content_image/376009.jpg content_image/375998.jpg content_image/375999.jpg content_image/376000.jpg content_image/376003.jpg content_image/376010.jpg content_image/376001.jpg content_image/376004.jpg content_image/376005.jpg content_image/376006.jpg content_image/376007.jpg content_image/376002.jpg content_image/376008.jpg
## II. LINEARIZED EQUATION OF MOTION 

In this section we undertake the construction of an ap- proximate expression for the force density, Eq. (13), as a linear functional of u . The stress tensor P$_{µν}$ ( r , t ), whose divergence determines the force density, is defined as the expectation value of the stress tensor operator ˆ P$_{µν}$ ( r ) in the evolving quantum state | ψ ( t ) 〉 : 

An exact and unambiguous expression for the operator ˆ P$_{µν}$ ( r ) in an arbitrary system of coordinates is obtained by considering the universal many-body Hamiltonian 

(external potential not included) in the presence of a “metric tensor field” g$_{µν}$ ( r ). As is well known $^{23}$, the metric tensor g$_{µν}$ ( r ) allows us to express the length ds of an infinitesimal displacement from r to r + d r in terms the corresponding increments of the coordinates dr$_{µ}$ : 

In ordinary Euclidean space and in cartesian coordi- nates g$_{µν}$ ( r ) = δ$_{µν}$ , independent of position. In gen- eral, however, a non-Euclidean space is characterized by a position-dependent, symmetric g$_{µν}$ ( r ). A non-Euclidean metric can also be generated by a change of coordinates in an Euclidean space, as we will see shortly. As an im- portant technical point we also introduce the tensor g µν as the inverse of g$_{µν}$ , and we define g as the determinant of g$_{µν}$ (so g − 1 is the determinant of g $^{µν}$). 

The hamiltonian ˆ H$_{u}$ undergoes the following changes in the presence of a non-trivial metrics. First, the lapla- cian operator ∂$_{µ}$∂$_{µ}$ for the kinetic energy is replaced by 

Second, the Euclidean distance between two points (which controls the interaction energy) is replaced by the non-Euclidean length of the shortest path (geodesic) con- necting the two points. We denote by ˆ H$_{u}$ [ g ] the Hamil- tonian in the presence of the metric field g$_{µν}$ . Then the stress tensor operator is defined as the first varia- tion of ˆ H$_{u}$ [ g ] under an instantaneous virtual variation of the metric tensor g$_{µν}$ ( r ), i.e., 

The first-order change in the Hamiltonian due to a change δg$_{µν}$ in the metric tensor is given by 

<img src='content_image/376009.jpg'>

## A. Derivation of the force density 

<img src='content_image/375998.jpg'>

<img src='content_image/375999.jpg'>

<img src='content_image/376000.jpg'>

<img src='content_image/376003.jpg'>

<img src='content_image/376010.jpg'>

Notice that the stress tensor operator defined in this man- ner is itself a functional of the metrics. This definition is completely analogous to the standard definition of the current density operator as the derivative of the Hamil- tonian with respect to a vector potential. An explicit expression for ˆ P$_{µν}$ in Euclidean metrics is reported for completeness in Appendix A (see also Refs 18, 19, 24 and 25). We note that the definition of the quantum me- chanical stress tensor via the variational derivative with respect to the metric tensor has been also employed in Ref. 26. 

We will now focus on the calculation of P$_{µν,}$$_{1}$ – the correction to P$_{µν}$ of first order in u . In order to ex- press P$_{µν,}$$_{1}$ ( r , t ) and its divergence as functionals of the displacement field we resort to Tokatly’s recent formu- lation of quantum dynamics in the co-moving reference frame. 18–21 The co-moving frame is an accelerated refer- ence frame which, at each point and each time, moves with the velocity of the volume element of the fluid at that point and that time, so that the density is constant and equal to the ground-state density, while the current density is zero. The time-dependent transformation from the laboratory frame (coordinates r ) to the co-moving frame (coordinates ξ ) is defined by the solution of the equation 

where v ( r , t ) = j ( r ,t ) n ( r ,t ) is the velocity field. In the linear response regime the velocity is approximated as j ( r , t ) /n$_{0}$ ( r ), where n$_{0}$ ( r ) is the ground-state density. In this regime we can write 

where u ( ξ , t ) is the (small) displacement of a fluid ele- ment for its initial position ξ . Expressing ds 2 = d r · d r in terms of the new coordinates ξ and making use of Eq. (18) we see that the metric tensor in the co-moving frame is given by 

From Eq. (23), to first order in the displacement field, we immediately get 

and 

where 

is the strain tensor . Also to first order in u the determi- nant of the metric tensor is easily seen to be 

<img src='content_image/376001.jpg'>

<img src='content_image/376004.jpg'>

<img src='content_image/376005.jpg'>

<img src='content_image/376006.jpg'>

<img src='content_image/376007.jpg'>

<img src='content_image/376002.jpg'>

<img src='content_image/376008.jpg'>

