 } \over { 4{\lambda ^2 } } } } - 1 } \over { ( 1 - { v_0}/{c_{\rm{s } } } ) } } \simeq { 1 \over { ( 1 - { v_0}/{c_{\rm{s}}})}}{{\lambda _ c^2 } \over { 8{\lambda ^2}}}.$$\end{document } as we will discuss in section 5.1.3 , this is the reason why sonic horizons in a bec can exhibit different features from those in standard general relativity . the dispersion relation ( 228 ) exhibits a contribution due to the background flow \documentclass[12pt]{minimal } 
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				 \begin{document}$v _ 0^i{k_i}$\end{document } , plus a quartic dispersion at high momenta . the group velocity is 230\documentclass[12pt]{minimal } 
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				 \begin{document}$$v_g^i = { { \partial \omega } \over { \partial { k_i } } } = v_0^i \pm { { \left({{c^2 } + { { { \hbar ^2 } } \over { 2{m^2}}}{k^2 } } \right ) } \over { \sqrt { { c^2}{k^2 } + { { \left({{\hbar \over { 2m}}{k^2 } } \right)}^2}}}}{k^i}.$$\end{document } dispersion relations of this type ( but in most cases with the sign of the quartic term reversed ) have been used by corley and jacobson in analysing the issue of trans - planckian modes in the hawking radiation from general relativistic black holes [ 185 , 186 , 88 ] .
the existence of modified dispersion relations ( mdr ) , that is , dispersion relations that break lorentz invariance , can be taken as a manifestation of new physics showing up at high energies / short wavelengths . in their analysis
( unruh s analysis of this problem used a slightly different toy model in which the dispersion relation saturated at high momentum . ) in our case , however , the group velocity grows without bound allowing high - momentum modes to escape from behind the horizon .
thus the acoustic horizon is not absolute in these models , but is instead frequency dependent , a phenomenon that is common once non - trivial dispersions are included .
indeed , with hindsight the fact that the group velocity goes to infinity for large k was preordained : after all , we started from the generalised nonlinear schrdinger equation , and we know what its characteristic curves are . like the diffusion equation the characteristic curves of the schrdinger equation ( linear or nonlinear ) move at infinite speed . if we then approximate this generalised nonlinear schrdinger equation in any manner , for instance by linearization , we can not change the characteristic curves
: for any well behaved approximation technique , at high frequency and momentum we should recover the characteristic curves of the system we started with .
however , what we certainly do see in this analysis is a suitably large region of momentum space for which the concept of the effective metric both makes sense , and leads to finite propagation speed for medium - frequency oscillations .
they found that this escape of modes from behind the horizon often leads to self - amplified instabilities in systems possessing both an inner horizon as well as an outer horizon , possibly causing them to disappear in an explosion of phonons .
this is also in partial agreement with the stability analysis performed by garay et al .
let us however leave further discussion regarding these developments to the section 5.1.3 on horizon stability .
its structural richness confers on helium a paradigmatic character regarding the emergence of many and varied macroscopic properties from the microscopic world ( see   and references therein ) . here
, we are interested in the emergence of effective geometries in helium , and their potential use in testing aspects of semiclassical gravity .
helium four , a bosonic system , becomes superfluid at low temperatures ( 2.17 k at vapour pressure ) .
this superfluid behaviour is associated with the condensation in the vacuum state of a macroscopically large number of atoms .
a superfluid is automatically an irrotational and inviscid fluid , so in particular one can apply to it the ideas worked out in section 2 .
the propagation of classical acoustic waves ( scalar waves ) over a background fluid flow can be described in terms of an effective lorentzian geometry : the acoustic geometry . however , in this system one can naturally go considerably further , into the quantum domain . for long wavelengths ,
one can separate the classical behaviour of a background flow ( the effective geometry ) from the behaviour of the quantum phonons over this background . in this way one can reproduce , in laboratory settings , different aspects of quantum field theory over curved backgrounds .
the speed of sound in the superfluid phase is typically of the order of cm / sec .
therefore , at least in principle , it should not be too difficult to establish configurations with supersonic flows and their associated ergoregions .
helium three , the fermionic isotope of helium , in contrast becomes superfluid at very much lower temperatures ( below 2.5 milli - k ) .
the reason behind this rather different behaviour is the pairing of fermions to form effective bosons ( cooper pairing ) , which are then able to condense . in the so - called he  a phase , the structure of the fermionic vacuum is such that it possesses two fermi points , instead of the more typical fermi surface . in an equilibrium configuration
one can choose the two fermi points to be located at { px = 0 , pz = pf } ( in this way , the z - axis signals the direction of the angular momentum of the pairs ) .
close to either fermi point the spectrum of quasiparticles becomes equivalent to that of weyl fermions . from the point of view of the laboratory ,
there is a speed for the propagation of quasiparticles along the z - axis , c  cm / sec , and a different speed , c  10 
 c , for propagation perpendicular to the symmetry axis .
real  , but can be made to disappear by an appropriate rescaling of the coordinates .
therefore , in the equilibrium case , we are reproducing the behaviour of weyl fermions over minkowski spacetime .
these collective excitations will be experienced by the weyl quasiparticles as the introduction of an effective electromagnetic field and a curved lorentzian geometry .
apart from the standard way to provide a curved geometry based on producing non - trivial flows , there is also the possibility of creating topologically non - trivial configurations with a built - in non - trivial geometry .
for example , it is possible to create a domain - wall configuration [ 200 , 199 ] ( the wall contains the z - axis ) such that the transverse velocity c acquires a profile in the perpendicular direction ( say along the x - axis ) with c passing through zero at the wall ( see figure 8) .
this particular arrangement could be used to reproduce a black hole - white hole configuration only if the soliton is set up to move with a certain velocity along the x - axis .
this configuration has the advantage than it is dynamically stable , for topological reasons , even when some supersonic regions are created . 

a third way in which superfluid helium can be used to create analogues of gravitational configurations is the study of surface waves ( or ripplons ) on the interface between two different phases of he [ 415 , 417 ] . in particular ,
if we have a thin layer of he  a in contact with another thin layer of he  b , the oscillations of the contact surface  see  an effective metric of the form [ 415 , 417 ] 231\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } = { 1 \over { \left({1 - { \alpha _ a}{\alpha _ b}{u^2 } } \right)}}\left [ { - \left({1 - { w^2 } - { \alpha _ a}{\alpha _ b}{u^2 } } \right)\;\;{\rm{d}}{t^2 } - 2{\bf{w } } \cdot { \bf{dx}}\;{\rm{d}}t + { \bf{dx } } \cdot { \bf{dx } } } \right],$$\end{document } where 232\documentclass[12pt]{minimal } 
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				 \begin{document}$${\bf{w } } \equiv { \alpha _ a}{{\bf{v}}_{\bf{a } } } + { \alpha _ b}{{\bf{v}}_{\bf{b}}};\quad \quad { \bf{u } } \equiv { { \bf{v}}_{\bf{a } } } - { { \bf{v}}_{\bf{b}}};$$\end{document } and 233\documentclass[12pt]{minimal } 
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				 \begin{document}$${\alpha _ a } \equiv { { { h_b}{\rho _ a } } \over { { h_a}{\rho _ b } + { h_b}{\rho _ a}}};\quad \quad { \alpha _ b } \equiv { { { h_a}{\rho _ b } } \over { { h_a}{\rho _
b } + { h_b}{\rho _ a}}}.$$\end{document } ( all of this provided that we are looking at wavelengths larger than the layer thickness , k ha  1 and k hb  1 . ) the advantage of using surface waves instead of bulk waves in superfluids is that one could create horizons without reaching supersonic speeds in the bulk fluid .
this could alleviate the appearance of dynamical instabilities in the system , that in this case are controlled by the strength of the interaction of the ripplons with bulk degrees of freedom [ 415 , 417 ] .
the geometrical interpretation of the motion of light in dielectric media leads naturally to conjecture that the use of flowing dielectrics might be useful for simulating general relativity metrics with ergoregions and black holes .
unfortunately , these types of geometry require flow speeds comparable to the group velocity of the light . since typical refractive indexes in non - dispersive media are quite close to unity , it is then clear that it is practically impossible to use them to simulate such general relativistic phenomena .
in particular the achievement of controlled slowdown of light , down to velocities of a few meters per second ( or even down to complete rest ) [ 383 , 204 , 52 , 211 , 309 , 374 , 348 ] , has opened a whole new set of possibilities regarding the simulation of curved - space metrics via flowing dielectrics . 

the key effect used to achieve this takes the name of electromagnetically induced transparency ( eit ) .
a laser beam is coupled to the excited levels of some atom and used to strongly modify its optical properties . in particular
one generally chooses an atom with two long - lived metastable ( or stable ) states , plus a higher energy state that has some decay channels into these two lower states .
the coupling of the excited states induced by the laser light can affect the transition from a lower energy state to the higher one , and hence the capability of the atom to absorb light with the required transition energy .
the system can then be driven into a state where the transitions between each of the lower energy states and the higher energy state exactly cancel out , due to quantum interference , at some specific resonant frequency . in this way
eit is characterised by a transparency window , centred around the resonance frequency , where the medium is both almost transparent and extremely dispersive ( strong dependence on frequency of the refractive index ) .
this in turn implies that the group velocity of any light probe would be characterised by very low real group velocities ( with almost vanishing imaginary part ) in proximity to the resonant frequency .
one can start by considering a medium in which an eit window is opened via some control laser beam which is oriented perpendicular to the direction of the flow .
one then illuminates this medium , now along the flow direction , with some probe light ( which is hence perpendicular to the control beam ) .
this probe beam is usually chosen to be weak with respect to the control beam , so that it does not modify the optical properties of the medium . in the case
in which the optical properties of the medium do not vary significantly over several wavelengths of the probe light , one can neglect the polarization and can hence describe the propagation of the latter with a simple scalar dispersion relation [ 235 , 124 ] 234\documentclass[12pt]{minimal } 
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				 \begin{document}$${k^2 } - { { { \omega ^2 } } \over { { c^2}}}[1 + \chi ( \omega)],$$\end{document } where x is the susceptibility of the medium , related to the refractive index n via the simple relation \documentclass[12pt]{minimal } 
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				 \begin{document}$n = \sqrt { 1 + { \rm{\chi}}}$\end{document}. it is easy to see that in this case the group and phase velocities differ 235\documentclass[12pt]{minimal } 
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				 \begin{document}$${v_{\rm{g } } } = { { \partial \omega } \over { \partial k } } = { c \over { \sqrt { 1 + \chi } + { \omega \over { 2n}}{{\partial \chi } \over { \partial \omega}}}};\quad \quad { v_{{\rm{ph } } } } = { \omega \over k } = { c \over { \sqrt { 1 + \chi}}}.$$\end{document } so even for small refractive indexes one can get very low group velocities , due to the large dispersion in the transparency window , and in spite of the fact that the phase velocity remains very near to c. ( the phase velocity is exactly c at the resonance frequency 0 ) . in an ideal eit regime
the probe light experiences a vanishing susceptibility  near the the critical frequency 0 , this allows us to express the susceptibility near the critical frequency via the expansion 236\documentclass[12pt]{minimal } 
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				 \begin{document}$$\chi ( \omega ) = { { 2\alpha } \over { { \omega _ 0}}}(\omega - { \omega _ 0 } ) + o\left [ { { { ( \omega - { \omega _ 0})}^3 } } \right],$$\end{document } where  is sometimes called the  group refractive index  .
the parameter  depends on the dipole moments for the transition from the metastable states to the high energy one , and most importantly depends on the ratio between the probe - light energy per photon , 0 , and the control - light energy per atom .
this might appear paradoxical because it seems to suggest that for a dimmer control light the probe light would be further slowed down .
however this is just an artificial feature due to the extension of the eit regime beyond its range of applicability . in particular in order to be effective the eit requires the control beam energy to dominate all processes and hence it can not be dimmed at will . at resonance we have 237\documentclass[12pt]{minimal } 
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				 \begin{document}$${v_{\rm{g } } } = { { \partial \omega } \over { \partial k } } \rightarrow { c \over { 1 + \alpha } } \approx { c \over \alpha};\quad \quad { v_{{\rm{ph } } } } = { \omega \over k } \rightarrow c.$$\end{document } we can now generalise the above discussion to the case in which our highly dispersive medium flows with a characteristic velocity profile u(x , t ) . in order to find the dispersion relation of the probe light in this case we just need to transform the dispersion relation ( 234 ) from the comoving frame of the medium to the laboratory frame .
let us consider for simplicity a monochromatic probe light ( more realistically a pulse with a very narrow range of frequencies  near 0 ) .
the motion of the dielectric medium creates a local doppler shift of the frequency 238\documentclass[12pt]{minimal } 
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				 \begin{document}$$\omega \rightarrow \gamma ( { \omega _ 0 } - { \bf{u } } \cdot { \bf{k}}),$$\end{document } where  is the usual relativistic factor .
given that k  /c is a lorentz invariant , it is then easy to see that this doppler detuning affects the dispersion relation ( 234 ) only via the susceptibility dependent term
. given further that in any realistic case one would deal with non - relativistic fluid velocities u
 c we can then perform an expansion of the dispersion relation up to second order in u / c . expressing the susceptibility via ( 236 ) we can then rewrite the dispersion relation in the form   239\documentclass[12pt]{minimal } 
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				 \begin{document}$${g^{\mu \nu}}{k_\mu}{k_\nu } = 0,$$\end{document } where 240\documentclass[12pt]{minimal } 
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				 \begin{document}$${k_\nu } = \left({{{{\omega _ 0 } } \over c } , - { \bf{k } } } \right),$$\end{document } and ( most of the relevant articles adopt the signature ( +    ) , as we also do for this particular section ) 241 the inverse of this tensor will be the covariant effective metric experienced by the probe light , whose rays would then be null geodesics of the line element ds = gdxdx . in this sense
explicitly one finds the covariant metric to be 242 where 243\documentclass[12pt]{minimal } 
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				 \begin{document}$$a = { { 1 - 4\alpha { u^2}/{c^2 } } \over { 1 + ( { \alpha ^2 } - 3\alpha){u^2}/{c^2 } - 4{\alpha ^2}{u^4}/{c^4}}};$$\end{document } 
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				 \begin{document}$$b = { 1 \over { 1 + ( { \alpha ^2 } - 3\alpha){u^2}/{c^2 } - 4{\alpha ^2}{u^4}/{c^4}}};$$\end{document } 
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				 \begin{document}$$c = { { 1 - ( 4/\alpha + 4{u^2}/{c^2 } ) } \over { 1 + ( { \alpha ^2 } - 3\alpha){u^2}/{c^2 } - 4{\alpha ^2}{u^4}/{c^4}}}.$$\end{document } several comments are in order concerning the metric ( 242 ) . first of all it is clear that although more complicated than an acoustic metric it will be still possible to cast it into the arnowitt - deser - misner - like form   246 where the effective speed ueff is proportional to the fluid flow speed u and the three space effective metric geff is ( rather differently from the acoustic case ) non - trivial . in any case , the existence of this adm form already tells us that an ergoregion will always appear once the norm of the effective velocity exceeds the effective speed of light ( which for slow light is approximately c/ where  can be extremely large due to the huge dispersion in the transparency window around the resonance frequency 0 ) .
however a trapped surface ( and hence an optical black hole ) will form only if the inward normal component of the effective flow velocity exceeds the group velocity of light . in the slow light setup
so far considered such a velocity turns out to be \documentclass[12pt]{minimal } 
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				 \begin{document}$u = c/(2\sqrt \alpha)$\end{document}. the realization that ergoregions and event horizons can be simulated via slow light may lead one to the ( erroneous ) conclusion that this is an optimal system for simulating particle creation by gravitational fields .
however , as pointed out by unruh in [ 284 , 379 ] , such a conclusion would turn out to be over - enthusiastic . in order to obtain particle creation an inescapable requirement
is to have so - called  mode mixing  , that is , mixing between the positive and negative frequency modes of the incoming and outgoing states .
this is tantamount to saying that there must be regions where the frequency of the quanta as seen by a stationary observer at infinity ( laboratory frame ) becomes negative beyond the ergosphere at g00 = 0 . in a flowing medium
this can in principle occur thanks to the tilting of the dispersion relation due to the doppler effect caused by the velocity of the flow equation ( 238 ) , but this also tells us that the condition 0  u 
k < 0 can be satisfied only if the velocity of the medium exceeds 0/k which is the phase velocity of the probe light , not its group velocity .
since the phase velocity in the slow light setup we are considering is very close to c , the physical speed of light in vacuum , not very much hope is left for realizing analogue particle creation in this particular laboratory setting .
however it was also noticed by unruh and schtzhold   that a different setup for slow light might deal with this and other issues ( see   for a detailed summary ) . in the setup suggested by these authors
the field describing the beat fluctuations of this electromagnetic background can be shown to satisfy , once the dielectric medium is in motion , the same wave equation as that on a curved background . in this particular situation
the phase velocity and the group velocity are approximately the same , and both can be made small , so that the previously discussed obstruction to mode mixing is removed .
so in this new setup it is concretely possible to simulate classical particle creation such as , e.g. , super - radiance in the presence of ergoregions .
nonetheless the same authors showed that this does not open the possibility for a simulation of quantum particle production ( e.g. , hawking radiation ) .
this is because that effect also requires the commutation relations of the field to generate the appropriate zero - point energy fluctuations ( the vacuum structure ) according to the heisenberg uncertainty principle .
this is not the case for the effective field describing the beat fluctuations of the system we have just described , which is equivalent to saying that it does not have a proper vacuum state ( i.e. , analogue to any physical field ) . hence one has to conclude that any simulation of quantum particle production is precluded .
we feel that the catalogue we have just presented is reasonably complete and covers the key items . for additional background on any of these topics
, we would suggest sources such as the books  artificial black holes    and  the universe in a helium droplet  . for more specific detail , check this review s bibliography , and use spires to check for recent developments .
of course , the entire motivation for looking at analogue models is to be able to learn more physics .
one could start studying analogue models with the idea of seeing whether it is possible ( either theoretically or in practice ) to reproduce in the laboratory various gravitational phenomena whose real existence in nature can not be currently checked .
these are phenomena that surpass our present ( and foreseeable ) observational capabilities , but yet , we believe in their existence because it follows from seemingly strong theoretical arguments within the standard frameworks of general relativity and field theory in curved space . however , the interest of this approach is not merely to reproduce these gravitational phenomena in some formal analogue model , but to see which departures from the ideal case show up in real analogue models , and to analyse whether similar deviations are likely to appear in real gravitational systems .
when one thinks about emergent gravitational features in condensed matter systems , one immediately realises that these features only appear in the low - energy regime of the analogue systems . when the systems are probed at high energies ( short length scales ) the effective geometrical description of the analogue models break down , as one starts to be aware that the systems are actually composed of discrete pieces ( atoms and molecules ) .
this scenario is quite similar to what one expects to happen with our geometrical description of the universe , when explored with microscopic detail at the planck scale .
that is , the study of analogue models of general relativity is giving us insights as to how the standard theoretical picture of different gravitational phenomena could change when taking into account additional physical knowledge coming from the existence of an underlying microphysical structure .
quite robustly , these studies are telling us already that the first deviations from the standard general relativity picture can be encoded in the form of high - energy violations of lorentz invariance in particle dispersion relations . beyond these first deviations ,
the analogue models of general relativity provide well understood examples ( the underlying physics is well known ) in which a description in terms of fields in curved spacetimes shows up as a low - energy - regime emergent phenomena .
the analogue models are being used to shed light on these general questions through a number of specific routes .
let us now turn to discussing several specific physics issues that are being analysed from this perspective .
as is well known , in 1974 stephen hawking announced that quantum mechanically even a spherical distribution of matter collapsing to form a black hole should emit radiation ; with a spectrum approximately that of a black body [ 159 , 160 ] .
a black hole will tend to evaporate by emitting particles from its horizon toward infinity .
the existence of radiation emission is a kinematic effect that does not rely on einstein equations .
therefore , one can aim to reproduce it in a condensed matter system . within standard field theory ,
a minimal requirement for having hawking radiation is the existence in the background configuration of an apparent horizon .
so , in principle , to be able to reproduce hawking radiation in a laboratory one would have to fulfil at least two requirements : 
 to choose an adequate analogue system ; it has to be a quantum analogue model ( see section 4 ) such that its description could be separated into a classical effective background spacetime plus some standard relativistic quantum fields living on it ( it can happen that the quantum fields do not satisfy the appropriate commutation or anti - commutation relations ) .to configure the analogue geometry such that it includes an apparent horizon . that is , within an appropriate quantum analogue model , the formation of an apparent horizon for the propagation of the quantum fields should excite the fields as to result in the emission of a thermal distribution of field particles.22 . to choose an adequate analogue system ; it has to be a quantum analogue model ( see section 4 ) such that its description could be separated into a classical effective background spacetime plus some standard relativistic quantum fields living on it ( it can happen that the quantum fields do not satisfy the appropriate commutation or anti - commutation relations ) . to configure the analogue geometry such that it includes an apparent horizon . that is , within an appropriate quantum analogue model , the formation of an apparent horizon for the propagation of the quantum fields should excite the fields as to result in the emission of a thermal distribution of field particles.22 .
this is a straight and quite naive translation of the standard hawking effect derivation to the condensed matter realm .
however , in reality , this translation process has to take into account additional issues that , as we are trying to convey , instead of problems , are where the interesting physics lies . 
 the effective description of the quantum analogue systems as fields over a background geometry breaks down when probed at sufficiently short length scales .
in fact , immediately after the inception of the idea that black holes radiate , it was realised that there was a potential problem with the calculation .
it strongly relies on the validity of quantum field theory on curved backgrounds up to arbitrary high energies . following a wave packet with a certain frequency at future infinity backwards in time
, we can see that it had to contain arbitrarily large frequency components with respect to a local free fall observer ( well beyond the planck scale ) when it was close to the horizon . in principle
any unknown physics at the planck scale could strongly influence the hawking process so that one should view it with suspicion .
this is the so - called trans - planckian problem of hawking radiation . to create an analogue model exhibiting hawking radiation
will be , therefore , equivalent to giving a solution to the trans - planckian problem.in order to clearly observe hawking radiation , one should first be sure that there is no other source of instabilities in the system that could mask the effect . in analogue models such as liquid helium or becs the interaction of a radial flow
( with speed of the order of the critical landau speed , which in these cases coincides with the sound speed ) with the surface of the container ( an electromagnetic potential in the becs case ) might cause the production of rotons and quantised vortices , respectively .
thus , in order to produce an analogue model of hawking radiation , one has to be somewhat ingenious .
for example in the liquid helium case , instead of taking acoustic waves in a supersonic flow stream as the analogue model , it is preferable to use as analogue model ripplons in the interface between two different phases , a and b phases , of helium-3 .
then , the topological stability of the configuration prevents its destruction when creating a horizon [ 199 , 200 ] . in the case of becs
a way to suppress the formation of quantised vortices is to take effectively one - dimensional configurations .
if the transverse dimension of the flow is smaller than the healing length then there is no space for the existence of a vortex . in either liquid helium or becs ,
there is also the possibility of creating an apparent horizon by rapidly approaching a critical velocity profile ( see figure 10 ) , but without actually crossing into the supersonic regime , softening in this way the appearance of dynamical instabilities.real analogue models can not , strictly speaking , reproduce eternal black - hole configurations .
an analogue model of a black hole has always to be created at some finite laboratory time .
therefore , one is forced to carefully analyse the creation process , as it can greatly influence the hawking effect .
depending on the procedure of creation , one could end up in quite different quantum states for the field and only some of them might exhibit hawking radiation . this becomes more important when considering that the analogue models incorporate modified dispersion relations .
an inappropriate preparation , together with modified dispersion relation effects , could completely eliminate hawking radiation .another important issue is the need to characterise  how quantum  a specific analogue model is . even though , strictly speaking , one could say that any system undergoes quantum fluctuations , the point is how important they are in its description . in trying to build an analogue model of hawking s quantum effect , the relative value of hawking temperature with respect to the environment is going to tell us whether the system can be really thought as a quantum analogue model or as effectively classical .
for example , in our standard cosmological scenario , for a black hole to radiate at temperatures higher than that of the cosmic microwave background ,  3 k , the black hole should have a diameter of the order of micrometers or less .
we would have to say that such black holes are no longer classical , but semiclassical .
the black holes for which we have some observational evidence are of much higher mass and size , so their behaviour can be thought of as completely classical .
this is telling us that regarding the hawking process , becs can be considered to be highly - quantum analogue models.there is also the very real question of whether one should trust semiclassical calculations at all when it comes to dealing with back - reaction in the hawking effect . see for instance the arguments presented by helfer ( [ 167 , 168 , 169 ] , and references therein ) . 

figure 10rippions in the interface between two sliding superfluids . 
  the effective description of the quantum analogue systems as fields over a background geometry breaks down when probed at sufficiently short length scales .
in fact , immediately after the inception of the idea that black holes radiate , it was realised that there was a potential problem with the calculation .
it strongly relies on the validity of quantum field theory on curved backgrounds up to arbitrary high energies .
following a wave packet with a certain frequency at future infinity backwards in time , we can see that it had to contain arbitrarily large frequency components with respect to a local free fall observer ( well beyond the planck scale ) when it was close to the horizon . in principle any unknown physics at the planck scale could strongly influence the hawking process so that one should view it with suspicion .
this is the so - called trans - planckian problem of hawking radiation . to create an analogue model exhibiting hawking radiation
will be , therefore , equivalent to giving a solution to the trans - planckian problem . in order to clearly observe hawking radiation
, one should first be sure that there is no other source of instabilities in the system that could mask the effect . in analogue models such as liquid helium or becs the interaction of a radial flow
( with speed of the order of the critical landau speed , which in these cases coincides with the sound speed ) with the surface of the container ( an electromagnetic potential in the becs case ) might cause the production of rotons and quantised vortices , respectively .
thus , in order to produce an analogue model of hawking radiation , one has to be somewhat ingenious .
for example in the liquid helium case , instead of taking acoustic waves in a supersonic flow stream as the analogue model , it is preferable to use as analogue model ripplons in the interface between two different phases , a and b phases , of helium-3 .
then , the topological stability of the configuration prevents its destruction when creating a horizon [ 199 , 200 ] . in the case of becs
a way to suppress the formation of quantised vortices is to take effectively one - dimensional configurations .
if the transverse dimension of the flow is smaller than the healing length then there is no space for the existence of a vortex . in either liquid helium or becs ,
there is also the possibility of creating an apparent horizon by rapidly approaching a critical velocity profile ( see figure 10 ) , but without actually crossing into the supersonic regime , softening in this way the appearance of dynamical instabilities .
real analogue models can not , strictly speaking , reproduce eternal black - hole configurations .
an analogue model of a black hole has always to be created at some finite laboratory time .
therefore , one is forced to carefully analyse the creation process , as it can greatly influence the hawking effect .
depending on the procedure of creation , one could end up in quite different quantum states for the field and only some of them might exhibit hawking radiation . this becomes more important when considering that the analogue models incorporate modified dispersion relations .
an inappropriate preparation , together with modified dispersion relation effects , could completely eliminate hawking radiation .
another important issue is the need to characterise  how quantum  a specific analogue model is .
even though , strictly speaking , one could say that any system undergoes quantum fluctuations , the point is how important they are in its description . in trying to build an analogue model of hawking s quantum effect , the relative value of hawking temperature with respect to the environment is going to tell us whether the system can be really thought as a quantum analogue model or as effectively classical . for example , in our standard cosmological scenario , for a black hole to radiate at temperatures higher than that of the cosmic microwave background ,  3 k , the black hole should have a diameter of the order of micrometers or less . we would have to say that such black holes are no longer classical , but semiclassical .
the black holes for which we have some observational evidence are of much higher mass and size , so their behaviour can be thought of as completely classical .
this is telling us that regarding the hawking process , becs can be considered to be highly - quantum analogue models .
there is also the very real question of whether one should trust semiclassical calculations at all when it comes to dealing with back - reaction in the hawking effect .
see for instance the arguments presented by helfer ( [ 167 , 168 , 169 ] , and references therein ) .
because of its importance , let us now review what we know about the effects of high - energy dispersion relations on the hawking process .
we saw in the introduction to this section that the trans - planckian problem of hawking radiation was one of the strongest motivations for the modern research into analogue models of gravity .
in fact it was soon realised that such models could provide a physical framework within which a viable solution of the problem could be found .
as we have said , the requirement of a reservoir of ultra - high frequency modes nearby the horizon seems to indicate a possible ( and worrisome ) sensitivity of the black hole radiation to the microscopic structure of spacetime . actually by assuming exact lorentz invariance one could in principle
always locally transform the problematic ultra high frequency modes to low energy ones via some appropriate lorentz transformation .
however in doing so it would have to rely on the physics of reference frames moving ultra fast with respect to us , as the reference frame needed would move arbitrarily close to the speed of light .
hence we would have to apply lorentz invariance in a regime of arbitrary large boosts yet untested and in principle never completely testable given the non - compactness of the boost subgroup .
the assumption of an exact boost symmetry is linked to a scale - free nature of spacetime given that unbounded boosts expose ultrashort distances .
hence the assumption of exact lorentz invariance needs , in the end , to rely on some idea on the nature of spacetime at ultra - short distances .
it was this type of reasoning that led in the nineties to a careful reconsideration of the crucial ingredients required for the derivation of hawking radiation [ 185 , 186 , 377 ] .
in particular investigators explored the possibility that spacetime microphysics could provide a short distance , lorentz - breaking cutoff , but at the same time leave hawking s results unaffected at energy scales well below that set by the cutoff .
of course ideas about a possible cutoff imposed by the discreteness of spacetime at the planck scale had already been discussed in the literature well before unruh s seminal paper .
however such ideas were running into serious difficulties given that a naive short distance cutoff posed on the available modes of a free field theory results in a complete removal of the evaporation process ( see e.g. , jacobson s article   and references therein , and the comments in [ 167 , 168 , 169 ] ) .
indeed there are alternative ways through which the effect of the short scales physics could be taken into account , and analogue models provide a physical framework where these ideas could be put to the test .
in fact analogue models provide explicit examples of emergent spacetime symmetries , they can be used to simulate black hole backgrounds , they may be endowed with quantizable perturbations and , in most of the cases , they have a well known microscopic structure .
given that hawking radiation can be , at least in principle , simulated in such systems one might ask how and if the trans - planckian problem is resolved in these cases .
the general feature that most of the work on this subject has focussed on is that in analogue models the quasi - particles propagating on the effective geometry are actually collective excitations of atoms .
this generically implies that their dispersion relation will be a relativistic one only at low energies ( large scales),23 and in each case there will be some short length scale ( e.g. , intermolecular distance for a fluid , coherence length for a superfluid , healing length for a bec ) beyond which deviations will be non - negligible
. in general such microphysics induced corrections to the dispersion relation take the form 247\documentclass[12pt]{minimal } 
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				 \begin{document}$${e^2 } = { c^2}\left({{m^2}{c^2 } + { k^2 } + \delta ( k , k ) } \right)$$\end{document } where k is the scale that describes the transition to the full microscopic system ( what we might call the  analogue planck scale  ) . in general
the best one can do is to expand (k , k ) around k = 0 , obtaining an infinite power series ( of which it will be safe to retain only the lowest order terms ) , although in some special models ( like bec ) the series is automatically finite due to intrinsic properties of the system .
( in any case one can see that most of the analogue models so far considered lead to modifications of the form k / k or k / k . ) depending on the sign in front of the modification the group velocity at high energy can be larger ( + ) or smaller (  ) than the low energy speed of light c. these cases are usually referred in the literature as  superluminal  and  subluminal  dispersion relations .
most of the work on the trans - planckian problem in the nineties focussed on studying the effect on hawking radiation due to such modifications of the dispersion relations at high energies in the case of acoustic analogues [ 185 , 186 , 377 , 378 , 88 ] , and the question of whether such phenomenology could be applied to the case of real black holes ( see e.g. , [ 50 , 188 , 88 , 299]).24 in all the aforementioned works hawking radiation can be recovered under some suitable assumptions , as long as neither the black hole temperature nor the frequency at which the spectrum is considered are too close to the scale of microphysics k. however , the applicability of these assumptions to the real case of black hole evaporation is an open question . also , in the case of the analogue models the mechanism by which the hawking radiation is recovered is not always the same .
we concisely summarise here the main results ( but see e.g. ,   for further details ) .
this was the case originally considered by unruh [ 377 , 378 ] , 248\documentclass[12pt]{minimal } 
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				 \begin{document}$$\omega = k{\left({\tanh { { ( k / k)}^n } } \right)^{1/n}},$$\end{document } and by corley and jacobson   249\documentclass[12pt]{minimal } 
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				 \begin{document}$${\omega ^2 } = { k^2 } - { k^4}/{k^2},$$\end{document } where both dispersion relations are given in the co - moving frame .
the key feature is that in the presence of a subluminal modification the group velocity of the modes increases with k only up to some turning point ( which is equivalent to saying that the group velocity does not asymptote to c , which could be the speed of sound , but instead is upper bounded ) . for values of k beyond the turning point
the group velocity decreases to zero ( for ( 248 ) ) or becomes imaginary ( for ( 249 ) ) . in the latter case
this can be interpreted as signifying the breakdown of the regime where the dispersion relation ( 249 ) can be trusted .
the picture that emerged from these analyses concerning the origin of the outgoing hawking modes at infinity is quite surprising .
in fact , if one traces back in time an outgoing mode , as it approaches the horizon it decreases its group velocity below the speed of sound . at some point before reaching the horizon , the outgoing mode will appear as an ingoing one dragged into the black hole by the flow . stepping further back in time
it is seen that such a mode was located at larger and larger distances from the horizon , and tends to a very high energy mode far away at early times . in this case one
, where the origin of the outgoing hawking quanta seems to originate from ingoing modes which have  bounced off  the horizon without reaching trans - planckian frequencies in its vicinities .
several detailed analytical and numerical calculations have shown that such a conversion indeed happens [ 378 , 50 , 88 , 87 , 170 , 330 , 380 ] and that the hawking result can be recovered for k  k where k is the black hole surface gravity .
the case of a superluminal dispersion relation is quite different and , as we have seen , has some experimental interest given that this is the kind of dispersion relation that arises in some promising analogue models ( e.g. , becs ) . in this situation , the outgoing modes are actually seen as originating from behind the horizon
this implies that these modes somehow originate from the singularity ( which can be a region of high turbulence in acoustic black hole analogues ) , and hence it would seem that not much can be said in this case .
however it is possible to show that if one imposes vacuum boundary conditions on these modes near the singularity , then it is still possible to recover the hawking result , i.e. , thermal radiation outside the hole .
it is particularly interesting to note that this recovering of the standard result is not always guaranteed in the presence of superluminal dispersion relations .
corley and jacobson   in fact discovered a very peculiar type of instability due to such superluminal dispersion in the presence of black holes with inner horizons .
the net result of the investigation carried out in   is that the compact ergo - region characterizing such configurations is unstable to self - amplifying hawking radiation .
the presence of such an instability seems to be confirmed by the analysis carried on in [ 136 , 137 ] where a bose - einstein condensate analogue black hole was considered .
is it possible to reduce the rather complex phenomenology just described to a few basic assumptions that must be satisfied in order to recover hawking radiation in the presence of lorentz violating dispersion relations ?
a tentative answer is given in , where the robustness of the hawking result is considered for general modified ( subluminal as well as superluminal ) dispersion relations .
the authors of   assume that the geometrical optics approximation breaks down only in the proximity of the event horizon ( which is equivalent to saying that the particle production happens only in such a region ) . here
, the would - be trans - planckian modes are converted in sub - planckian ones .
then , they try to identify the minimal set of assumptions that guarantees that such  converted modes 
are generated in their ground states ( with respect to a freely falling observer ) , as this is a well known condition in order to recover hawking s result .
they end up identifying three basic assumptions that guarantee such emergence of modes in the ground state at the horizon .
first , the preferred frame selected by the breakdown of lorentz invariance must be the freely falling one instead of the rest frame of the static observer at infinity ( which coincides in this limit with the laboratory observer ) .
second , the planckian excitations must start off in the ground state with respect to freely falling observers .
finally , they must evolve in an adiabatic way ( i.e. , the planck dynamics must be much faster than the external sub - planckian dynamics ) . of course
, although several systems can be found in which such conditions hold , it is also possible to show   that realistic situations in which at least one of these assumptions is violated can be imagined .
it is hence still an open question whether real black hole physics does indeed satisfy such conditions , and whether it is hence robust against modifications induced by the violation of lorentz invariance . in spite of the remarkable insight given by the models discussed above ( based on modified dispersion relations ) it is not possible to consider them fully satisfactory in addressing the trans - planckian problem .
in particular it was soon recognised [ 89 , 189 ] that in this framework it is not possible to explain the origin of the short wavelength incoming modes which are  progenitors  of the outgoing modes after bouncing off in the proximity of the horizon .
for example , in the unruh model ( 248 ) , one can see that if one keeps tracking a  progenitor  incoming mode back in time , then its group velocity ( in the co - moving frame ) drops to zero as its frequency becomes more and more blue shifted ( up to arbitrarily large values ) , just the situation one was trying to avoid .
this is tantamount to saying that the trans - planckian problem has been moved from the region near the horizon out to the region near infinity . in the corley - jacobson model ( 249 )
this unphysical behaviour is removed thanks to the presence of the physical cutoff k. however it is still true that in tracking the incoming modes back in time one finally sees a wave packet so blue shifted that k = k. at this point one can no longer trust the dispersion relation ( 249 ) ( which anyway in realistic analogue models is emergent and not fundamental ) , and hence the model has no predictive power regarding the ultimate origin of the relevant incoming modes .
these conclusions regarding the impossibility of clearly predicting the origin at early times of the modes ultimately to be converted into hawking radiation are not specific to the particular dispersion relations ( 248 ) or ( 249 ) one is using .
the killing frequency is in fact conserved on a static background , thus the incoming modes must have the same frequency as the outgoing ones , hence there can be no mode - mixing and particle creation .
this is why one has actually to assume that the wkb approximation fails in the proximity of the horizon and that the modes are there in the vacuum state for the co - moving observer . in this sense
the need for these assumptions can be interpreted as evidence that these models are not fully capable of solving the trans - planckian problem .
it was to overcome this type of issue that alternative ways of introducing an ultra - violet cutoff due to the microphysics were considered [ 318 , 319 , 89 ] .
in particular in   the transparency of the refractive medium at high frequencies has been used to introduce an effective cutoff for the modes involved in hawking radiation in a classical refractive index analogue model ( see section 4.1.3 of this review ) . in this model an event horizon for the electromagnetic field modes can be simulated by a surface of singular electric and magnetic permeabilities .
this would be enough to recover hawking radiation but it would imply the unphysical assumption of a refractive index which is valid at any frequency
. however it was shown in   that the hawking result can be recovered even in the case of a dispersive medium which becomes transparent above some fixed frequency k ( which we can imagine as the plasma frequency of the medium ) , the only ( crucial ) assumption being again that the  trans - planckian  modes with k > k are in their ground state near the horizon .
there a lattice description of the background was used for imposing a cutoff in a more physical way with respect to the continuum dispersive models previously considered . in such a discretised spacetime the field takes values only at the lattice points , and wavevectors are identified modulo 2/ where  is the lattice characteristic spacing , correspondingly one obtains a sinusoidal dispersion relation for the propagating modes . hence the problem of recovering a smooth evolution of incoming modes to outgoing ones is resolved by the intrinsically regularised behaviour of the wave vectors field . in   the authors explicitly considered the hawking process for a discretised version of a scalar field , where the lattice is associated to the free - fall coordinate system ( taken as the preferred system ) .
furthermore the requirement of a fixed short distance cutoff leads to the choice of a lattice spacing constant at infinity , and that the lattice points are at rest at infinity and fall freely into the black hole.25 in this case the lattice spacing grows in time and the lattice points spread in space as they fall toward the horizon .
however this time dependence of the lattice points is found to be of order 1/ , and hence unnoticeable to long wavelength modes and relevant only for those with wavelengths of the order of the lattice spacing .
the net result is that on such a lattice long wavelength outgoing modes are seen to originate from short wavelength incoming modes via a process analogous to the bloch oscillations of accelerated electrons in crystals [ 89 , 189 ] .
although closely related , as we will soon see , we have to distinguish carefully between the mode analysis of a linear field theory ( with or without modified dispersion relations  mdr ) over a fixed background and the stability analysis of the background itself .
let us consider a three - dimensional irrotational and inviscid fluid system with a stationary sink - type of flow ( see figures 1 , 2 ) .
the details of the configuration are not important for the following discussion , only the fact that there is a spherically symmetric fluid flow accelerating towards a central sink , that sink being surrounded by a sphere acting as a sonic horizon .
then , as we have discussed in section 2 , linearizing the euler and continuity equations leads to a massless scalar field theory over a black - hole - like spacetime .
( we are assuming that the hydrodynamic regime remains valid up to arbitrarily short length scales ; for instance , we are neglecting the existence of mdr ) . to be specific ,
let us choose the geometry of the canonical acoustic black - hole spacetime described in : 250\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } = - { c^2}\left({1 - { { r_0 ^ 4 } \over { { r^4 } } } } \right)\;{\rm{d}}{\tau ^2 } + { \left({1 - { { r_0 ^ 4 } \over { { r^4 } } } } \right)^{- 1}}\;{\rm{d}}{r^2 } + { r^2}\left({{\rm{d}}{\theta ^2 } + { { \sin}^2}\theta \;{\rm{d}}{\varphi ^2 } } \right).$$\end{document } in this expression we have used the schwarzschild time coordinate  instead of the lab time t ; c is constant .
if we expand the field in spherical harmonics , 251\documentclass[12pt]{minimal } 
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				 \begin{document}$${\phi _ { lm}}(\tau , r , \theta , \varphi ) \equiv { e^{- i\omega \tau}}{{{\chi _ { lm}}(r ) } \over r}{y_{lm}}(\theta , \varphi),$$\end{document } we obtain the following equation for the radial part of the field : 252\documentclass[12pt]{minimal } 
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				 \begin{document}$${{{\omega ^2 } } \over { { c^2}}}\chi = \left({- { { { { \rm{d}}^2 } } \over { { \rm{d}}{r^{\ast 2 } } } } + { v_l}(r ) } \right)\chi;$$\end{document } where 253\documentclass[12pt]{minimal } 
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				 \begin{document}$${v_l}(r ) = \left({1 - { { r_0 ^ 4 } \over { { r^4 } } } } \right)\left [ { { { l(l + 1 ) } \over { { r^2 } } } + { { 4r_0 ^ 4 } \over { { r^6 } } } } \right].$$\end{document } here 254\documentclass[12pt]{minimal } 
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				 \begin{document}$${r^{\ast } } \equiv r - ( { r_0}/4)\{\ln ( r + { r_0})/(r - { r_0 } ) + 2\arctan \;r/{r_0}\}$$\end{document } is a  tortoise  coordinate . in a normal mode analysis one requires boundary conditions such that the field is regular everywhere , even at infinity .
however , if one is analysing the solutions of the linear field theory as a way of probing the stability of the background configuration , one can consider less restrictive boundary conditions .
for instance , one can consider the typical boundary conditions that lead to quasinormal modes : these modes have to be purely out - going at infinity and purely in - going at the horizon ; but one does not require , for example , the modes to be normalizable . the quasinormal modes associated to this sink configuration
the results found are qualitatively similar to those in the classical linear stability analysis of the schwarzschild black hole in general relativity [ 384 , 385 , 317 , 431 , 267 ] .
of course , the gravitational field in general relativity has two dynamical degrees of freedom  those associated with gravitational waves  that have to be analysed separately ; these are the  axial  and  polar  perturbations .
in contrast , in the present situation we only have scalar perturbations . nevertheless , the potentials associated with 
axial  and  polar  perturbations of schwarzschild spacetime , and that associated with scalar perturbations of the canonical acoustic black hole , produce qualitatively the same behaviour : there is a series of damped quasinormal modes  proving the linear stability of the system  with higher and higher damping rates .
an important point we have to highlight in here is that although in the linear regime the dynamical behaviour of the acoustic system is similar to general relativity , this is no longer true once one enters the non - linear regime .
the differences are so profound , that in the general case of acoustic geometries constructed from compressible fluids , there exist sets of perturbations that , independently of how small they are initially , can lead to the development of shocks , a situation completely absent in vacuum general relativity .
now , given an approximately stationary , and at the very least metastable , classical black - holelike configuration , a standard quantum mode analysis leads to the existence of hawking radiation in the form of phonon emission .
this shows , among other things , that quantum corrections to the classical behaviour of the system must make the configuration with a sonic horizon dynamically unstable against hawking emission .
moreover , in an analogue system with quantum fluctuations that maintain strict adherence to the equivalence principle ( no mdr ) it must then be impossible to create an isolated truly stationary horizon by external means  any truly stationary horizon must be provided with an external power source to stabilise it against hawking emission .
that is , in an analogue system one could in principle , by manipulating external forces , compensate for the backreaction effects that in a physical general relativity scenario cause the horizon to shrink ( or evaporate ) and thus become non - stationary .
a wonderful physical system that has mdr explicitly incorporated in its description is the bose - einstein condensate .
the macroscopic wave function of the bec behaves as a classical irrotational fluid but with some deviations when short length scales become involved .
( for length scales of the order of or shorter than the healing length . )
what are the effects of the mdr on the dynamical stability of a black - hole - like configuration in a bec ?
the stability of a sink configuration in a bec has been analysed in [ 136 , 137 ] but taking the flow to be effectively one - dimensional .
what they found is that these configurations are dynamically unstable : there are modes satisfying the appropriate boundary conditions such that the imaginary parts of their associated frequencies are positive .
the dynamical tendency of the system to evolve is suggestively similar to that in the standard evaporation process of a black hole in semiclassical general relativity .
this observation alone could make us question whether the first signatures of a quantum theory underlying general relativity might show up in precisely this manner .
interest in this question is reinforced by a specific analysis in the  loop quantum gravity 
approach to quantizing gravity that points towards the existence of fundamental mdr at high energies .
the formulation of effective gravitational theories that incorporate some sort of mdr at high energies is currently under investigation ( see for example [ 247 , 2 ] ) ; these exciting developments are however beyond the scope of this review . before continuing with the discussion on the stability of configurations with horizons , and in order not to cause confusion between the different wording used when talking about the physics of becs and the emergent gravitational notions on them ,
let us write down a quite loose but useful translation dictionary : 
 the  classical  or macroscopic wave function of the bec represents the classical spacetime of gr , but only when probed at long enough wavelengths.the  classical  long - wavelength perturbations to a background solution of the gross - pitaevskii equation correspond to classical gravitational waves in gr
. of course this analogy does not imply that these are spin 2 waves , it only points out that the perturbations are made from the same  substance  as the background configuration itself.the macroscopic wave function of the bec , without the restriction of being probed only at long wavelengths , corresponds to some sort of semiclassical vacuum gravity .
its  classical  behaviour ( in the sense that does not involve any probability notion ) is already taking into account its underlying quantum origin.the bogoliubov quantum quasiparticles over the 
classical  wave function correspond to a further step away from semiclassical gravity in that they are analogous to the existence of quantum gravitons over a ( semiclassical ) background spacetime .
the  classical  or macroscopic wave function of the bec represents the classical spacetime of gr , but only when probed at long enough wavelengths .
classical  long - wavelength perturbations to a background solution of the gross - pitaevskii equation correspond to classical gravitational waves in gr . of course
this analogy does not imply that these are spin 2 waves , it only points out that the perturbations are made from the same  substance  as the background configuration itself .
the macroscopic wave function of the bec , without the restriction of being probed only at long wavelengths , corresponds to some sort of semiclassical vacuum gravity .
its  classical  behaviour ( in the sense that does not involve any probability notion ) is already taking into account its underlying quantum origin .
classical  wave function correspond to a further step away from semiclassical gravity in that they are analogous to the existence of quantum gravitons over a ( semiclassical ) background spacetime . at this point
we would like to remark , once again , that the analysis based on the evolution of a bec has to be used with care .
for example , they can not directly serve to shed light on what happens in the final stages of the evaporation of a black hole , as the bec does not fulfil , at any regime , the einstein equations . now continuing the discussion , what happens when treating the perturbations to the background bec flow as quantum excitations ( bogoliubov quasiparticles ) ? what we certainly know
is that the analysis of modes in a collapsing - to - form - a - black - hole background spacetime leads to the existence of radiation emission very much like hawking emission [ 50 , 87 , 91 ] .
( this is why it is said that hawking s process is robust against modifications of the physics at high energies . )
the comparison of these calculations with that of hawking ( without mdr ) , tells us that the main modification to hawking s result is that now the hawking flux of particles would not last forever but would vanish after a long enough time ( this is why , in principle , we can dynamically create a configuration with a sonic horizon in a bec ) .
the emission of quantum particles reinforces the idea that the supersonic sink configurations are unstable . summarizing : 
 if the perturbation to the bec background configuration have  classical seeds  ( that is , are describable by the linearised gross - pitaevskii equation alone ) , then , one will have  classical  instabilities.if the perturbations have  quantum seeds  ( that is , are described by the bogoliubov equations ) , then , one will have  quantum  instabilities .
if the perturbation to the bec background configuration have  classical seeds  ( that is , are describable by the linearised gross - pitaevskii equation alone ) , then , one will have  classical  instabilities .
if the perturbations have  quantum seeds  ( that is , are described by the bogoliubov equations ) , then
, one will have  quantum  instabilities . in the light of the acoustic analogies it is natural to ask whether there are other geometric configurations with horizons of interest , besides the sink type of configurations
( these are the most similar to the standard description of black holes in general relativity , but probably not the simplest in terms of realizability in a real laboratory ; for an entire catalogue of them see ) . here
, we are going to specifically mention two effectively one - dimensional black hole - white hole configurations , one in a straight line and one in a ring ( see figures 11 and 12 , respectively ) . 

figure 11one - dimensional velocity profile with a black hole horizon and a white hole horizon . 

figure 12one - dimensional velocity profile in a ring ; the fluid flow exhibits two sonic horizons , one of black hole type and the other of white hole type .
one - dimensional velocity profile with a black hole horizon and a white hole horizon .
one - dimensional velocity profile in a ring ; the fluid flow exhibits two sonic horizons , one of black hole type and the other of white hole type . a quantum mode analysis of the black hole - white hole configuration in a straight line , taking into account the existence of superluminal dispersion relations ( similar to those in a bec ) , led to the conclusion that the emission of particles in this configuration proceeds in a self - amplified ( or runaway ) manner .
we can understand this effect as follows : at the black horizon a virtual pair of phonons are converted into real phonons , the positive energy phonon goes towards infinity while the negative energy pair falls beyond the black horizon .
however , the white horizon makes this negative energy phonon bounce back towards the black horizon ( thanks to superluminal motion ) stimulating the emission of additional phonons .
although related to hawking s process this phenomenon has a quite different nature : for example , there is no temperature associated with it .
a stability analysis of a configuration like this in a bec would lead to strong instabilities .
this same configuration , but compactified into a ring configuration , has been dynamically analysed in [ 136 , 137 ] .
what they found is that there are regions of stability and instability depending on the parameters characterizing the configuration .
we suspect that the stability regions appear because of specific periodic arrangements of the modes around the ring . among other reasons ,
these arrangements are interesting because they could be easier to create in laboratory with current technology and their instabilities easier to detect than hawking radiation itself .
to conclude this subsection , we would like to highlight that there is still much to be learned by studying the different levels of description of an analogue system , and how they influence the stability or instability of configurations with horizons .
in addition , among the many papers using analogue spacetimes as part of their background mindset when addressing these issues we mention : 
  top down  calculations of hawking radiation starting from some idealised model of quantum gravity [ 1 , 201 , 276 , 277].bottom up  calculations of hawking radiation starting from curved space quantum field theory [ 22 , 23 , 61 , 62 , 71 , 111 , 112 , 128 , 253 , 254 , 255 , 256 , 257 , 298 , 321].trans - hawking versions of hawking radiation , either as reformulations of the physics , or as alternative scenarios [ 30 , 68 , 130 , 131 , 150 , 156 , 167 , 168 , 169 , 233 , 289 , 291 , 325 , 329 , 333 , 334 , 340 , 341 , 350 , 364].black hole entropy viewed in the light of analogue spacetimes .hawking radiation interpreted as a statement about particles travelling along complex spacetime trajectories [ 292 , 350 , 360 ] .
 top down  calculations of hawking radiation starting from some idealised model of quantum gravity [ 1 , 201 , 276 , 277 ] . 
bottom up  calculations of hawking radiation starting from curved space quantum field theory [ 22 , 23 , 61 , 62 , 71 , 111 , 112 , 128 , 253 , 254 , 255 , 256 , 257 , 298 , 321 ] .
trans - hawking versions of hawking radiation , either as reformulations of the physics , or as alternative scenarios [ 30 , 68 , 130 , 131 , 150 , 156 , 167 , 168 , 169 , 233 , 289 , 291 , 325 , 329 , 333 , 334 , 340 , 341 , 350 , 364 ] .
black hole entropy viewed in the light of analogue spacetimes . hawking radiation interpreted as a statement about particles travelling along complex spacetime trajectories [ 292 , 350 , 360 ] .
another phenomenon that has been ( and is being ) analysed from the analogue gravity perspective is super - radiance .
the rotational kinetic energy accumulated in a rotating black hole can be extracted from it by scattering into it waves of sufficiently low frequency and high angular momentum . in general , in order that the wave can extract rotational energy from the system it has to satisfy the condition 255\documentclass[12pt]{minimal } 
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				 \begin{document}$$\omega < m\;\omega$$\end{document } where  is the angular speed of the black hole at the event horizon and m is the harmonic azimuthal number of the wave .
when dealing with quantum fields , as opposed to classical fields , this process can proceed spontaneously : quantum mechanically , a rotating black hole will tend to radiate away all of its angular momentum , eventually approaching a non - rotating schwarzschild black hole [ 429 , 430 ] .
( the term super - radiance was already used in condensed matter to describe processes in which there was some coherent emission of radiation from an otherwise disordered system . )
again , these processes have a purely kinematical origin so they are perfectly suitable for being reproduced in an analogue model .
regarding these processes , the simplest geometry that one can reproduce , thinking of analogue models based on fluid flows , is that of the draining bathtub of section 2 . of course , this metric does not exactly correspond to kerr geometry , nor even to a section of it
however , it is qualitatively similar . it can be used to simulate both penrose s classical process and quantum super - radiance as these effects do not depend on the specific multipole decomposition of kerr s geometry , but only on its rotating character .
a specific experimental setup has been put forward by schtzhold and unruh using gravity waves in a shallow basin mimicking an ideal draining bathtub .
equivalently to what happens with kerr black holes , this configuration is classically stable ( in the linear regime ) .
a word of caution is in order here : interactions of the gravity surface waves with bulk waves ( neglected in the analysis ) could cause the system to become unstable .
this instability has no counterpart in standard general relativity ( though it might have one in braneworld theories ) .
super - resonant scattering of waves in this rotating sink configuration , or in a simple purely rotating vortex , could in principle be observed in this and other analogue models .
there are already several articles dealing with this problem [ 25 , 27 , 26 , 64 , 113 , 237 ] .
one would only have to surround the rotating configuration by a mirror for it to become grossly unstable .
what causes the instability is that those in - going waves that are amplified when reflected in the ergosphere would then in turn be reflected back toward the ergoregion , due to the exterior mirror , thus being amplified again , and so on . an interesting phenomenon that appears in many condensed matter systems is the existence of quantised vortices .
the angular momentum of these vortices comes in multiples of some fundamental unit ( typically  or something proportional to  ) .
the extraction of rotational energy by a penrose process in these cases could only proceed via finite - energy transitions .
in such a highly quantum configuration it is also important to look for the effect of having high - energy dispersion relations .
for example , in becs , the radius of the ergoregion of a single quantised vortex is of the order of the healing length , so one can not directly associate an effective lorentzian geometry to this portion of the configuration .
any analysis that neglects the high energy terms is not going to give any sensible result in these cases .
the key items of interest are the friedmann - robertson - walker ( frw ) geometries , more properly called the friedmann - lematre - robertson - walker ( flrw ) geometries .
the simulation of such geometries has been considered in various works such as [ 17 , 18 , 116 , 115 , 114 , 242 , 58 , 59 , 424 ] with a specific view to enhancing our understanding of  cosmological particle production  driven by the expansion of the universe .
the acoustic metric can be written as 256\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } = { \rho \over { { c_{\bf{s}}}}}\left [ { - ( c_{\rm{s}}^2 - { v^2})\;{\rm{d}}{t^2 } - 2{\bf{v } } \cdot { \bf{dx}}\;dt + { \bf{d}}{{\bf{x}}^2 } } \right].$$\end{document } essentially there are two ways to use this metric to reproduce cosmological spacetimes : one is based on physical explosion , the other on rapid variations in the  effective speed of light  .
following [ 18 , 115 , 202 , 73 , 203 ] one can take a homogeneous system (t ) , cs(t ) and a radial profile for the velocity v = (  / b)r , with b a scale factor depending only on t. then , defining a new radial coordinate as rb = r / b the metric can be expressed as 257\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } = { \rho \over { { c_{\bf{s}}}}}\left [ { - c_{\rm{s}}^2{\rm{d}}{t^2 } + { b^2}({\rm{d}}r_b^2 + r_b^2\;{\rm{d}}\omega _ 2 ^ 2 ) } \right].$$\end{document } introducing a hubble - like parameter , 258\documentclass[12pt]{minimal } 
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				 \begin{document}$${h_b}(t ) = { { \dot b(t ) } \over { b(t)}},$$\end{document } the equation of continuity can be written as 259\documentclass[12pt]{minimal } 
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				 \begin{document}$$\dot \rho + 3{h_b}(t)\;\rho = 0;\quad \quad \rightarrow \quad \quad \rho ( t ) = { { { \rho _ 0 } } \over { { b^3}(t)}},$$\end{document } with 0 constant .
finally we arrive at the metric of a flat flrw geometry 260\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } = - { t^2}(t)\;{\rm{d}}{t^2 } + a_s^2(t)\;({\rm{d}}r_b^2 + r_b^2\;{\rm{d}}\omega _ 2 ^ 2),$$\end{document } with 261\documentclass[12pt]{minimal } 
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				 \begin{document}$$t(t ) \equiv \sqrt { \rho \;{c_{\rm{s } } } } ; \quad \quad { a_s}(t ) \equiv \sqrt { { \rho \over { { c_{\rm{s } } } } } } \;b.$$\end{document } the proper friedmann time ,  , is related to the laboratory time , t , by 262\documentclass[12pt]{minimal } 
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				 \begin{document}$$\tau = \int { t(t)\;{\rm{d}}t}.$$\end{document } the other avenue starts from a fluid at rest v = 0 with respect to the laboratory at all times : 263\documentclass[12pt]{minimal } 
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				 \begin{document}$${\rm{d}}{s^2 } = - \rho \;{c_{\bf{s}}}\;{\rm{d}}{t^2 } + { \rho \over { { c_{\rm{s}}}}}{\bf{d}}{{\bf{x}}^2}.$$\end{document } now it is not difficult to imagine a situation in which  remains constant , in a sufficiently large region of space , while the speed of sound decreases with time ( this can be made in becs for example by changing with time the value of the scattering length [ 17 , 18 ] ) .
models considered to date focus on variants of the bec - inspired analogues : 
 fedichev and fischer [ 115 , 114 ] have investigated wkb estimates of the cosmological particle production rate and ( 1 + 1 ) dimensional cosmologies , both in expanding becs.lidsey , and fedichev and fischer   have focussed on the behaviour of cigar - like condensates in grossly asymmetric
traps.barcelo et al . [ 17 , 18 ] have focussed on the central region of the bec and thereby tried ( at least locally ) to mimic flrw behaviour as closely as possible.fischer and schtzhold   propose the use of two - component becs to simulate cosmic inflation.weinfurtner [ 425 , 424 ] has concentrated on the approximate simulation of de sitter spacetimes .
fedichev and fischer [ 115 , 114 ] have investigated wkb estimates of the cosmological particle production rate and ( 1 + 1 ) dimensional cosmologies , both in expanding becs .
lidsey , and fedichev and fischer   have focussed on the behaviour of cigar - like condensates in grossly asymmetric traps .
[ 17 , 18 ] have focussed on the central region of the bec and thereby tried ( at least locally ) to mimic flrw behaviour as closely as possible .
fischer and schtzhold   propose the use of two - component becs to simulate cosmic inflation .
weinfurtner [ 425 , 424 ] has concentrated on the approximate simulation of de sitter spacetimes . in all of these models
the general expectations of the relativity community have been borne out  theory definitely predicts particle production , and the very interesting question is the extent to which the formal predictions are going to be modified when working with real systems experimentally .
we expect that these analogue models provide us with new insights as to how their inherent modified dispersion relations affect cosmological processes such as the generation of a primordial spectrum of perturbations ( see for example [ 42 , 41 , 43 , 44 , 45 , 46 , 47 , 70 , 107 , 158 , 177 , 178 , 207 ] , [ 229 , 230 , 244 , 252 , 249 , 250 , 251 , 274 , 275 , 296 , 349 , 361 , 362 , 363 , 371 ] where analogue - like ideas are applied to cosmological inflation ) .
an interesting side - effect of the original investigation , is that birefringence can now be used to model
( vsl ) geometries [ 28 , 108 ] . since analogue models quite often lead to two or more  excitation cones  , rather than one , it is quite easy to obtain a bimetric or multi - metric model .
if one of these metrics is interpreted as the  gravitational  metric and the other as the  photon  metric , then vsl cosmologies can be given a mathematically well - defined and precise meaning [ 28 , 108 ] . as we have seen in the previous sections , analogue models have in the past been very useful in providing new , condensed matter physics inspired , ideas about how to solve longstanding problems of semiclassical gravity . in closing this section , it is interesting to briefly discuss what perhaps represents , so far , the only attempt to use analogue models in the reverse direction ; that is to import well known concepts of semiclassical gravity into condensed matter frameworks .
this is an experiment dealing with a gas of a few million   rb atoms at a temperature of about 3 nk .
the condensate is rendered unstable by exploiting the possibility of tuning the interaction ( more precisely the scattering length ) between the atoms via a magnetic field .
reversing the sign of the interaction , making it attractive , destabilises the condensate . after a brief waiting time ( generally called tcollapse ) , the condensate implodes and loses a sizeable fraction of its atoms in the form of a  nova burst  .
if left to evolve undisturbed , the number of atoms in the burst stabilises and a remnant condensate is left .
however if the condensate interaction is again made repulsive after some time tevolve , before the condensate has sufficient time to stabilise , then the formation of  jets  of atoms is observed , these jets being characterised by lower kinetic energy and a distinct shape with respect to the burst emission .
interestingly , an elegant explanation of such a phenomenology was proposed in [ 58 , 59 ] , based on the well known semiclassical gravity analysis of particle creation in an expanding universe .
in fact the dynamics of quantum excitations over the collapsing bec was shown to closely mimic that for quantum excitations in a time - reversed ( collapsing instead of expanding ) scenario for cosmological particle creation .
this is not so surprising as the quantum excitations above the bec ground state feel a time - varying background during the collapse , and as a consequence one then expects squeezing of the vacuum state and mode mixing which are characteristic of quantum field theory in variable external fields .
in fact in [ 58 , 59 ] a key role in explaining the observed burst and jets is played by the concepts of  frozen  versus  oscillating  modes  borrowed from cosmology  ( although with a reverse dynamics with respect to the standard ( expanding ) cosmological case ) . in the case of bose novae
the modes which are amplified are those for which the physical frequency is smaller than the collapse rate , while modes with higher frequencies remain basically unaffected and their amplitudes obey a harmonic oscillator equation . as the collapse rate decreases , more and more modes stop growing and start oscillating , which is equivalent to a creation of particles from the quantum vacuum . in the case of a sudden stop of the collapse by a new reversal of the sign of the interaction all of the previously growing modes
are suddenly converted into particles , explaining in this way the generation of jets and their lower energy ( they correspond to modes with lower frequencies with respect to those generating the bursts ) .
although this simple model can not explain all the details of the bose novae phenomenology , we think it is remarkable how far it can go in explaining several observed features by exploiting the language and techniques so familiar to quantum cosmology . in this sense
the analysis presented in [ 58 , 59 ] primarily shows a possible new application of analogue models where they could be used to lend ideas and techniques developed in the context of gravitational physics to the explanation of condensed matter phenomena . for more details on the trans - planckian problem
some of the key papers are the relatively early papers of unruh [ 377 , 378 ] , and jacobson [ 185 , 186 ] . for super - radiance and cosmological issues
there seems to be considerable ongoing interest , and one should carefully check spires for the most recent articles .
the key question one should ask at this stage is :  where can we go from here ? 
apart from continuing with the analysis of the issues described in the previous section , there are additional interesting routes , both theoretical and experimental , worth of future exploration .
in particular , the following topics come to mind : 
 back reaction , equivalence principle , emergent gravity , quantum gravity phenomenology , quantum gravity , experimental analogue gravity . 

some work has already been done dealing with these topics in the context of analogue gravity .
there are important phenomena in gravitational physics whose understanding needs analysis well beyond classical general relativity and field theory on curved background spacetimes . the black hole evaporation process can be considered as paradigmatic among these phenomena . here , we particularise our discussion to this case . since we are currently unable to analyse the entire process of black hole evaporation within a complete quantum theory of gravity , a way of proceeding is to analyse the simpler ( but still extremely difficult ) problem of semiclassical back reaction ( see for example [ 99 , 79 , 34 , 132 , 51 , 257 ] ) . one takes a background black hole spacetime , calculates the expectation value of the quantum energy - momentum tensor of matter fields in the appropriate quantum state ( the unruh vacuum state for a radiating black hole ) , and then takes this expectation value as a source for the perturbed einstein equations .
this calculation gives us information about the tendency of spacetime to evolve under vacuum polarization effects .
a nice feature of analogue models of general relativity is that although the underlying classical equations of motion have nothing to do with einstein equations , the tendency of the analogue geometry to evolve due to quantum effects is formally equivalent ( approximately , of course ) to that in semiclassical general relativity .
therefore , the onset of the back reaction effects ( if not their precise details ) can be simulated within the class of analogue models .
an example of the type of back - reaction calculations one can perform are those in [ 9 , 10 ] .
these authors started from an effectively one - dimensional acoustic analogue model , configured to have an acoustic horizon by using a laval nozzle to control the flow s speed .
they then considered the effect of quantizing the acoustic waves over the background flow . to calculate the appropriate back reaction terms they took advantage of the classical conformal invariance of the ( 1 + 1)-dimensional reduction of the system . in this case , we know explicitly the form of the expectation value of the energy - momentum tensor trace ( via the trace anomaly ) .
the other two independent components of the energy - momentum tensor were approximated by the polyakov stress tensor . in this way
, what they found is that the tendency of a left - moving flow with one horizon is for it to evolve in such a manner as to push the horizon down - stream at the same time that its surface gravity is decreased .
this is a behaviour similar to what is found for near - extremal reissner - nordstrm black holes .
( however , we should not conclude that acoustic black holes are in general closely related to near - extremal reissner - nordstrm black holes , rather than to schwarzschild black holes .
can we expect to learn something new about gravitational physics by analysing the problem of back reaction in different analogue models ? as we have repeatedly commented , the analyses based on analogue models force us to consider the effects of modified high - energy dispersion relations .
for example , in becs , they affect the  classical  behaviour of the background geometry as much as the behaviour of the quantum fields living on the background .
in seeking a semiclassical description for the evolution of the geometry , one would have to compare the effects caused by the modified dispersion relations to those caused by pure semiclassical back reaction ( which incorporates deviations from standard general relativity as well ) .
in other words , one would have to understand the differences between the standard back reaction scheme in general relativity , and that based on equations ( 186 ) and ( 187 ) . to end this subsection
, we would like to comment that one can go beyond the semiclassical back - reaction scheme by using the so - called stochastic semiclassical gravity programme [ 179 , 180 , 181 ] .
this programme aims to pave the way from semiclassical gravity toward a complete quantum - gravitational description of gravitational phenomena .
this stochastic gravity approach not only considers the expectation value of the energy - momentum tensor but also its fluctuations , encoded in the semiclassical einstein - langevin equation . in a very interesting paper ,
parentani showed that the effects of the fluctuations of the metric ( due to the in - going flux of energy at the horizon ) on the out - going radiation led to a description of hawking radiation similar to that obtained with analogue models .
it would be interesting to develop the equivalent formalism for quantum analogue models and to investigate the different emerging approximate regimes .
analogue models are of particular interest in that the analogue spacetimes that emerge often violate , to some extent , the einstein equivalence principle [ 16 , 399 ] . since the einstein equivalence principle ( or more precisely the universality of free fall )
is experimentally tested to the accuracy of about 1 part in 10 , it is important to build this principle into realistic models of gravitation  most likely at a fundamental level .
one way of interpreting the einstein equivalence principle is as a  principle of universality  for the geometrical structure of spacetime .
whatever the spacetime geometrical structure is , if all excitations  see  the same geometry one is well on the way to satisfying the observational and experimental constraints . in a metric theory , this amounts to the demand of mono - metricity : a single universal metric must govern the propagation of all excitations .
if one is dealing with a single degree of freedom then mono - metricity is no great constraint .
but with multiple degrees of freedom , analogue spacetimes generally lead to refringence  that is the occurrence of fresnel equations that often imply multiple propagation speeds for distinct normal modes . to even obtain a bi - metric model ( or more generally , a multi - metric model ) ,
requires an algebraic constraint on the fresnel equation that it completely factorises into a product of quadratics in frequency and wavenumber .
if one further wishes to impose mono - metricity then the fresnel equation must be some integer power of some single quadratic expression , an even stronger algebraic statement [ 16 , 399 ] .
faced with this situation , there are two ways in which the analogue gravity community might proceed : 
 try to find a broad class of analogue models ( either physically based or mathematically idealised ) that naturally lead to mono - metricity .
little work along these lines has yet been done ; at least partially because it is not clear what features such a model should have in order to be 
compelling.accept refringence as a common feature of the analogue models and attempt to use refringence to ones benefit in one or more ways : 
 there are real physical phenomena in non - gravitational settings that definitely do exhibit refringence and sometimes multi - metricity . though situations of this type are not directly relevant to the gravity community , there is significant hope that the mathematical and geometrical tools used by the general relativity community might in these situations shed light on other branches of physics.use the refringence that occurs in many analogue models as a way of  breaking  the einstein equivalence principle , and indeed as a way of  breaking  even more fundamental symmetries and features of standard general relativity , with a view to exploring possible extensions of general relativity . while the analogue models are not themselves primary physics , they can nevertheless be used as a way of providing hints as to how more fundamental physics might work .
try to find a broad class of analogue models ( either physically based or mathematically idealised ) that naturally lead to mono - metricity .
little work along these lines has yet been done ; at least partially because it is not clear what features such a model should have in order to be 
accept refringence as a common feature of the analogue models and attempt to use refringence to ones benefit in one or more ways : 
 there are real physical phenomena in non - gravitational settings that definitely do exhibit refringence and sometimes multi - metricity . though situations of this type are not directly relevant to the gravity community , there is significant hope that the mathematical and geometrical tools used by the general relativity community might in these situations shed light on other branches of physics.use the refringence that occurs in many analogue models as a way of  breaking  the einstein equivalence principle , and indeed as a way of  breaking  even more fundamental symmetries and features of standard general relativity , with a view to exploring possible extensions of general relativity . while the analogue models are not themselves primary physics , they can nevertheless be used as a way of providing hints as to how more fundamental physics might work .
there are real physical phenomena in non - gravitational settings that definitely do exhibit refringence and sometimes multi - metricity .
though situations of this type are not directly relevant to the gravity community , there is significant hope that the mathematical and geometrical tools used by the general relativity community might in these situations shed light on other branches of physics .
use the refringence that occurs in many analogue models as a way of  breaking  the einstein equivalence principle , and indeed as a way of  breaking  even more fundamental symmetries and features of standard general relativity , with a view to exploring possible extensions of general relativity .
while the analogue models are not themselves primary physics , they can nevertheless be used as a way of providing hints as to how more fundamental physics might work .
one of the more fascinating approaches to  quantum gravity  is the suggestion , typically attributed to sakharov [ 332 , 393 ] , that gravity itself may not be  fundamental physics  .
indeed it is now a relatively common opinion , maybe not mainstream but definitely a strong minority opinion , that gravity ( and in particular the whole notion of spacetime and spacetime geometry ) might be no more  fundamental  than is fluid dynamics .
fundamental  is here used in a rather technical sense  fluid mechanics is not fundamental because there is a known underlying microphysics , that of molecular dynamics , of which fluid mechanics is only the low - energy low - momentum limit . indeed the very concepts of density and velocity field , which are so central to the euler and continuity equations , make no sense at the microphysical level and emerge only as one averages over timescales and distance - scales larger than the mean free time and mean free path . in the same way , it is plausible ( even though no specific and compelling model of the relevant microphysics has yet emerged ) that the spacetime manifold and spacetime metric might arise only once one averages over suitable microphysical degrees of freedom .
sakharov had in mind a specific model in which gravity could be viewed as an 
elasticity  of the spacetime medium , and was  induced  via one - loop physics in the matter sector [ 332 , 393 ] . in this way
sakharov had hoped to relate the observed value of newton s constant ( and the cosmological constant ) to the spectrum of particle masses .
more generally the phrase  emergent gravity  is now used to describe the whole class of theories in which the spacetime metric arises as a low - energy approximation , and in which the microphysical degrees of freedom might be radically different .
analogue models , and in particular analogue models based on fluid mechanics or the fluid dynamic approximation to becs , are specific examples of  emergent physics  in which the microphysics is well understood . as such they are useful for providing hints as to how such a procedure might work in a more fundamental theory of quantum gravity . over the last few years
a widespread consensus has emerged that observational tests of quantum gravity are for the foreseeable future likely to be limited to precision tests of dispersion relations and their possible deviations from lorentz invariance .
the key point is that at low energies ( well below the planck energy ) one expects the locally minkowskian structure of the spacetime manifold to guarantee that one sees only special relativistic effects ; general relativistic effects are negligible at short distances .
however as ultra high energies are approached ( although still below planck scale energies ) several quantum gravity models seem to predict that the locally euclidean geometry of the spacetime manifold will break down .
there are several scenarios for the origin of this breakdown ranging from string theory [ 214 , 109 ] to brane worlds   and loop quantum gravity .
common to all such scenarios is that the microscopic structure of spacetime is likely to show up in the form of a violation of lorentz invariance leading to modified dispersion relations for elementary particles .
such dispersion relations are characterised by extra terms ( with respect to the standard relativistic form ) which are generally expected to be suppressed by powers of the planck energy .
remarkably the last years have seen a large wealth of work in testing the effects of such dispersion relations and in particular strong constraints have been cast by making use of high energy astrophysics observations ( see for example [ 3 , 82 , 195 , 194 , 196 , 197 , 261 , 355 ] and references therein ) . several of the analogue models are known to exhibit similar behaviour , with a low - momentum effective lorentz invariance eventually breaking down at high momentum once the microphysics is explored.26 thus some of the analogue models provide controlled theoretical laboratories in which at least some forms of subtle high - momentum breakdown of lorentz invariance can be explored .
as such the analogue models provide us with hints as to what sort of modified dispersion relation might be natural to expect given some general characteristics of the microscopic physics .
hopefully investigation of appropriate analogue models might be able to illuminate possible mechanisms leading to this kind of quantum gravity phenomenology , and so might be able to provide us new ideas about other effects of physical quantum gravity that might be observable at sub - planckian energies . when it comes to dealing with 
fundamental  theories of quantum gravity , the analogue models play an interesting role which is complementary to the more standard approaches .
the search for a quantum theory of gravity is fundamentally a search for an appropriate mathematical structure in which to simultaneously phrase both quantum questions and gravitational questions .
more precisely , one is searching for a mathematical framework in which to develop an abstract quantum theory which then itself encompasses classical einstein gravity ( the general relativity ) , and reduces to it in an appropriate limit [ 65 , 356 , 129 ] .
m - models  ) ,  loop space  ( and the related  spin foams  ) , and  lattice models  ( euclidean or lorentzian ) all
pre - geometry  as a replacement for classical differential geometry ( which is the natural and very successful mathematical language used to describe einstein gravity ) [ 65 , 356 , 129 , 327 , 326 , 40 ] .
the basic idea is that the mooted replacement for differential geometry would be relevant at extremely small distances ( where the quantum aspects of the theory would be expected to dominate ) , while at larger distances ( where the classical aspects dominate ) one would hope to recover both ordinary differential geometry and specifically einstein gravity or possibly some generalization of it .
approaches to quantum gravity differ in detail in that they emphasise different features of the long - distance model , and so obtain rather different short - distance replacements for classical differential geometry . because the relevant mathematics is extremely difficult , and by and large not particularly well understood , it is far from clear which if any of these three standard approaches will be preferable in the long run .
we feel it likely that analogue models can shed new light on this very confusing field by providing a concrete specific situation in which the transition from the short - distance  discrete  or  quantum 
theory to the long - distance  continuum  theory is both well understood and non - controversial .
here we are specifically referring to fluid mechanics , where at short distances the system must be treated using discrete atoms or molecules as the basic building blocks , while at large distances there is a well - defined continuum limit that leads to the euler and continuity equations .
furthermore once one is in the continuum limit , there is a well - defined manner in which a notion of  lorentzian differential geometry  , and in particular a 
lorentzian effective spacetime metric  can be assigned to any particular fluid flow [ 376 , 389 , 284 ] .
indeed , the  analogue gravity programme  is extremely successful in this regard , providing a specific and explicit example of a 
what the  analogue gravity programme  does not seem to do as easily is to provide a natural direct route to the einstein equations of general relativity , but that merely indicates that current analogies have their limits and therefore , one should not take them too literally [ 389 , 284 ] .
fluid mechanics is a guide to the mathematical possibilities , not an end in itself .
the parts of the analogy that do work well are precisely the steps where the standard approaches to quantum gravity have the most difficulty , and so it would seem useful to develop an abstract mathematical theory of the  discrete    continuum 
  differential geometry  chain using this fluid mechanical analogy ( and related analogies ) as inspiration . beyond the various theoretical issues we have discussed above
currently several experimental groups are interested , but to the best of our knowledge no actual experiments are currently underway .
broadly speaking , for any experimental group interested in analogue spacetimes the two key issues to address are : 
 identify a particular analogue model easily amenable to laboratory investigation , and double check the extent to which the model provides a theoretically robust and clean analogue to general relativistic curved spacetime.identify the technical issues involved in actually setting up a laboratory experiment . identify a particular analogue model easily amenable to laboratory investigation , and double check the extent to which the model provides a theoretically robust and clean analogue to general relativistic curved spacetime . identify the technical issues involved in actually setting up a laboratory experiment .
while the consensus in the theoretical community is that bose - einstein condensates are likely to provide the best working model for analogue gravity , it is possible that we might still be surprised by experimental developments . we leave this as an open challenge to the experimental community .
in this review article we have seen the interplay between standard general relativity and various analogies that can be used to capture aspects of its behaviour .
these analogies have ranged from rather general but very physical analogue models based on fluid - acoustics , geometrical optics , and wave optics , to highly specific models based on becs , liquid helium , slow light , etc .
additionally , we have seen several rather abstract mathematical toy models that bring us to such exotic structures and ideas as birefringence , bimetricity , finsler spaces , and sakharov s induced gravity .
the primary reason that these analogies were developed within the general relativity community was to help in the understanding of general relativity by providing very down - to - earth models of otherwise subtle behaviour in general relativity .
secondary reasons include the rather speculative suggestion that there may be more going on than just analogy  it is conceivable ( though perhaps unlikely ) that one or more of these analogue models could suggest a relatively simple and useful way of quantizing gravity that side - steps much of the technical machinery currently employed in such efforts . a tertiary concern ( at least as far as the general relativity community is concerned ) is the use of relativity and differential geometric techniques to improve understanding of various aspects of condensed matter physics .
the authors expect interest in analogue models to continue unabated , and suspect that there are several key but unexpected issues whose resolution would be greatly aided by the analysis of appropriate analogue models though every practicing scientist already knows this , for the sake of any student reading this we mention the following resources : 
 http://www.slac.stanford.edu/spires/  the bibliographic database for keeping track of ( almost all of ) the relevant literature.http://www.arxiv.org  the electronic - preprint ( e - print ) database for accessing the text of ( almost all , post 1992 ) relevant articles.http://www.livingreviews.org/  the living reviews portal . 

those three access points should allow you to keep abreast of what is going on in the field .
http://www.slac.stanford.edu/spires/  the bibliographic database for keeping track of ( almost all of ) the relevant literature .
http://www.arxiv.org  the electronic - preprint ( e - print ) database for accessing the text of ( almost all , post 1992 ) relevant articles .
 analogue models of ( and for ) gravity have a long and distinguished history dating back to the earliest years of general relativity . in this review article 
 we will discuss the history , aims , results , and future prospects for the various analogue models . 
 we start the discussion by presenting a particularly simple example of an analogue model , before exploring the rich history and complex tapestry of models discussed in the literature . 
 the last decade in particular has seen a remarkable and sustained development of analogue gravity ideas , leading to some hundreds of published articles , a workshop , two books , and this review article . 
 future prospects for the analogue gravity programme also look promising , both on the experimental front ( where technology is rapidly advancing ) and on the theoretical front ( where variants of analogue models can be used as a springboard for radical attacks on the problem of quantum gravity ) . 
Introduction
The Simplest Example of an Analogue Model
History and Motivation
A Catalogue of Models
Lessons from Analogue Models
Future Directions
Conclusions