text:
\section{Introduction}

Along last years, several branches of theoretical physics have predicted the existence of extra dimensions.
Supporting this idea, we have the well--known string theory, higher dimensional black holes and higher dimensional stellar distributions, to name a few.

In this concern, higher dimensional stellar distributions have been widely studied in the literature. See for example references \cite{Chilambwe:2015rra, Dadhich:2016wtb,Molina:2016xeu,Bhar:2014tqa,Paul:2018vho,Estrada:2018vrl}. It is worth to mention that in all these references the topology of a $D$--dimensional stellar distributions corresponds to $S^{D-2}$ topology. For example, the topology of a five dimensional stellar distribution corresponds to a $S^3$ topology. However, it is still an open problem to understand how to represent $4D$ spherically symmetric and static compact stellar distributions within the framework of extra dimensions, where the topology should differ from the $S^3$ topology.

One the the first models in representing $4D$ geometries, immersed in extra dimensions was the well--known  Kaluza--Klein (KK) model. In this model, the higher dimensional space--time representation corresponds to
the product between the Minkowski space--time and a compactified extra dimension. The most common example is $\mathcal{M}_{4}\times S^{1}$, being $\mathcal{M}_{4}$ the Minkowski manifold and $S^{1}$ a sphere of radius $\mathcal{R}$. In this case, $\mathcal{M}_{4}$ corresponds to the horizontal direction, whilst the sphere $S^{1}$ is the boundary. Therefore, the whole space--time is a cylinder (see figure \ref{FigKK} of reference \cite{Shifman:2009df} and figure \ref{FigKK1}).

\begin{figure}
\centering
\includegraphics[scale=0.85]{Fig1.jpg}
\caption{KK set-up \cite{Shifman:2009df}\label{FigKK}}.
\end{figure}

\begin{figure}
\centering
\includegraphics[scale=0.85]{Fig11.jpg}
\caption{KK set-up \label{FigKK1}}.
\end{figure}

Other models admitting a $(D-1)$--dimensional geometry representation immersed in extra dimensions, is the so--called black strings (BS). This scenario, is the simplest extension of black hole solutions. Specifically, BS consists in slice replications of a $(D-1)$--dimensional black hole along a compact extra dimension (see for example figure \ref{FigSlides}). It is worth mentioning that the BS setup, is different from the KK mechanism, because the KK approach includes more degrees of freedom (vector and scalar fields) in addition to the gravitational ones.

In the simplest BS scenario (the case $D=5$), the radial coordinate of a 4--dimensional black hole corresponds to the horizontal direction, while the extra coordinate corresponds to the boundary of a $S^{1}$ sphere. The reasons why this is the simplest scheme, lie on the fact that the horizontal direction always coincides with the 4--dimensional space--time and not only with the radial component. So, this scenario corresponds to the situation shown by figure \ref{FigCilindro}.

\begin{figure}
\centering
\includegraphics[scale=0.85]{Fig2.jpg}
\caption{BS set-up \cite{Kleihaus}\label{FigSlides}}.
\end{figure}

\begin{figure}
\centering
\includegraphics[scale=0.65]{Fig3.jpg}
\caption{Cylindrical set--up of BS \label{FigCilindro}}.
\end{figure}

The simplest version of BS is given by the line element
\begin{equation} \label{Kalusa}
    ds_{5D}^2=d{s}_{4D}^2+ dz^2,
\end{equation}
 where $d{s}_{4D}^2$ is the 4--dimensional Schwarzschild space--time. The line element \eqref{Kalusa} corresponds to the Kaluza--Klein black string (KKBS). This solution has $S^2 \times R$ topology for non compact $z$. The representation of this BS for a compact extra coordinate $z$, can be visualized in the figure \ref{FigKKBS} (see reference \cite{Kol:2004ww} for further details). In this case, the topology corresponds to the product between the horizontal direction, whose topology is $S^2$ and the compact extra dimension, with topology $S^1$, {\it i.e.} the topology is $S^2 \times S^1$. So, for $r<r_0$, being $r_0$ the Schwarzschild radius, the interior region is denominated as {\it cylindrical event horizon}. So, as was claimed in reference \cite{Kleihaus:2016kxj}, the horizon completely wraps the compact extra dimension.

Following figures \ref{FigKK}, \ref{FigKK1} and \ref{FigSlides}, similar slices of the 4--dimensional surface $(t,r,\theta,\phi)$ (see figures \ref{FigCilindro} and \ref{FigKKBS}, for a simplified point of view with similar radial points $r$) are replicated in different locations of the compact extra dimension $z$. In this respect, in the reference \cite{Kleihaus:2006ee} was argued that the KKBS solution shows translational symmetry along the extra coordinate direction.

 \begin{figure}
\centering
\includegraphics[scale=0.85]{Fig4.jpg}
\caption{KK BS \cite{Kol:2004ww} \label{FigKKBS}}.
\end{figure}

 It is worth mentioning that in the KKBS the 4--dimensional Schwarzschild black hole is represented (whose topology is $S^2$) into a compact extra dimension (whose topology is $S^1$). So, the $S^2 \times S^1$ topology of the KKBS differs from the usual $S^3$ topology of the 5--dimensional Schwarzschild--Tangherlini solution \cite{Kol:2002xz}.
 On the other hand, in the KKBS the topology of the central singularity corresponds to the product between the Schwarzschild singularity and the $S^1$ manifold for a compact extra coordinate.

 Recently, a new methodology has been provided for the representation of a 4--dimensional regular black hole into an extra dimension using the black string framework \cite{Estrada:2021naf}. An interesting extension of this work was developed in \cite{Culetu:2021zun}. So, motivated by this methodology, here the main aim of the present work, is to test the possibility to carry out the representation of a 4--dimensional compact stellar distribution into an extra dimension, using the black strings setup. So, instead the usual $S^3$ topology of the 5--dimensional stellar distributions, the topology of our solution, which corresponds to a 4--dimensional spherically symmetric and static stellar configuration into an extra dimension, is $S^2 \times R(S^1)$ for a non compact extra dimension.

So, in this article we provide a specific form for the 4--dimensional and 5--dimensional line elements, along with a suitable form for the energy--momentum tensor. We start from the 5--dimensional Einstein's field equations and, after a suitable choice on the function $A(z)$, which deforms the 4--dimensional geometry, the 5--dimensional field equations are remarkably reduced to the usual form of the 4--dimensional equations of motion.
Moreover, as toy a model, we apply the mentioned framework to the well--known  Buchdahl type geometry \cite{Buchdahl:1959zz} as 4--dimensional manifold, representing the compact stellar distribution.

\section{The Model} \label{sec3}

In this section we present in brief the strategy to represent a 4--dimensional manifold in a 5--dimensional space--time, by employing the BS framework. The underlying 4--dimensional geometry will represent to a stellar distribution. The starting point are the 5--dimensional Einstein's field equations given by
\begin{equation} \label{EcuacionDeEinstein}
    G^M_{N}+\Lambda_{5D} \delta^M_{N}= 8 \pi \tilde{G}_{5D} T^M_{N},
\end{equation}
where $M,N=0,1,2,3,5$ and $\tilde{G}_{5D}$ is the 5--dimensional Newton's constant. The line element is:
\begin{align}
ds_{5D}^2=& W(z) \cdot d\bar{s}_{4D}^2+ dz^2, \label{elementodelinea} \\
=& e^{-2A(z)} \cdot g_{\mu \nu}^{(4D)} dx^\mu dx^\nu + dz^2, \label{elementodelineaIntermedio} \\
=& e^{-2A(z)}  \left[-e^{\nu(r)}dt^2+\mu^{-1}(r)dr^2+r^2 d\Omega^2 \right]+ dz^2, \label{Elementodelinea1}
\end{align}
with $\mu,\nu=0,1,2,3$. In order to have a regular 5--dimensional geometry \cite{Estrada:2021naf}, the extra coordinate $z$ should be compact at the BS style, in sense as was explained before. In this work we consider $z \in [-\pi,\pi]$. Besides, the function $W$ depends only on the extra coordinate $z$. Thus, the 5--dimensional geometry is modified by the function $W(z)$. Although, similar slices of the 4--dimensional surface are replicated along the extra dimension $z$, the 5--dimensional geometry no longer has traslational symmetry along the $z$ coordinate, as occurs in the KKBS \cite{Kleihaus:2006ee}.

Along with previous considerations, here we propose the following energy--momentum tensor
\begin{equation} \label{EnergiaMomentum1}
    T^M_{N}=\mbox{diag}\big (-\rho(r,z),p_r(r,z),p_\theta(r,z),p_\phi(r,z),p_z(r,z) \big),
    \end{equation}
where $\rho$ denotes the energy density while, $p_r$, $p_\theta$, $p_\phi$ and $p_z$ are representing the pressure components. Some examples in $5D$, where the energy--momentum tensor depends on the extra coordinate can be found at \cite{Culetu:2021zun}, and energy--momentum tensors depending on the radial coordinate in a BSs scenario surrounded by quintessence matter can be found at \cite{Ali:2019mxs}.

\subsection{Relation between $4D$ and $5D$ Newton's constants}

In order to establish a relation between the 4--dimensional and $D$--dimensional Newton's constants, we are going to follow the approach given in: ``heuristics of higher-dimensional gravity'' \cite{Maartens:2010ar}. Then we will see that this analysis serves to reduce the 5--dimensional field equations to 4--dimensions. In this reference, for a $D$ dimensional space--time, the effective four dimensional Planck scale $M_4$ is given by
\begin{equation} \label{Constantes1}
    M_4^2=M_D^{D-2}L^{D-4},
\end{equation}
where $M_D$ represents the $D$ dimensional Planck scale and where $L$ has units of length. In this context, as indicates the reference \cite{Maartens:2010ar}, the Newton constant in $D$ dimensions is given by:

\begin{equation} \label{Constantes2}
    \tilde{G}_D = \frac{1}{M_D^{D-2}}.
\end{equation}

So, equation \eqref{Constantes1} can be re--written as:
\begin{equation} \label{Constantes3}
    \tilde{G}_D=\tilde{G}_4 L^{D-4}.
\end{equation}

Using Planck units, the $D$ dimensional Newton's constant has the following units $[\tilde{G}_D]=\ell_p^{D-2}$ \cite{Garraffo:2008hu}, where $\ell_p$ corresponds to the Planck length. Also, this can be obtained from equation \eqref{Constantes2}, assuming that $[M_D]=\ell_p^{-1}$. Besides, using the same system of units in equation \eqref{Constantes1}, one gets $[L]=\ell_p$.

From equation \eqref{Constantes3}:
\begin{equation} \label{Constantes31}
    \tilde{G}_5=\tilde{G}_4 L.
\end{equation}

Since the units of the Ricci scalar and the Einstein tensor are $[R]=[G^M_N]=\ell_p^{-2}$ and $[\sqrt{g}d^Dx]=\ell_p^D$, respectively. Then, the gravitational action is dimensionless $[\tilde{G}_D^{-1} \sqrt{g}d^dxR]=\ell_p^0$.

Following \cite{Aros:2019quj}, the energy--momentum tensor has units of $[T^M_N]=\ell_p^{-D}$, and then, $[G^M_N]=[\tilde{G}_D \cdot T^M_N]=\ell_p^{-2}$.

So, in order to describe the 5--dimensional energy--momentum tensor in terms of 4--dimensional quantities, we define a 4--dimensional energy--momentum as $\bar{T}^\mu_\nu$, where $[\bar{T}^\mu_\nu]=\ell_p^{-4}$, and constant $K$ with dimensions $[K]=\ell_p^{4-D}$, such that:

\begin{equation} \label{Constantes4}
    [\bar{T}^\mu_{\nu}]=[\delta^\mu_M \delta^N_\nu \cdot K^{-1} \cdot T^M_{N}]=\ell_p^{-4}
\end{equation}
{\it i.e.}, for $D=5$, $[K]=\ell_p^{-1}$ and $[T^M_{N}]=\ell_p^{-5}$.

\subsection{Equations of motion}

For the line element \eqref{Elementodelinea1} and the energy--momentum tensor \eqref{EnergiaMomentum1}, Einstein's field equations are:

\begin{eqnarray}\label{EqMovimiento5D}
&&\hspace{-0.5cm} 6(\dot{A})^2-3 \ddot{A} + \Lambda_{5D}+ e^{2A(z)}\bigg\{\frac{\mu'}{r}-\frac{1}{r^2}+\frac{\mu}{r^2}\bigg\}=- 8 \pi \tilde{G}_{5D} \rho(r,z), \label{1tt} \\
&&\hspace{-0.5cm} 6(\dot{A})^2-3 \ddot{A}+ \Lambda_{5D}+e^{2A(z)}\bigg\{\frac{\mu \nu'}{r}-\frac{1}{r^2}+\frac{\mu}{r^2}\bigg\}=8 \pi \tilde{G}_{5D} p_r (r,z), \label{1rr} \\
&&\hspace{-0.5cm} 6(\dot{A})^2-3 \ddot{A} + \Lambda_{5D}+e^{2A(z)}\bigg\{ \frac{\mu \nu'}{2r}+\frac{\mu'}{2r} +\frac{1}{4}\mu'\nu'+\frac{1}{2}\mu \nu''+\frac{1}{4}\mu (\nu')^2 \bigg\}=8 \pi \tilde{G}_{5D} p_\theta (r,z), \label{1thetatheta} \\
&&\hspace{-0.5cm} 6(\dot{A})^2+\Lambda_{5D}+e^{2A(z)}\bigg\{  \frac{1}{4}\mu'\nu'+\frac{1}{2}\mu \nu''+\frac{1}{4}\mu (\nu')^2+\frac{\mu \nu'}{r}+\frac{\mu'}{r}-\frac{1}{r^2}+\frac{\mu}{r^2} \bigg\}=8 \pi \tilde{G}_{5D} p_z (r,z),~~~~~~\label{1zz}
\end{eqnarray}
where dot indicates derivation with respect to the extra coordinate $z$ and the primes represent derivation with respect to the radial coordinate $r$.

Following the same ansatz for $A(z)$ given in \cite{Estrada:2018vrl}, one obtains

\begin{equation} \label{1EcuacionA}
        6(\dot{A})^2 =- \Lambda_{5D},
    \end{equation}
with
\begin{equation}
    \Lambda_{5D}=-\frac{6}{l^2},
\end{equation}
then
\begin{equation} \label{warpfactor1}
e^{-2A(z)}=C e~^{\mp z/l},
\end{equation}
where $C$ a positive and dimensionless constant and, $l$ corresponds to the AdS radius when the space--time corresponds to an AdS space--time. For the sake of simplicity, we consider that the magnitude of $C=1$. Therefore, for this particular case we will consider the positive sign in equation \eqref{warpfactor1}.

Using the ansatz \eqref{1EcuacionA}, the  three first terms in left member of Eqs. \eqref{1tt}, \eqref{1rr} and \eqref{1thetatheta}, and the two first terms in the left  member of the equation \eqref{1zz}  are cancel--out.

In order to write the equations of motion in terms of the $4D$ Newton's constant, we use the relation \eqref{Constantes3}. Thus, Einstein's field equations are:

\begin{eqnarray}\label{EqMovimiento5D}
&&\hspace{-0.5cm}  W^{-1}\bigg\{\frac{\mu'}{r}-\frac{1}{r^2}+\frac{\mu}{r^2}\bigg\}=- 8 \pi L \tilde{G}_4 \rho(r,z), \label{2tt} \\
&&\hspace{-0.5cm} W^{-1}\bigg\{\frac{\mu \nu'}{r}-\frac{1}{r^2}+\frac{\mu}{r^2}\bigg\}=8 \pi L \tilde{G}_4 p_r (r,z), \label{2rr} \\
&&\hspace{-0.5cm}W^{-1}\bigg\{ \frac{\mu \nu'}{2r}+\frac{\mu'}{2r} +\frac{1}{4}\mu'\nu'+\frac{1}{2}\mu \nu''+\frac{1}{4}\mu (\nu')^2 \bigg\}=8 \pi L \tilde{G}_4 p_\theta (r,z), \label{2thetatheta} \\
&&\hspace{-0.5cm} W^{-1}\bigg\{  \frac{1}{4}\mu'\nu'+\frac{1}{2}\mu \nu''+\frac{1}{4}\mu (\nu')^2+\frac{\mu \nu'}{r}+\frac{\mu'}{r}-\frac{1}{r^2}+\frac{\mu}{r^2} \bigg\}=8 \pi L \tilde{G}_4 p_z (r,z),~~~~~~\label{2zz}
\end{eqnarray}

Besides, we propose the following energy--momentum tensor (\ref{EnergiaMomentum1})
\begin{align} \label{EnergiaMomentum2}
T^M_{N} =& F \big ( W  (z) \big) \cdot K \cdot \mbox{diag}[-\bar{\rho}(r),\bar{p}_r(r),\bar{p}_\theta(r),\bar{p}_\phi(r),\hat{p}_z(r)],
\end{align}
where $F \big ( W  (z) \big)$ represents a function depending on the extra coordinate and $[K]=\ell_p^{-1}$ in order to be consistent with the equation \eqref{Constantes4}.

For the sake of simplicity, the above energy--momentum tensor can be written as follows
\begin{equation} \label{EnergiaMomentum}
    T^M_{N}=\delta^M_\mu \delta^\nu_N \cdot F \big ( W  (z) \big) \cdot K \cdot \bar{T}^\mu_{\nu} + \delta^{M}_{5} \delta^{5}_{N} T^5_{5},
\end{equation}
As we will see in the equation \eqref{EcuacionReducida}, $\bar{T}^\mu_{\nu}$ represents the energy--momentum tensor associated to the 4--dimensional geometry, given by
\begin{equation} \label{EnergiaMomentum4D}
    \bar{T}^\mu_{\nu}=   \mbox{diag}(-\bar{\rho}(r),\bar{p}_r(r),\bar{p}_\theta(r),\bar{p}_\phi(r))
\end{equation}
Thus $\bar{\rho}(r),\bar{p}_r(r),\bar{p}_\theta(r),\bar{p}_\phi(r)$ are representing the energy density and the $r$, $\theta$, $\phi$ pressure components of the 4--dimensional geometry.

It is worth to mentioning that in our case, the 4--dimensional geometry is stretched along the extra dimension, then the 4--dimensional matter given by  $\bar{T}^\mu_\nu$ spread over the whole extra dimension. So, the value of the 5--dimensional energy--momentum tensor varies for different values of $z$ through the function $F \big ( W  (z) \big)$. This is also consistent with the fact that although similar slices of the 4--dimensional surface are replicated along the extra dimension $z$, the 5--dimensional geometry no longer has traslational symmetry along the $z$ coordinate, as occurs for the KKBS \cite{Kleihaus:2006ee}.

On the other hand, $T^5_{5}$ corresponds to the energy--momentum tensor associated to the extra dimension
\begin{equation} \label{EnergiaMomentum5}
    T^5_{5}= \delta^5_5  F \big ( W  (z) \big) K \hat{p}_z(r),
\end{equation}
where $\hat{p}_z(r)$ represents the radial dependency of $T^5_{5}$.

The components $\rho$, $p_r$, $p_\theta$, $p_\phi$, $p_z$ of the 5--dimensional energy--momentum tensor, are given in equation \eqref{EnergiaMomentum2}, such that the components of the 4--dimensional energy--momentum tensor $\bar{\rho}$, $\bar{p}_r$, $\bar{p}_\theta$, $\bar{p}_\phi$ have units of $[\bar{T}^\mu_\nu]=\ell_p^{-4}$ (for further details see equation \eqref{Constantes4}).

At this point it is important to stress that the function $W(z)$ modifies both, the geometry and the components of the energy--momentum tensor.

It is worth mentioning that, in an arbitrary way and for sake of simplicity, we impose that constants $L$ and $K$ have magnitude equal to the unity, $L=K=1$. Moreover, hereinafter we shall consider $\tilde{G}_5=\tilde{G}_4=1$ \cite{Alcubierre:2018ahf,Spallucci:2021ljr}.

Now, for the line element \eqref{Elementodelinea1} and the energy--momentum tensor \eqref{EnergiaMomentum} the units of $[L \cdot K]= \ell_p^0$ are cancelled. Thus, the $(t,t)$, $(r,r)$, $(\theta,\theta)$ and $(z,z)$ components of the field equations are given by
\begin{eqnarray}\label{EqMovimiento5D}
&&\hspace{-0.5cm} W^{-1}\bigg\{\frac{\mu'}{r}-\frac{1}{r^2}+\frac{\mu}{r^2}\bigg\}=- 8 \pi F(W) \bar{\rho}, \label{3tt} \\
&&\hspace{-0.5cm} W^{-1}\bigg\{\frac{\mu \nu'}{r}-\frac{1}{r^2}+\frac{\mu}{r^2}\bigg\}=8 \pi F(W) \bar{p}_r, \label{3rr} \\
&&\hspace{-0.5cm} W^{-1}\bigg\{ \frac{\mu \nu'}{2r}+\frac{\mu'}{2r} +\frac{1}{4}\mu'\nu'+\frac{1}{2}\mu \nu''+\frac{1}{4}\mu (\nu')^2 \bigg\}=8 \pi F(W) \bar{p}_\theta, \label{3thetatheta} \\
&&\hspace{-0.5cm} W^{-1}\bigg\{  \frac{1}{4}\mu'\nu'+\frac{1}{2}\mu \nu''+\frac{1}{4}\mu (\nu')^2+\frac{\mu \nu'}{r}+\frac{\mu'}{r}-\frac{1}{r^2}+\frac{\mu}{r^2} \bigg\}=8 \pi F(W) \hat{p}_5,~~~~~~\label{3zz}
\end{eqnarray}
where we impose that:
\begin{equation} \label{EcuacionW}
    F \left(W (z) \right) = W^{-1}= e^{2A(z)}.
\end{equation}

Thus, by means of equations \eqref{1EcuacionA} and \eqref{EcuacionW} it is straightforward to check that the components \eqref{3tt}, \eqref{3rr} and \eqref{3thetatheta} of the 5--dimensional equations of motion are reduced to the usual ones corresponding to the 4--dimensional geometry $d\bar{s}_{4D}^2$. This latter is given by equations \eqref{elementodelinea}, \eqref{elementodelineaIntermedio} and \eqref{Elementodelinea1}
\begin{eqnarray}\label{EqMovimiento5D1}
&&\frac{\mu'}{r}-\frac{1}{r^2}+\frac{\mu}{r^2}=- 8 \pi  \bar{\rho} \label{tt1} \\
&&\frac{\mu \nu'}{r}-\frac{1}{r^2}+\frac{\mu}{r^2}=8 \pi  \bar{p}_r, \label{rr1} \\
&& \frac{\mu \nu'}{2r}+\frac{\mu'}{2r} +\frac{1}{4}\mu'\nu'+\frac{1}{2}\mu \nu''+\frac{1}{4}\mu (\nu')^2 =8 \pi \bar{p}_\theta. \label{thetatheta1}
\end{eqnarray}

Thus, the above equations have the following form
\begin{equation} \label{EcuacionReducida}
\bar{G}^\mu_\nu \propto \bar{T}^\mu_\nu.
\end{equation}

An important consequence in employing the BS setup, lies on the fact that using the line element \eqref{Elementodelinea1}, for the functions \eqref{warpfactor1} \eqref{EcuacionW}, the $(\mu,\nu)$ components of the 5--dimensional equations of motion adopt the usual 4--dimensional ones \eqref{EcuacionReducida}, corresponding to the line element $d\bar{s}_{4D}^2$ .

On the other hand, a direct computation reveals that the $z$--$z$ component, equation \eqref{3zz}, takes the form
\begin{equation}
     \bar{p}_\theta- \frac{\bar{\rho}}{2}+\frac{\bar{p}_r}{2}=\hat{p}_5,
\end{equation}

In order to test the sign of the pressure along the extra dimension, $\hat{p}_5$, we rewrite the last equation as:
\begin{equation} \label{ecuacionp5}
2 \hat{p}_5 = - (\bar{\rho}-\bar{p}_r-2\bar{p}_\theta)=\bar{T}.
\end{equation}

Thus, under our assumptions, the sign of the pressure $\hat{p}_5$ is given by the sign of the trace of the 4--dimensional energy--momentum tensor $\bar{T}^{\mu\nu}$. This is known as the Trace Energy Condition (TEC), and should be negative when the signature is $(-,+,+,+)$ \cite{Martin-Moruno:2017exc}. However, as mentioned in \cite{Martin-Moruno:2017exc}, the study of TEC has been abandoned after the 70s decade.

It is worth to mentioning that, for the chosen energy--momentum \eqref{EnergiaMomentum2}, along with the line element \eqref{Elementodelinea1}, the relation \eqref{EcuacionW} and, for the present case where $\bar{p}_\theta=\bar{p}_\phi$, the conservation equation $\nabla_M T^M_N=0$ is:
\begin{eqnarray}\label{tov}
&& \hspace{-1cm} - \frac{1}{2} \dot{F} \bigg [  2 \hat{p}_5 +  \Big(\bar{\rho} - \bar{p}_r- 2\bar{p}_\theta \Big)   \bigg ] \nonumber \\
&& - F \bigg[ - \frac{\nu'}{2} (\bar{\rho}+\bar{p}_r) - \frac{d \bar{p}_r}{dr} +\frac{2}{r} (\bar{p}_\theta-\bar{p}_r) \bigg] =0.
\end{eqnarray}

The condition given by equation \eqref{ecuacionp5}, allows us to eliminate the first bracket in the left hand side of (\ref{tov}). On the other hand, since the extra coordinate $z$ is a compact coordinate, from the condition \eqref{EcuacionW} the factor $F \neq 0$, hence the conservation equation adopts the form of the usual 4--dimensional conservation law
\begin{equation}
- \frac{\nu'}{2} (\bar{\rho}+\bar{p}_r) - \frac{d \bar{p}_r}{dr} +\frac{2}{r} (\bar{p}_\theta-\bar{p}_r) =0 .
\end{equation}

\begin{figure}
\centering
\includegraphics[scale=0.65]{rho.jpg}\\
\includegraphics[scale=0.65]{pre.jpg}
\caption{\textbf{Upper panel}: The trend of the density against the dimensionless radial coordinate $r/\bar{R}$. \textbf{Lower panel}: The behavior of the pressure versus $r/\bar{R}$. These plots consider that the extra coordinate $z$ is taking values in the interval $[-\pi,\pi]$. }\label{f1}
\end{figure}

It is remarkable to note that, although the present form to solve the equations of motion is simple in its right own, this way allows that the 5--dimensional equations of motion are  reduced to the usual Einstein's 4--dimensional field equations. Therefore, this methodology could serve to represent other types of 4--dimensional objects into an extra dimension in future works.

\subsection{About the five dimensional matching conditions} \label{Matching}

We assume that the 4--dimensional geometry is stretched between $z \in [-\pi,\pi]$. So, we call $z_0$ to each point belonging to the domain of the extra coordinate $z$.

In to order to find all constant parameters characterizing the model, we shall employ the well--known Israel--Darmois junction conditions. Those are given by

\begin{itemize}
   \item The 4--dimensional line element must be continuous,{\it i.e} at the exterior of the star the geometry must be described by the exterior Schwarzschild space--time. This is consistent with the fact that the induced metric, for all point $z_0 \in z$, where is stretched the $4D$ geometry along the extra dimension, is given by \cite{Aros:2012xi}
 \begin{equation}   \label{metricIanducida}
    \tilde{g}_{\mu \nu} = g_{\mu \nu}^{(4D)} W(z_0).
\end{equation}

So, at the boundary of the star, $r=\bar{R}$, for a fixed value $z=z_0$, the internal geometry of the star \eqref{elementodelinea} is
\begin{equation}
     (ds_{5D}^2)^-= W(z_0) \cdot d\bar{s}_{4D}^2 ~~~(\mbox{Star}).
     \end{equation}
It must be matched with
\begin{equation}
(ds_{5D}^2)^+= W(z_0) \cdot d{s}_{4D}^2 ~~~(\mbox{Schwarzschild}).
\end{equation}
So, the induced line element must be continuous {\it i.e.} there must be no jumps in the metric:
\begin{equation} \label{MatchingStar}
    (g_{\mu \nu}^{(4D)})^-_{\mbox{star}}=(g_{\mu \nu}^{(4D)})^-_{\mbox{Schw}}.
\end{equation}
Thus, inside the stellar distribution, the topology corresponds to the product between the 4--dimensional star and $S^1$. However, outside the stellar distribution, the topology corresponds to the product between the Schwarzschild geometry and the $S^1$ manifold. So, at the stellar interior and the exterior region, the topology is $S^2 \times S^1$.
\item From previous arguments, the equation \eqref{EcuacionReducida} at the stellar surface $\Sigma$, defined by $r=\bar{R}$, must satisfy
\begin{equation}\label{secondform}
    \left[\bar{G}_{\mu\nu}r^{\nu}\right]_{\Sigma}=0,
\end{equation}
where $r_{\nu}$ is an unit radial vector and  $\left[F\right]_{\Sigma}\equiv F\left(r \rightarrow \bar{R}^{+}\right)- F\left(r \rightarrow \bar{R}^{-}\right)$, for any function $F=F(r)$.
Using the equation (\ref{secondform}) and general Einstein's field equations, one finds
\begin{equation}
    \left[\bar{T}_{\mu\nu}r^{\nu}\right]_{\Sigma}=0,
\end{equation}
which leads to
\begin{equation}\label{secondformpr}
    \left[\bar{p}_r\right]_{\Sigma}=0.
\end{equation}
This is so because the radial pressure is zero for the external vacuum Schwarzchild solution. Then, the last equation yields
\begin{equation}
\bar{p}_r(\bar{R})=0.    \label{p4}
\end{equation}
\end{itemize}

\subsection{A 4-- dimensional geometry example: The Buchdahl space--time}  \label{ToyModel}

As stated before, one of the main aims here is to provide a methodology to explore the consequences of representing a 4--dimensional spherical symmetric compact object into an extra dimension using the BS setup. In this opportunity, to see the effects of the reminiscent compactified extra coordinate $z$ on the 4--dimensional manifold, we have choice the well--known Buchdahl space--time \cite{Buchdahl:1959zz}. So, in this section we revisited in short the principal aspects of the Buchdahl solution and how (\ref{warpfactor1}) is altering the main salient features of the model.

\begin{figure}
\centering
\includegraphics[scale=0.65]{ecs.jpg}\\
\includegraphics[scale=0.65]{gam.eps}
\caption{\textbf{Upper panel}: The trend of the energy conditions against the dimensionless radial coordinate $r/\bar{R}$. \textbf{Lower panel}: The relativistic adiabatic index versus $r/\bar{R}$, for different values of the extra coordinate $z$.}\label{f2}
\end{figure}

\subsubsection{The Buchdahl metric}
The ansatz corresponds to the well--known Buchdahl space--time, whose metric potentials are given by
\begin{eqnarray}
\hspace{-0.5cm} e^{\nu(r) \over 2} &=& A \left\{B \left(2 c r^2+5\right) \sqrt{2-c r^2}+\left(cr^2+1\right)^{3/2}\right\}\\
\hspace{-0.5cm} \mu(r) &=& \frac{2-c r^2}{2 \left(c r^2+1\right)},
\end{eqnarray}
where, $A$ and $B$ are dimensionless integration constants and $c$ is an arbitrary constant with units of $\ell_p^{-2}$. Since this solution is isotropic, we will have $p_r=p_\theta$. Using the field equations \eqref{3tt}--\eqref{3thetatheta}, and previous arguments, we get for the 5--dimensional components of the energy--momentum tensor,
\begin{equation}
\label{density}
8\pi \rho = e^{z/l} K \dfrac{3 \alpha c  \left(c r^2+3\right)}{2 \left(c r^2+1\right)^2},
\end{equation}
\begin{equation}\label{pressure}
\resizebox{1.04\hsize}{!}{$
\begin{split}
8\pi p_r = 8\pi p_\theta = e^{z/l}  K   \frac{9 \alpha c  \left[B \left(2 c^2 r^4-3 c r^2-2\right)+\left(1-c r^2\right) \sqrt{-c^2 r^4+c r^2+2}\,\right]}{2 \sqrt{2-c r^2} \left(c r^2+1\right) \left[B \left(2 c r^2+5\right) \sqrt{2-c r^2}+\left(c r^2+1\right)^{3/2}\right]}~~~
\end{split}$},
\end{equation}
where the constant $\alpha$ has units of $\ell_p^{-2}$.

It is easy to check that, these 5--dimensional quantities (without bar), which correspond to equation \eqref{EnergiaMomentum1}, are related with the 4--dimensional quantities (with bar) through the relation \eqref{EnergiaMomentum2}, where the function $F \big ( W  (z) \big)=e^{z/l}$ is given by equations \eqref{EcuacionW} and \eqref{warpfactor1}. Moreover, the component $p_z$ of equation \eqref{EnergiaMomentum1}, can be computed from equations \eqref{EnergiaMomentum2} and \eqref{ecuacionp5}. Since the function $F \big ( W  (z) \big)=e^{z/l}$ is positive, and the TEC is negative, hence the component $p_z$ is negative too.

In order to explore the consequences of representing the 4--dimensional Buchdahl space--time into an extra dimension, using the BSs setup, we study ( using a graphical analysis), how the values of the salient thermodynamics properties for different values of $z_0 \in z$ are varying. Following \cite{Aros:2012xi}, a direct computation shows that the induced 4--dimensional energy density and the induced 4--dimensional pressure components for all fixed point $z_0 \in z$, are given by
\begin{align}
    &\tilde{\rho}=\exp \left (2A(z_0)\right) \bar{\rho}= \rho(r,z_0) \\
    &\tilde{p}_i=\exp \left (2A(z_0)\right) \bar{p}_i =p_i(r,z_0)
\end{align}
with $i=0,1,2,3$.

As can be appreciated, from the above expressions (\ref{density})--(\ref{pressure}), the function (\ref{warpfactor1}) is quite involved in the behavior of the density and pressure. Although, the solution obtained for the function (\ref{warpfactor1}) is simple, the way it affects the thermodynamic properties of the fluid describing the energy--momentum tensor of the 4--dimensional structure, is in principle non--trivial. Of course, different strategies will lead to more complex functions in their shape, perhaps having a more predominant effect on the essential characteristics of the fluid sphere. Next, we shall explore in brief by means of a graphical analysis the impact of the mentioned function (\ref{warpfactor1}) on the configuration.
\begin{figure}
\centering
\includegraphics[scale=0.65]{M_R.eps}\\
\includegraphics[scale=0.65]{MI.eps}
\caption{\textbf{Upper panel}: The $M$--$\bar{R}$ curve for different values of $z$. \textbf{Lower panel}: The moment of inertia $I$ versus the mass $M$ for different values of the extra coordinate $z$. }\label{f3}
\end{figure}

 \subsubsection{Some properties}

As it is well--known, the Buchdhal solution of the Einstein field equations describes a fluid sphere threading by an isotropic matter distribution. Particularly, this space--time meets all the necessary requirements to recreate a realistic toy model for compact stars \cite{Delgaty:1998uy}:
\begin{itemize}
\item The pressure $\bar{p}$ must be a decreasing function such that this one has a finite and positive value at the
centre $r=0$ and it must vanish at the surface $r/\bar{R}=1$.
\item It is required that the energy density be monotonically decreasing towards the boundary $r/\bar{R}=1$.
\item The following energy conditions are required: $\bar{p}+\bar{\rho} \ge 0$, $3\bar{p}+\bar{\rho} \ge 0$, $\bar{\rho}-\bar{p} \ge 0$.
\item The Adiabatic index, $\Gamma$, which determines the stability of the star, must be positive.
\end{itemize}

In order to obtain Figures \ref{f1}--\ref{f3}, previously we have employed junction condition process, Eqs. \eqref{MatchingStar} and \eqref{p4}, to determine the set of constant parameters $\{A,B,c\}$. So, from figures \ref{f1}--\ref{f3} we can highlight the following points

\begin{enumerate}
    \item The Fig. \ref{f1} is displaying the density $\rho$ (upper panel) and pressure \footnote{The scheme presented in Sec. \ref{sec3} concerns a more general situation where the pressures along the radial and tangential directions are unequal \i.e., $p_{r}\neq p_{\theta}$. However, this case corresponds to an isotropic fluid $p_{r}=p_{\theta}$, hence along this section the pressure will be denoted as $p$ only.} $p$ (lower panel) behavior versus the dimensionless radial coordinate $r/\bar{R}$. It is observed that in moving the extra coordinate $z=z_{0}$ from $-\frac{\pi}{8}$ to $\frac{\pi}{8}$, both $\rho$ and $p$ are taking higher values at $r=0$ when $z_{0}\geq0$ and they are decreasing in magnitude when $z_{0}\leq 0$. Then, it is clear that increasing $z_{0}$ in magnitude the core of the structure becomes denser. Furthermore, as the upper panel of Fig. \ref{f2} depicts, the energy conditions are also satisfied, what is more the chance of having an energy--momentum satisfying these conditions increases when $z_{0}$ takes higher values.
    It is worth mentioning that the units of $\mbox{Energy}/L^4$ can be easily related with Planck units, using $[\mbox{Energy}]=\ell_p^{-1}$ and $[L]=\ell_p$.

    \item Interestingly, the relativistic adiabatic index $\Gamma$ is not affected by function $W$ (see lower panel of Fig. \ref{f2}). Indeed, by definition the relativistic adiabatic index $\Gamma$ is given by \cite{10.1093/mnras/239.1.91}
    \begin{equation}
        \Gamma=\frac{\rho+p}{p}\frac{dp}{d\rho}.
    \end{equation}
    So, as the function $W$ is just a global factor, after factorize it from the corresponding expressions for $\rho$ and $p$, it is cancel out
    \begin{equation}
        \Gamma=\frac{\bar{\rho}+\bar{p}}{\bar{p}}\frac{d\bar{p}}{d\bar{\rho}}. \nonumber
    \end{equation}
    Then, in measuring the stability of the model by means of $\Gamma$ this quantity is not altered by the function $W$ (\ref{warpfactor1}). So, in this concern the extra coordinate is not playing any role.
    \item An important curve in this kind of studies, is the so--called M--$\bar{R}$ curve. As the upper panel of Fig. \ref{f3} exhibits, as $z$ increases in magnitude the mass and radii of the compact object increase too. Moreover, when $z=\frac{\pi}{8}$ the mass of the configuration is above $2M_{\odot}$ whereas for $z=-\frac{\pi}{8}$, the mass is just above $1M_{\odot}$. Then, it is clear that for increasing and positive values of $z$, it is possible to build up more massive and compact objects in comparison with the case where $z=0$. On the other hand, the lower panel in Fig. \ref{f3} is showing the  moment of inertia $I$ versus the mass $M$. This plot has been obtained by considering the following expression
    \begin{equation}
    I=\frac{2}{5}\left[1+\frac{(M/\bar{R})\cdot km}{M_{\odot}}\right]M\bar{R}^{2},
    \end{equation}
    proposed in \cite{Bejger:2002ty} to covert a static model to a rotating one. As can be seen, the sensitivity of $I$ increases when $z$ moves from $-\frac{\pi}{8}$ to $\frac{\pi}{8}$, reaching its maximum at $z=\frac{\pi}{8}$.

It is worth to stress that, when we increase the radius of the compact star the mass content within it also increases and hence we have an increasing curve, however, at a certain radius the mass within it will be large enough, so its gravity will overcome the increase in radius and the additional mass add--on will contracted to a small radius. This kind of nature can seen only in compact stellar systems. Of course, the nature of the $M-\bar{R}$ curve is links with the matter compositions. When the mass increase with radius, the energy density at the center also increases as well as the stiffness corresponding to the equation of state (EoS). However, when the interior density reaches a certain density the stiffness of the EoS suddenly changes due to some kind of phase transitions making the system more compact.

\end{enumerate}
So, as primary conclusion it is evident that the function (\ref{warpfactor1}) is affecting in a positive way the main salient features of the compact structure, specially when the extra coordinate $z$ takes positive values. It is worth mentioning that, to obtain the above results and graphs, the constant appearing in (\ref{warpfactor1}) \i.e., $c$, $\alpha$ and $l$ have been taken to be one.

\section{Concluding Remarks}

We have provided a way of representing a 4--dimensional stellar distribution using the black string framework. For this, we have chosen a specific form for the 4--dimensional and 5--dimensional line elements and, for the 4--dimensional and 5--dimensional energy--momentum tensor, such that the 4--dimensional and  5--dimensional quantities are related by a function $A(z)$, where $z$ represents  the extra dimension.

One important consequence of the proposed methodology, is that, using the line element \eqref{Elementodelinea1}, the function \eqref{warpfactor1} and the function \eqref{EcuacionW}, the 5--dimensional equations of motion are reduced to the usual 4--dimensional equations of motion \eqref{EcuacionReducida}, which correspond to the line element $d\bar{s}_{4D}^2$. Also, the 5--dimensional conservation equation adopts the form of the usual 4--dimensional conservation equation.

It is worth mentioning that, although the presented methodology is simple, this reduction form could serve to represent other types of 4--dimensional objects into an extra dimension in future works. This is a not minor point, since in the literature the most $D$--dimensional stellar distributions studied, are those whose topology correspond to $S^{D-2}$ \cite{Chilambwe:2015rra, Dadhich:2016wtb,Molina:2016xeu,Bhar:2014tqa,Paul:2018vho,Estrada:2018vrl}. For example, the topology of a 5--dimensional stellar distribution corresponds to a $S^3$. However, our methodology shows a new way for representing 4--dimensional stellar distributions (whose topology is $S^2$) into an extra compact dimension (whose topology is $S^1$), using the black string setup. So, the topology of the 5--dimensional solution is $S^2 \times S^1$ instead the usual $S^3$ topology.

The schematic representation of our model follows the black string setup, see figures \ref{FigSlides},\ref{FigCilindro} and \ref{FigKKBS}. However, in our case, the horizontal direction is representing the radial coordinate of a 4--dimensional stellar distribution, instead the radial coordinate of a black hole solution as occurs in the usual black strings models.

In the present model, due to the action of the function $A(z)$, the 5--dimensional geometry no longer has traslational symmetry along the  compact coordinate $z$, as occurs for the Kaluza--Klein black string. On the other hand, under our assumptions the sign of the pressure along the extra dimension is given by the trace of the 4--dimensional energy--momentum tensor.

In the subsection \eqref{Matching} we have provide the matching condition for our form of representation of $4D$ stellar distribution using the black string style, which are reduced to the usual four dimensional matching condition.

In order to explore the consequences of representing 4--dimensional stellar distribution into an extra dimension in the black string framework, we have characterized the 4--dimensional Buchdahl geometry \cite{Buchdahl:1959zz} as toy model. One of the most relevant characteristics of this scheme, is that allows the construction of more stable and compact objects. This is quite relevant from the astrophysical point of view, because it is clear that this mechanism works as a mass generator. Particularly, in this case, as the coordinate $z=z_{0}$ moves from $-\frac{\pi}{8}$ to $\frac{\pi}{8}$ the mass of the object increases, leading to more compact configurations than in the normal ($z=0$) scenario. We wish to emphasize that this protocol is not a tool to find analytical solutions to 4--dimensional Einstein's field equations, but rather to establish guidelines for determining how a 4--dimensional solution is immersed in an extra dimension, when the approach at the black string style is applied.

For a fixed value of $z=z_0$, inside the stellar distribution, the topology corresponds to the product between the Buchdahl geometry and $S^1$. Outside the stellar distribution the topology corresponds to the product between the Schwarzschild geometry and $S^1$. So, inside and outside of the stellar distribution, the topology is $S^2 \times S^1$.


text:
\section{Introduction}

Important correlative characteristics have been attributed over the years to pulses in gamma-ray burst (GRB) prompt emission. Examples of these characteristics include (1) temporal asymmetry characterized by longer decay than rise rates, (2) hard-to-soft spectral evolution, and (3) broadening at lower energies (e.g. \cite{nor96,rrf00,nor02,ryd05}). Despite these suggestive correlations, GRB prompt emission studies have focused on the measurement of bulk properties, with pulses often regarded as nothing more than minor variations to the overall signal.

Recently, Hakkila et al. \cite{hak08a, hak08b} demonstrated that bulk GRB characteristics are composite properties formed from characteristics belonging to distinct yet unresolved pulses. Each pulse in a small sample of GRBs with known redshift was found to have its own lag, and the lag associated with the bulk gamma-ray emission (using the cross correlation function \cite{ban97}) was shown to be a heterogeneous composite of the individual pulse lags favoring the high intensity, short duration pulses. Since bulk GRB lags are correlated with GRB peak luminosities \cite{nor00}, this implies that pulse properties must be highly correlated to produce measurable correlations in bursts composed of often many pulses. Pulses belonging to the small sample of BATSE GRBs with known redshifts used in this study have unique, individual pulse lags that correlate with the pulse durations and that anti-correlate with pulse peak luminosity.  In order to verify that this is not a statistical discrepancy unique to this data set, we have set out to expand our GRB pulse database by sequentially analyzing bursts in the Current BATSE Catalog (\url{http://www.batse.msfc.nasa.gov/batse/grb/catalog/current/}). The results allow us to determine how pervasive correlated pulse properties are while sampling the BATSE Catalog data in a relatively unbiased way.

\section{Procedure}

We use a semi-automated procedure and code to identify and fit pulses in multi-channel observations of GRB prompt emission using 64 ms data \cite{hak08a, hak08b}. The approach starts with a search for potential pulses in summed four channel data. Candidate time intervals potentially containing pulses are identified using the Bayesian Blocks methodology \cite{sca98}; a pulse model is optimized within each interval to see if it contains a statistically-significant pulse. The data from intervals containing statistically-insignificant model pulses are merged into the surrounding intervals, and the fitting process begins again. Iteration eventually produces an optimal set of model pulses for the summed four-channel data.

The four-parameter pulse model used \cite{nor05} assumes the following functional form for each pulse:
\begin{equation}
I(t) = A \lambda \exp^{[-\tau_1/(t - t_s) - (t - t_s)/\tau_2]},
\end{equation}
where $t$ is time since trigger, $A$ is the pulse amplitude, $t_s$ is the pulse start time,
$\tau_1$ and $\tau_2$ are characteristics of the pulse rise and pulse decay, and
$\lambda = \exp{[2 (\tau_1/\tau_2)^{1/2}]}$). A two-parameter background model is
simultaneously fitted along with any pulses.
The resulting 4-channel pulse characteristics are used as starting points from
which individual energy channel pulse fits are obtained.
The aforementioned process is repeated until convergent solutions
are obtained for pulses in each energy channel.

A variety of model-dependent pulse properties are extracted from the aforementioned pulse parameters in the fitting process; these include pulse durations, pulse peak fluxes (256 ms timescale), pulse peak lags, pulse asymmetries, and pulse spectral hardnesses. The {\em pulse duration} $w$ is obtained from the summed four-channel data and is defined as $w = [9 + 12\sqrt{\tau_1/\tau_2}]^{1/2}$; this is the interval between times when the pulse amplitude is $A {e^{-3}}$. The {\em pulse peak flux} $p_{256}$ is defined on the 256 ms timescale in terms of the summed four-channel data. {\em Pulse peak lags} $l$ are the differences between the pulse peak times in different energy channels (pulse peak times are given by $\tau_{\rm peak} = t_s + \sqrt{\tau_1 \tau_2}$). Pulse peak lags can be obtained for any pulse between two energy channels, although we define the standard pulse peak lag $l_{31}$ as that measured between energies of 100 to 300 keV (BATSE channel 3) and 25 to 50 keV (BATSE channel 1). The pulse asymmetry $\kappa$ is defined as $\kappa = w/(3 + 2\sqrt{\tau_1/\tau_2}).$ {\em Hardness ratios} $HR$ are constructed by dividing pulse fluences in two different energy channels; we use a hardness ratio $HR_{31}$ defined by $HR_{31}=S_3/S_1$, where $S_3$ is the channel 3 fluence and $S_1$ is the channel 1 fluence.

This analysis approach has been successfully applied to both BATSE and Swift bursts. However, difficulty in normalizing BAT data to BATSE data has thus far prevented us from merging measurements from the two datasets.

The procedure is fairly successful at pulse extraction, even though the pulse signal-to-noise ratio is often low in more than one of the BATSE energy channels (most often in channel 4). Pulse properties are cleanly extracted in a relatively unambiguous manner for GRBs containing non-overlapping or isolated pulses. Typically, these tend to be low luminosity, long duration bursts \cite{nor05, hak07}. However, the process is also successful at identifying and fitting many pulses in complex GRBs containing overlapping pulses. The approach is less successful at fitting low intensity pulses, pulses that strongly overlap, and very short pulses, which can be indistinguishable from Poisson noise. Ambiguous pulse identifications often result in poor $\chi^2$ goodness-of-fit measures, in fits that appear to merge pulses separable to the eye, and/or in pulses that have disparate properties or are not observed in contiguous energy channels. We exclude from our analysis  overlapping and low fluence pulses that are overtly ambiguous, but recognize existence of the latter by tentatively identifying them as low fluence events (LFEs).

We also classify the bursts that we study. Although the definition of the Short and Long class of GRBs has become more complex during the Swift era than it was, we use a classification scheme based on duration and spectral hardness originally obtained from BATSE bursts using machine learning algorithms \cite{hak07}. This scheme \citep{hak03} classifies GRBs as Short if they satisfy the inequality  (T90 $<1.954$) OR ($1.954 \le T90 < 4.672$ AND $HR_{321} > 3.01$), where $HR_{321} = S_3/(S_2 + S_1)$ \citep{muk98}, where $S_2$ is the channel 2 fluence. If GRBs do not satisfy the aforementioned inequality, then they are classified as Long.

When pulses have been fitted and the GRBs to which they belong have been classified, we are able to seek out possible correlations among the properties of pulses in Long and Short BATSE GRBs.

\section{Analysis}

Our current database consists of 307 pulses in 106 Long and 46 Short GRBs. We have started simultaneously at the ends of the Current BATSE Catalog and are working our way through the rest of the Catalog sequentially. The results obtained with this larger database are consistent with those obtained previously for smaller datasets \cite{hak08a, hak08b}: pulse properties, rather than bulk properties of the prompt emission, underly GRB measurements.

Figure 1 demonstrates an example of pulse fits for BATSE Trigger 0214. This is a Long GRB with
three fitted pulses. Pulse properties are extracted for this and many other BATSE burst, and compiled in a pulse database.

\begin{figure}
  \includegraphics[height=.27\textheight]{00214_1.eps}
  \includegraphics[height=.27\textheight]{00214_3.eps}
  \caption{Example of pulse fits for a Long BATSE GRB (Trigger 0214): (a) channel 1, and (b) channel 3. The pulses have the following lags: pulse 1 lag = 0.679 s, pulse 2 lag = 0.368 s, and pulse 3 lag = 0.352 s. }
\end{figure}

\subsection{Correlations}

Pulse properties are compared and correlated for both the Long and Short GRB classes. The most rigorous correlation found is the one between pulse duration and pulse peak lag: the probability that these are uncorrelated in Long GRB pulses is $2.3 \times 10^{-19}$. Such a strong correlation suggests that the pulse duration and the pulse peak lag are two different representations of the same phenomenon; these also appear related to hard-to-soft pulse evolution. Both characteristics have been previously found to anti-correlate with pulse peak luminosity \cite{hak08a} and are thus luminosity indicators. We note that it is much easier to accurately measure pulse durations than pulse peak lags, since pulse durations are longer and since the smoothly-varying pulse fitting function allows the pulse duration to be accurately determined.

Although the pulse duration vs.\ pulse peak lag relation is generally very tight, there are pulses with specific pulse properties for which this relation may not correlate. Pulse peak lags of Short GRB pulses, for example, do not strongly correlate with pulse durations; this could be because it is difficult to measure pulse peak lags for Short GRB pulses using the 64 ms timescale. The lags of Short GRB pulses are so short that roughly half of them are negative; this is consistent with a distribution centered around zero seconds and is likely due primarily to measurement error. Additionally, a small number of Long GRB pulses are found to have short lags but long durations; these appear to be long pulses with sharply-defined pulse peaks. These pulses also appear to be inconsistent with the Pulse Scale Conjecture of Nemiroff \cite{nem00} which assumes that all pulses have temporal structures that are similar when properly scaled. Hakkila et al. \cite{hak08a} have previously suggested that these pulses might be external shock signatures, but it is possible that these represent inaccurately-measured prorpeties of overlapping pulses.

We note that many Long GRBs contain short duration pulses; these typically have pulse peak lags near zero seconds. This indicates that a short duration pulse does not necessarily identify a GRB belonging to the Short class. Likewise, there are many Long GRB pulses with pulse lags that are demonstrably negative. Figure 3 (BATSE trigger 1807) is one of these: the negative-lag pulse does not appear to be a single pulse with homogeneous properties, but instead has two peaks indicative of overlapping pulses. Given the relative rarity of negative-lag pulses, we suggest that many of these are merely unresolved overlapping pulses, although some could be pulses of short duration whose lags cannot be accurately measured.

\begin{figure}
  \includegraphics[height=.27\textheight]{w_v_l.eps}
  \includegraphics[height=.27\textheight]{w_v_l_big.eps}
  \caption{(a) Pulse duration vs. pulse peak lag for Long (square) and Short (*) GRBs in logarithmic units, showing the strong correlation between these parameters. (b) A subset of these data types in linear units, indicating that some pulses have negative lags. Many of these lag measurements are consistent with pulse lags near $l_{31}=0$, but some have abnormally-negative values. Short GRB pulses are typically found in region $S$ (short lags, short durations), while normal Long GRB pulses are found in region $L$ (following the correlation shown in Figure 2(a)). Region $H$ contains Long GRB pulses that have extremely negative lags and which might be marred by hidden pulses, and Region $E$ contains Long GRB pulses that have characteristics potentially associated with external shocks (short pulse lags coupled with long pulse durations).}
\end{figure}

\begin{figure}
  \includegraphics[height=.27\textheight]{01807_1.eps}
  \includegraphics[height=.27\textheight]{01807_3.eps}
  \caption{Example of a negative-lag pulse (BATSE Trigger 1807): (a) channel 1, and (b) channel 3. The second fit pulse (with a measured lag of -4.2 s) appears to be composed of two pulses, with the second one being spectrally harder than the first. Merged pulses such as this would explain many of the pulses having negative lags.}
\end{figure}

Figure 4 demonstrates strong anti-correlations between pulse duration and pulse peak flux for both Long and Short GRB pulses. Both Long and Short GRB pulses exhibit anti-correlations, but these relations have different slopes: using a model where $\log p_{256} = A \log w + B$, we find that $A_{\rm Long}=-0.27$ (with correlation coefficient $R=0.36$) and $A_{\rm Short} = -0.67$ ($R=0.64$). Different correlative relations could indicate different pulse mechanisms for the two burst classes. Figure 4 also demonstrates an anti-correlation between spectral hardness and pulse duration for Long GRB pulses. Long, low-luminosity pulses are softer than short, luminous pulses, supporting the argument that intrinsic properties dominate over cosmological effects.

\begin{figure}
  \includegraphics[height=.27\textheight]{p256_v_w.eps}
  \includegraphics[height=.27\textheight]{hr_v_w.eps}
  \caption{(a) 256 ms peak flux vs.\ pulse duration for Long (square) and Short (*) GRBs. Best-fit lines are given for Long (solid line) and Short (dashed line) relationships. (b) Spectral hardness vs.\ pulse duration for Long (square) and Short (*) GRBs.}
\end{figure}

Figure 5a demonstrates a correlation between Long GRB pulse duration and pulse asymmetry: shorter pulses tend to be more symmetric than longer ones. The asymmetries of Short GRB pulses are not plotted, since the short durations of these pulses (often only a few temporal bins wide) tend to make them be observed as being very symmetric or very asymmetric. Since duration is a luminosity indicator for Long GRB pulses, both spectral hardness and pulse asymmetry are also luminosity indicators.

\begin{figure}
  \includegraphics[height=.27\textheight]{w_v_kap.eps}
  \includegraphics[height=.3\textheight]{00809_1234_bw.eps}
  \caption{(a) Pulse duration vs.\ asymmetry for Long (square) GRBs, along with a best-fit line. (b) Summed four-channel pulse fits to BATSE trigger 0809, a Short GRB having potential characteristics of the Long GRB class (pulse 1 lag = 0.679 s, pulse 2 lag = 0.368 s, and pulse 3 lag = 0.352 s).}
\end{figure}

The correlative properties of GRB pulses are summarized in Table 1 (Long GRB pulses) and Table 2 (Short GRB pulses). Although Short GRB pulse properties are difficult to accurately measure with 64 ms resolution, there is evidence to suggest that Short GRB pulse properties are not as correlative as Long GRB pulse properties. Such a result could provide a mechanism for more accurate GRB classification. Similarly, pulse property correlations (or a lack thereof) could well provide constraints necessary for constructing more accurate models of GRB prompt emission for Long and Short GRBs.

\begin{table}
\begin{tabular}{lcccc}
\hline
  & \tablehead{1}{c}{b}{$w$}
  & \tablehead{1}{c}{b}{$P_{256}$}
  & \tablehead{1}{c}{b}{$HR_{31}$}
  & \tablehead{1}{c}{b}{$\kappa$}   \\
\hline
$l_{31}$ & ${\bf 2.3\times 10^{-19}}$ & ${\bf ^\dagger 4.8\times 10^{-3}}$ & ${\bf ^\dagger 2.8\times 10^{-3}}$ & $1.1\times 10^{-1}$\\
$w$ & --- & ${\bf ^\dagger 8.8\times 10^{-7}}$ & ${\bf ^\dagger 4.2\times 10^{-7}}$ & ${\bf 1.1\times 10^{-7}}$\\
$P_{256}$ & --- & --- & $1.7\times 10^{-1}$    & ${\bf ^\dagger 5.3\times 10^{-3}}$\\
$HR_{31}$ & --- & --- & ---    & ${\bf ^\dagger 3.3\times 10^{-3}}$\\
\hline
\end{tabular}
\caption{Long GRB pulse correlations. Spearman rank order correlation probabilities that the pulse characteristics in question are random. Anti-correlations are indicated by a $\dagger$, and a significant correlation or anti-correlation is indicated in \bf{boldface}.}
\label{tab:a}
\end{table}

\begin{table}
\begin{tabular}{lcccc}
\hline
  & \tablehead{1}{c}{b}{$w$}
  & \tablehead{1}{c}{b}{$P_{256}$}
  & \tablehead{1}{c}{b}{$HR_{31}$}
  & \tablehead{1}{c}{b}{$\kappa$}   \\
\hline
$l_{31}$ & $2.1\times 10^{-1}$ & $^\dagger 6.6\times 10^{-2}$ & $^\dagger 2.2\times 10^{-2}$ & $4.4\times 10^{-1}$\\
$w$ & --- & ${\bf ^\dagger 7.0\times 10^{-7}}$ & $^\dagger 1.3\times 10^{-1}$ & $^\dagger 8.2\times 10^{-2}$\\
$P_{256}$ & --- & --- & $7.3\times 10^{-2}$    & $7.9\times 10^{-1}$\\
$HR_{31}$ & --- & --- & ---    & $2.3\times 10^{-2}$\\
\hline
\end{tabular}
\caption{Short GRB pulse correlations.}
\label{tab:b}
\end{table}

Despite the insights gained from these pulse relationships, there is still ambiguity in the definition of Short and Long GRBs as obtained from bulk emission and pulse properties. Take, for example, BATSE trigger 0809 (Figure 5b): this burst is hard with a $T_{90}$ duration of less than two seconds, and would normally be classified as a Short GRB. However, the pulse properties of pulse duration, pulse peak flux, and pulse peak lag correlate with one another in a way that is consistent with it being a Long GRB. Thus, where many observers are now making claims to expand the Short GRB definition to reclassify many Long bursts as Short, we have found an ambiguous Short GRB that might actually be a Long GRB solely because it is lacking inter-pulse durations.

\section{Conclusions}

A large GRB pulse sample verifies that each GRB is characterized by its own lag. The lag of a Long GRB pulse is directly related to its duration, and both of these properties anti-correlate with the pulse peak luminosity. Via correlations with pulse peak lag and pulse peak duration, it also appears that pulse spectral hardness and pulse asymmetry are luminosity indicators. These relationships hold for most Long GRB pulses, but perhaps exclude long pulses with short, intense pulse peaks (external shocks?).

Short GRB pulses do not reflect the same correlative behaviors as those found for Long burst pulses, although this result might partly reflect the limited temporal resolution used in this study. Perhaps the most interesting behavior of Short burst pulses is an anti-correlation between pulse duration and pulse peak flux which might indicate a relationship between pulse duration and pulse peak luminosity.

There is a need for better measuring and understanding GRB pulse properties. In principle, the bulk characteristics of the prompt emission can be derived from knowledge of pulse properties and the pulse decomposition of a burst. The converse, however, is not true: we cannot infer the basic characteristics of the pulses, nor can we understand the relationship between bulk and (more fundamental) pulse properties, without explicit study of the constituent pulses.

\begin{theacknowledgments}
We thank Tom Loredo, Rob Preece, Tim Giblin, Chris Fragile, and Jay Norris for valuable discussions. R. Cumbee acknowledges financial support from the South Carolina Space Grant program and faculty support from Dr. J. Myers of Francis Marion University. We also gratefully thank USRA and the conference poster competition judges.
\end{theacknowledgments}

\bibliographystyle{aipprocl}

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\newcommand{\emll}[1]{{\color{red} #1}}
\title{Holographic Complexity from the Crofton's Formula in Lorentzian $\adst$}
\author[a,b]{Xing Huang \thanks{xingavatar@gmail.com}}
\author[a,b]{Le Zhang \thanks{leeviackerman@outlook.com}}
\affil[a]{Institute of Modern Physics, Northwest University, Xi'an 710069, China}
\affil[b]{Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710069, China}
\renewcommand*{\Affilfont}{\small\it}
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\date{}
\textwidth=6.0in \hoffset=-.55in
\textheight=9in \voffset=-.8in

\begin{document}

\maketitle
\begin{abstract}
We study the Crofton's formula in the Lorentzian AdS$_3$ and find that the area of a generic space-like two dimensional surface is given by the flux of space-like geodesics. The ``complexity=volume'' conjecture then implies a new holographic representation of complexity in terms of the number of geodesics. Finally, we explore the possible explanation of this result from the standpoint of information theory.
\end{abstract}

\newpage
\noindent\rule[0.1\baselineskip]{\textwidth}{0.8pt}
\tableofcontents
\noindent\rule[0.1\baselineskip]{\textwidth}{0.8pt}
\linespread{1.5}\selectfont

\section{Introduction}
As a standout of the holographic principle, the $\adscft$ correspondence \cite{maldacena} offers possibly the best playground for exploring quantum gravity as it indicates that the latter (likely to be string theory) in the Anti-de Sitter spacetime is equivalent to some well-defined conformal field theory on the boundary. It has long been conjectured that the bulk geometry shall emerge from the boundary entanglement as both should be universal \ie insensitive to
the discrepancies between the CFTs that admit gravity duals. A strong supporting evidence comes from the Ryu-Takayanagi formula \cite{ryu,ryu2}, which associates the area of a minimal surface anchored on the boundary with the entanglement entropy of some subregion in the boundary CFT.

Motivated by the Ryu-Takayanagi formula, Balasubramanian, et al. \cite{diffentropy} connected the length of a closed bulk curve on the time slice of $\adst$ to the differential entropy, which thus provides entropic interpretation for more general geometric objects. This has later been incorporated by Czech, et al. \cite{czech1} into the program of integral geometry, in which the length of a curve follows from the number of geodesics it has intersections with, which is known as the Crofton's formula in the mathematical literature \cite{santal}.

Complexity is another concept whose dual may help to uncover the information theoretic origin of gravitational physics. It is defined as the number of quantum gates (unitary operators) to produce a given state from a reference state. We can imagine the system to be a collection of qubits each located on a site of a spatial lattice. We would like the quantum gates to act only on a small number of neighboring qubits for our interests. Such restriction of locality is physically reasonable (as interactions are generally local particularly in the field theories we consider) and it is crucial to the definition of complexity. Accordingly the reference state is usually the one with no pre-existing long range correlation. For example, we can choose a state with all qubits set into the state $\st{0}$. The quantum gates form a quantum circuit \ie a unitary operator and hence it is natural to assign the complexity to the circuit itself. Such operator/circuit complexity can be defined in a geometric way pioneered by Nielsen and collaborators \cite{nielsen2006geometric, nielsen2006quantum, dowling2008geometry} (see also \cite{jefferson2017circuit} for recent developments), as the length of a geodesic in the space of all the unitary operators on the Hilbert space, measured by the cost function. Complexity provides ordering to the states in the Hilbert space. In this sense, it is argued that complexity can be understood as an analog of entropy (see \eg \cite{Susskind:2018pmk}). Naturally complexity can be also used as a probe of the chaotic behavior (see \eg \cite{Ali:2019zcj}).

It is difficult to define complexity in a field theory. We will instead work with those theories with holographic duals and focus on the holographic dual of complexity. There are two proposals on such duality that could turn out to be equivalent. Complexity may either be dual to the volume of the spatial slice (so-called complexity=volume, {\it a.k.a.} CV \cite{susskind2016computational}) or the gravity action inside the Wheeler-DeWitt patch (complexity=action, {\it a.k.a.} CA \cite{brown2016holographic, brown2016complexity}).

On the boundary CFT side, complexity can be defined using the Liouville action \cite{Miyaji:2016mxg, Caputa:2017urj, Caputa:2017yrh,Takayanagi:2018pml}. Recently the correspondence between a quantum circuit and a co-dimension one surface was proposed in \cite{Takayanagi:2018pml}. More precisely, the quantum circuit is realized as a path integral on the co-dimension one bulk surface $M[\xS(0)]$ whose boundary $\p M = \xS(0)$ is the constant time surface (taken to be $t = 0$) on the boundary. The evolution of the Euclidean time gives a one parameter family of quantum states, which correspond (in the sense of surface/state correspondence \cite{Miyaji:2015yva}) to the codimension-two surfaces $\xS(t)$ on $M[\xS(0)]$. This process is essentially the renormalization group flow and $t$ serves as the energy scale. Different surfaces $M[\xS(0)]$ lead to different induced metrics in the path integral, which can be written in the canonical form of $e^{2\phi} \eta_{\mu\nu}$. Due to conformal symmetry, all these path integrals are supposed to provide the same state with different normalization factors $e^{C(M[\xS(0)])}$ given by the Liouville action $C(M[\xS(0)]) = C_L(e^{2\phi} \eta_{\mu\nu})$ (the so-called PI complexity), which can be taken as an alternate definition of complexity (see \cite{caputa2019quantum,Camargo:2019isp} for the equivalence with the circuit complexity in Nielsen's approach). \footnote{In fact it might be possible to go one-step further to think that the bulk region enclosed by the surface is determined entirely by the path integral, which seems quite natural from the standpoint of surface/state correspondence even though the precise dictionary is unclear.} The CV conjecture then implies the areas of these surfaces are equal to the values of the cost function for the circuits, with the complexity being the minimum given by the optimized one \footnote{It should be noted that the area does not necessarily agree with PI complexity. In our opinion it is perfectly acceptable to have different definitions for complexity as long as they can be good enough approximation of each other. For example, the number of gates might be only polynomial in the circuit complexity defined geometrically \cite{nielsen2006quantum}. However, such generalization of CV does not seem to work in Lorentzian AdS as the extremal surface used to define complexity is the maximal one.}. Both the cost function and the path integral complexity can be defined for a generic circuit and we will loosely refer to them as the cost of a circuit.

It is known \cite{santal,huang} that the area of a codimension-one surface on a time slice or more generally in any Euclidean AdS-like space can be computed using the flux of geodesics. Although the same Crofton's formula is expected to hold in Lorentzian space, the precise form suitable for practical
computation is less clear. We study this problem in the Lorentzian AdS$_3$ and find the area of generic space-like two dimensional surface can be reproduced by the flux of space-like geodesics, which are associated with the entanglement entropies of intervals in the boundary CFT. In our opinions this may be the first step towards understanding the CV conjecture from the perspective of information theory. We analyze this result using some toy model and find it reasonable to interpret the geodesic number contributing to the complexity as counting the entanglement of a state constructed from the path integral with two boundaries. More importantly we realize that these geodesics are circuit-independent and hence contribute as lower bound on the cost of all the circuits.

In section~\ref{ig on e}, we review some basic facts in the integral geometry, particularly the Crofton's formula in the Euclidean space. In section~\ref{ig on m} we figure out the precise form of the Crofton's formula in the Lorentzian AdS$_3$ and show that the area of a generic 2d space-like surface is given by the number of space-like geodesics it intersects. Finally in section~\ref{sec:discussion}, we discuss the possible reason why the complexity can be expressed in terms of the geodesic number.

\section{Integral geometry in Euclidean space}\label{ig on e}
Integral geometry is not a new subject. In fact most of the conclusions we are going to use are probably well known to mathematicians \cite{santal}. It was not until very recently that \cite{czech1} they were introduced to the community of AdS/CFT and provided some interesting new insights. Here we will briefly touch on various useful results for later convenience.

\subsection{Kinematic space and invariant measure}\label{ssbks2}
In integral geometry, geometric objects are expressed in terms of the
integrals of some probe objects, the collection of which forms the so-called kinematic space. In the remainder of the paper, we mostly focus on the kinematic
space of the geodesics in which every point represents a single geodesic. In a symmetric space like $\ads_3$, the
geometry of the kinematic space can be determined entirely on symmetry ground
(as it can be expressed in terms of the coset of symmetry groups).

To see this explicitly, we can pick two points $x_1$ and $x_2$ on the boundary to denote a geodesic
and express the metric as follows
\be \rmd s^2=f_{\mu\nu}(x_1,x_2)\rmd x_1^{\mu}\rmd x_2^{\nu}\quad(\mu,\nu=0,1)\,.\ee
The precise form of $f_{\mu\nu}$ can be obtained using
the following trick in \cite{czech2}: Its transformation under the conformal group is the same as the two-point function of two spin-$1$ currents of conformal dimension $1$. The requirement of conformal invariance then fixes its
form to be
\be f_{\mu\nu}(x_1,x_2)=\frac{4I_{\mu\nu}(x_1-x_2)}{|x_1-x_2|^2},\label{fmunu}\ee
where $4$ is an normalization constant and $I_{\mu\nu}(x) = \eta_{\mu\nu}
- 2x_\mu x_\nu/x^2$.

For simplicity, we will work in the Poincare patch (\ie $x^{0,1}
= t, x$) with metric
\begin{equation}\label{metric}
   \rmd s^2=\frac{\rmd u^2+\rmd z\rmd\bar{z}}{u^2}=\frac{\rmd u^2+\rmd x^2-\rmd t^2}{u^2},
\end{equation}
where we introduce the null coordinates
\begin{equation}
    z=x+t\qquad \bar{z}=x-t\,.
\end{equation}
It is not different to check using eq.\eqref{fmunu} and the precise form
of $I_{\mu\nu}$ that the density for geodesics space becomes
\begin{equation}\label{3adsks}
\rmd s^2=\frac{2\rmd z_1\rmd z_2}{(z_1-z_2)^2}+\frac{2\rmd \bar{z}_{1}\rmd \bar{z}_{2}}{(\bar{z}_{1}-\bar{z}_{2})^2}.
\end{equation}
The same result can be deduced from the second derivatives of the geodesic length as $f_{\mu\nu} (x_1, x_2) =\p_\mu \p_\nu L (x_1, x_2)$, the latter of which in the null coordinates reads ($\xe$ being the cutoff)
\begin{equation}\label{3adsgeol}
  L(z_1,z_2)=\log\lsb\frac {(z_1-z_2)(\bar{z}_1-\bar{z}_2)} \xe\rsb\,.
\end{equation}

The Euclidean $\ads_2$ or rather $\BH^2$ can be regarded as a constant time slice of the $\ads_3$ like the one specified by $z_i=\bar{z}_i$, and we get the kinematic space on the $\BH^2$ with metric
\begin{equation}\label{2adsle}
    \rmd s^2=\frac{4\rmd z_1\rmd z_2}{(z_1-z_2)^2}\,.
\end{equation}

\subsection{Crofton's formula in the Euclidean $\ads_2$ }
In two-dimensional space, the Crofton's formula says that the length of a curve is given by the number of geodesics it meets. As explained above, this number
follows from a volume integral in the kinematic space. We leave the derivation in Euclidean plane in the appendix~\ref{e2}. The more interesting case to us is Euclidean
AdS$_2$ (E$\ads_2$), where the formula takes the following form
\begin{equation}\label{2adscf1}
  L_{\gamma} =\h \int_{M_1\cap L_1\neq0}\frac{2\sigma_{0}(M_1\cap L_1)}{(z_1-z_2)^2}\rmd z_2\wedge \rmd z_2\,,
\end{equation}
where the integration is over all geodesics $L_1$ with nonvanishing intersection
number $\sigma_{0}(M_1\cap L_1)$ with the curve $\xg$. As we can see, the denominator comes from the measure in eq.\er{2adsle}. In general the measure
is given by the second derivatives of the geodesic length and the $\cf$ goes
like
\begin{equation}\label{2adscf2}
  L_{\gamma} =\frac{1}{2}\int_{M_1\cap L_1\neq0} \sigma_{0}(M_1\cap L_1)\frac{\partial^2L(z_1,z_2)}{\partial z_1\partial z_2}\rmd z_1\wedge \rmd z_2.
\end{equation}
Moreover the length of a geodesic is related to the entanglement entropy via the RT formula:
\begin{equation}\label{rt}
 S(z_1,z_2)=\frac{ L(z_1,z_2)}{4G}
\end{equation}
where $S(z_1,z_2)$ is the entanglement entropy of an interval $(z_1,z_2)$ on the boundary. Putting eq.(\ref{rt}) into eq.(\ref{2adscf2}), we get
\begin{equation}\label{2adscf}
  \frac{L_{\gamma}}{4G} =\frac{1}{2}\int_{M_1\cap L_1\neq0} \sigma_{0}(M_1\cap L_1)\frac{\partial^2S(z_1,z_2)}{\partial z_1\partial z_2}\rmd z_1\wedge \rmd z_2.
\end{equation}
So the length $L_{\gamma}$ as a bulk geometric quantity is connected with the entanglement entropy $S(z_1,z_2)$. In fact, given the entanglement entropies of all intervals, one can reconstruct the geometry in the kinematic space and hence the geometry in the bulk. This is a perfect example of notion of ``entanglement=geometry''.

\subsection{The generic Crofton's formula}\label{gcf}
The generic  Crofton's formula first proposed in \cite{santal2} (see also \cite{santal,czech1,huang,convex}) establishes the relationship between $q$-dimensional target object $M_q$ and the sets of $r$-planes (geodesically complete submanifolds) for any constant curvature space. In a $d$ dimensional
Euclidean space, it takes the following form
\begin{equation}\label{gcf1}
\int_{M_q\cap L_r\neq0} \sigma_{q+r-d}(M_q\cap L_r) \epsilon_{\CK }=\frac{O_d...O_{d-r}O_{q+r-d}}{O_r...O_1O_0O_q}\sigma_q(M_q)\,.
\end{equation}
We note that the $r$-planes $L_r$ are unoriented \footnote{This convention is the same as \cite{huang,santal}, but different from \cite{czech1}. The difference leads to a factor $2$ in the Crofton's formula.}. The volume element $\epsilon_{\CK }$ of the kinematic space $\CK$ measures the number density of the $r$-planes. The symbols $\sigma_q(M_q)$ and $\sigma_{q+r-d}(M_q\cap L_r)$ denote the volumes of $n$ and $q+r-d$ dimensional objects, the latter of
which is the cross section between $M_q$ and $L_r$. The numeric factors $O_k$
are the area of $k$ dimensional unit-sphere,
\[O_k=\frac{2\pi^{\frac{k+1}{2}}}{\Gamma(\frac{k+1}{2})}.\]

In this paper, we only consider the kinematic space of geodesics (that is $r=1$) and in this case eq.\eqref{gcf1} reduces to
\be
\label{croftonq1}
\int_{M_{d-1}\cap L_1\neq0} N(M_{d-1}\cap L_1) \epsilon_{\CK }=\frac{O_d}{O_1}\sigma_{d-1}(M_{d-1})\,.
\ee
where $ N(M_{d-1}\cap L_1) \equiv \sigma_{0}(M_{d-1}\cap L_1)$ is the number of intersection points $M_{d-1}\cap L_1$. There are two special cases, $d=2$ and $d=3$, respectively,
\begin{equation}\label{gcf2}
\begin{split}
\left\{
  \begin{array}{ll}
    \sigma_1(M_1)=\frac{1}{2}\int_{M_1\cap L_1\neq0} N(M_1\cap L_1) \epsilon_{\CK }& \hbox{(d=2,r=1,q=1),} \\
    \sigma_2(M_2)=\frac{1}{\pi}\int_{M_2\cap L_1\neq0} N(M_2\cap L_1) \epsilon_{\CK }& \hbox{(d=3,r=1,q=2)}\,.
  \end{array}
\right.
\end{split}
\end{equation}

 One merit of this choice \footnote{Such a choice also extend the Crofton's formula to general Riemannian surface but this is irrelevant in
the current context.} is that the measure is always given by the
second derivative of the lengths of geodesics even in the absence of maximal
symmetry,
\be
\label{generalmeasure}
\xe_\CK = \det \lsb \frac {\p^2 L(\vec x_1, \vec x_2) }{\p \vec x_1 \p \vec x_2}\rsb \prod_{i=1}^{d-1} \rmd x^i_2 \wedge \rmd x^i_1  \,,
\ee
where we still use $L$ to denote the length of a geodesic (even though it no longer has any connection with entanglement entropy) and a geodesic is parameterized
using its coordinates $(\vec x_1, \vec x_2)$ ($x_{1,2}^i,\; i=1,\dots d-1$) of the end points.

\section{Integral geometry in the Lorentzian $\adst$}\label{ig on m}
In this section, we will study the  Crofton's formula on Lorentzian $\adst$. We will stick with the geodesics as the probe (\ie $r=1$) but now they are
no longer restricted to a time slice. One subtlety about the Lorentzian space is that there are three different types of geodesics (time-like, space-like and null). Moreover, the time-like geodesics never hit the boundary twice and they usually have no known information theoretic meaning in the boundary theory (neither are the null geodesics). Fortunately, it turns out that space-like geodesics are enough to see the space-like
2-surface and we have the following Crofton's formula similar to eq.(\ref{gcf2}) \begin{equation}\label{scf}
    \sigma_2(M_2)=\xk\int_{M^2\cap LS_1\neq0} \sigma_{0}(M^2\cap LS_1) \epsilon_{\CK},
\end{equation}
where $LS_1$ denote space-like geodesics and $M_2$ is a space-like 2-surface
and $\xk$ is a numeric factor to be determined later. In the remainder of this section, we will prove this formula.

\subsection{Geodesics}
To prove eq.(\ref{scf}) it is necessary to find all  geodesics passing through a given surface $M_2$. In Poincare coordinate \er{metric}, the parametrization of a geodesic from one boundary point $(z_1,\bar{z}_1,0)$ to another $(z_2,\bar{z}_2,0)$ is \cite{wittendiagram}
\begin{equation}\label{z}
z(\lambda)=\frac{z_1+z_2}{2}+\frac{z_1-z_2}{2}\tanh{\lambda},
\end{equation}
\begin{equation}\label{zbar}
\bar{z}(\lambda)=\frac{\bar{z}_{1}+\bar{z}_{2}}{2}+\frac{\bar{z}_{1}-\bar{z}_{2}}{2}\tanh{\lambda},
\end{equation}
\begin{equation}\label{u}
u(\lambda)=\frac{\sqrt{(z_1-z_2)(\bar{z}_{1}-\bar{z}_{2})}}{2\cosh{\lambda}},
\end{equation}
where $(z,\bar{z},u)$ denote a point along the geodesic and $\lambda$ is a parameter ranging from negative infinity to positive infinity.

A geodesic can provide a nonzero contribution to the integral when it hits  the target object $M_2$, which can be parameterized by the function $\bar{z}=\bar{z}(z,u)$. A bulk point $(z,\bar{z},u)$ on $M_2$ and a boundary point $(z_1,\bar{z}_1,0)$ determine the other
\begin{equation}\label{coor1}
  \begin{split}
    &z_2=z+\frac{u^2}{\bar{z}-\bar{z}_1},\\
   &\bar{z}_2=\bar{z}+\frac{u^2}{z-z_1}.
  \end{split}
\end{equation}
So we can instead use $(z,u,z_1,\bar z_1)$ to denote the geodesics, all of
which are space-like and therefore, we have
\begin{equation}\label{spacelike}
  \begin{split}
    &u^2+(z-z_1)(\bar{z}-\bar{z_1})>0,\\
 \zt{or equivalently}\quad & u^2+(z-x_1-t_1)(\bar{z}-x_1+t_1)>0,
  \end{split}
\end{equation}
Now we consider the target surface. Expressing the derivative of $\bar{z}$ with respect to $z$ and $u$ as
\[A = \frac {\p \bar z}{\p z},\quad B = \frac {\p \bar z}{\p u}\,,\] the space-like constraint of the surface requires that the normal dual vector $(A,-1,B)$ is time-like, that is
$$-4A+B^2<0.$$
\subsection{The Crofton's formula on the $\ads_3$}
We have already calculated the measure of kinematic space of Lorentzian $\ads_3$ in the sec~\ref{ssbks2}, which gives the following integral from eq.\eqref{scf},
\begin{equation}\label{3adsscf1}
\tilde \sigma_2(M_2)=\xk\int \rmd z_2\int \rmd\bar{z}_2\int \rmd z_1\int \rmd\bar{z}_1 \;\frac{\sigma_0(LS_1\cap M_2)}{(z_1-z_2)^2(\bar{z}_1-\bar{z}_2)^2}.
\end{equation}
Here we use the notation $\tilde \sigma_2(M_2)$ to denote the integral and eventually we will see that it is equal to the area of $M_2$. Under coordinate transformation (\ref{coor1}), the right hand side becomes
\begin{equation}\label{3adsscf2}
\begin{split}
 \tilde \sigma_2(M_2)
=\xk\int \rmd z\int \rmd u\int \rmd z_1\int \rmd\bar{z}_{1}\,
\frac{1}{2}\left|\frac{B[u^2-(z-z_1)(\bar{z}-\bar{z}_1)]+2u[A(z-z_1)-(\bar{z}-\bar{z}_1)]}{[u^2+(z-z_1)(\bar{z}-\bar{z_1})]^3}\right|
\end{split}
\end{equation}
The range of the parameters $z,u$ is determined by the surface $M_2$. Given $z$ and $u$, $(z_1, \bar{z}_1, 0)$ take all points satisfying eq.(\ref{spacelike}). It is noteworthy that we compute all the geodesics twice in the case, therefore the eq.(\ref{3adsscf2}) contains a factor $\frac{1}{2}$. The expression in eq.(\ref{3adsscf2}) only depends on the relative position of the bulk and boundary points, and therefore, we introduce the new coordinates $x,t$
\begin{equation}\label{zz}
\begin{split}
 &\hat{z}=z_1-z=x'+t' \\
 &\check{z}=\bar{z}_1-\bar{z}=x'-t'
\end{split}
\end{equation}
which satisfies
$$u^2+x'^2-t'^2>0.$$
After a boost transformation in $x',t'$, one gets
\begin{equation}\label{3adsscf3}
\begin{split}
  &\tilde \sigma_2(M_2)\\
=&\xk\int \rmd z\int \rmd u\int \rmd x'\int \rmd t' \frac{|B(u^2-x'^2+t'^2)-2ut'\sqrt{4A}|}{(u^2+x'^2-t'^2)^3}\\
=&\xk\int \rmd z\int \rmd u\int \rmd x'\int \rmd t' \frac{\sqrt{4A-B^2}|\sinh\xi(u^2-x'^2+t'^2)-2ut'\cosh\xi|}{(u^2+x'^2-t'^2)^3},
\end{split}
\end{equation}
where
\begin{equation}\label{xi}
  \begin{split}
\sinh\xi=\frac{B}{\sqrt{4A-B^2}}\,,\quad\cosh\xi=\frac{\sqrt{4A}}{\sqrt{4A-B^2}}.
  \end{split}
\end{equation}
To take care of the absolute value, we have to go to the angular coordinates
\begin{equation}\label{coor2}
  \begin{split}
&\sin\beta\sinh\alpha=\frac{2ut'}{u^2+x'^2-t'^2},\\
&\sin\beta\cosh\alpha=\frac{u^2-x'^2+t'^2}{u^2+x'^2-t'^2},\\
&\cos\beta=\frac{2ux'}{u^2+x'^2-t'^2}.
  \end{split}
\end{equation}
Physically, we pick three unit vectors $\hat x, \hat t, \hat u$ (vielbeins) along $\p_x, \p_t, \p_u$ and $\xa, \xb$ are the angles between the unit tangent vector $v$ of the geodesic and the vielbeins. More precisely, $\xb$ is the angle with $\hat x$ and $\xa$ is the (hyperbolic) angle between $v-(v \cdot \hat x) \hat x$ and $\hat u$. The Jacobian then reads
\begin{equation}\label{J2}
  \left|\frac{\partial(x',t')}{\partial(\alpha,\beta)}\right|=\frac{u^2|\sin\beta|}{(1+\cosh\alpha\sin\beta)^2}\,,
\end{equation}
and eq.(\ref{3adsscf3}) becomes
\begin{equation}\label{3adsscf4}
\begin{split}
&\tilde \sigma_2(M_2)\\
=&\xk\int \rmd z\int \rmd u\int_0^{2\pi} \rmd\beta\int_{-\infty}^{+\infty}\rmd\alpha \frac{\sqrt{4A-B^2}}{4 u^2}|\sin\beta||\sin\beta\sinh(\xi-\alpha)|\\
=&\xk\int \rmd z\int \rmd u \frac{\sqrt{4A-B^2}}{4u^2}(\cosh\alpha|^{-\frac{1}{\chi}}_0+\cosh\alpha|^{+\frac{1}{\chi}}_0)\\
=&\xk\int\rmd z\int \rmd u \frac{\sqrt{4A-B^2}}{2u^2}\lb \cosh\frac{1}{\chi}-1\rb,
\end{split}
\end{equation}
where $\chi$ is a cutoff for $\alpha$. We note that $|\sin \xb| \rmd \xb \rmd \xa$ is the volume element of the solid angle and $\sin \xb \sinh(\xi-\alpha)$ is the inner product between $v$ and the normal vector $\hat n$ of the surface, which implies this integral should be independent of the $\hat n$. Practically the parameter $\xi$ drops out after a shift in $\xa$ (which is equivalent to choosing new vielbeins with the normal vector as $\hat t$).

As a quick consistent check, we can perform the same integral in the Euclidean space. With $\hat n$ being one the of axes, the integral ($\xt$ being the angle with $\hat n$)
\be
\frac 1 2 \int |\cos \xt| \rmd \xO_{d-1} = \int_0^{\frac \pi 2} \cos \xt \sin^{d-2} \xt \rmd \xt \rmd \xO_{d-2} = \frac {O_d}{O_1} \,,
\ee
gives precisely the numeric factor on the right hand side of eq.\eqref{croftonq1}.

We can now compare the final result \eqref{3adsscf4} with the area of $M_2$. From the induced line element
\begin{equation}\label{induced line element}
  \rmd s^2=\frac{A\rmd z^2+B\rmd z\rmd u+\rmd u^2}{u^2},
\end{equation}
one may get
\begin{equation}\label{area}
\sigma_2(M_2)=\int \rmd z\int \rmd u \frac{\sqrt{4A-B^2}}{2u^2}\,,
\end{equation}
which agrees with (\ref{3adsscf4}) up to an infinite factor, which is canceled by $\xk$
\be
\label{infinitefactor}
\xk^{-1} = \h \int_0^{2\pi} \rmd\beta\int_{-\infty}^{+\infty}\rmd\alpha \sin^2\beta|\sinh(\alpha)| = \cosh\frac{1}{\chi}-1\,.
\ee
As a result, with the CV assumption complexity (whether it is that of a pure state or the reduced density matrix of a subregion \cite{Alishahiha:2015rta}) can be expressed in terms of the number of geodesics.

\section{Discussions}
\label{sec:discussion}
We examined the precise form of the Crofton's formula in the Lorentzian AdS$_3$ and showed that the area of a space-like two dimensional surface is given by the flux of space-like geodesics. Based on the validity of the Crofton's formula in general Euclidean AdS, we expect the same conclusion to hold for space-like codimension-one surfaces in higher dimensional Lorentzian asymptotically AdS spaces. In AdS$_3$, the geodesics have entropic interpretation and hence it is tempting to think that this conclusion may provide an information theoretic explanation of the CV conjecture. We would like to share some of our observations in that regard, leaving the more complete analysis to future study. For simplicity, we only consider the complexity of a pure state.

It was proposed in \cite{huang} that one can heuristically associate every geodesic with a Bell pair located at the two end points on the boundary. By no means this naive picture captures all the physics as the entanglement structure is not entirely bipartite. It does however offer a very nice interpretation of the entanglement entropy (of a single interval) as counting the number of Bell pairs crossing the entangling surface. Moreover, the length of a convex bulk curve (\ie differential entropy) can also be understood as the amount of long-range entanglement in this framework.

We find this picture also very illuminating in the current context and hence decide to stay with it in subsequent discussions despite its apparent flaw. To avoid the issue of Bell pairs, one can simply take the geodesic density as a type of measure of the two-point entanglement. Based on this picture, it was pointed out \cite{huang} that under renormalization group flow, the short-range entanglement is removed while the long-range entanglement is reshuffled to shorter scales. These two operations are the ``geodesic'' versions of disentangler and isometry in MERA. Complexity counts the total number of these two operations. From the perspective of one geodesic, it contributes one removing (disentangler) operation or one reshuffling (isometry) operation for each step of RG, with the total number proportional to the length of the geodesic (see Fig.\ref{complexityfig1}). Consequently, the complexity is given by counting the total number of geodesics weighted by the length of each, which is precisely the area of the codimension-one bulk surface according to the Crofton's formula applied to the constant time slice $\BH^2$ alone \cite{Abt:2018ywl}.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=8cm]{pic1}\\
  \caption{Contribution from a certain geodesic to complexity. Along the RG flow (depicted as horizontal straight-lines), a certain number of isometries (reshuffling operation) contribute to the complexity before the disentangler (removing operation) gets rid of the geodesic entirely. For simplicity we can consider the discrete case and assume that the number of sites is reduced by half at each step of RG and precisely one isometry is counted. A geodesic with $|x_2- x_1| = \ell$ can undergo $\log_2 \ell \sim \log \ell$ steps and hence the total number of gates is roughly the length of the geodesic.}\label{complexityfig1}
\end{figure}

The Crofton's formula in AdS$_3$ can reproduce not only the complexity (area of the corresponding optimized surface) of a state but also the cost of a non-optimized circuit, which may help to understand the CV conjecture from the viewpoint of information theory. It is however very unfortunate that henceforth we have to restrict ourselves to the Euclidean AdS. It is unlikely that the area of a generic codimension-one surface gives the cost of the circuit in the Lorentzian case. Obviously that is not the case for a time-like or null-like surface. Moreover, in the Lorentzian case it is usually the maximal surface that corresponds to the optimized circuit and gives the complexity. The infinite factor \er{infinitefactor} between the flux of geodesics and the area makes it difficult to connect the former to any information theoretic interpretation. Nevertheless, we still hope the subsequent discussions in the Euclidean case may shed some lights on the Lorentzian problem that we eventually have to tackle.

In the Euclidean case, the correspondence between circuit and co-dimension one surface, combined with the CV conjecture implies that the cost of a circuit is measured by the area of the surface, which in turn follows from the flux of geodesics via Crofton's formula. It is not clear to us why the number density of geodesic actually accounts for the cost. The good news is that the former does follow from entanglement entropy associated with the quantum circuit, which was computed in \cite{Caputa:2018xuf}. The conformal factors $e^{2\phi}$ at the end points provide corrections $c/6\, \phi$ to the entanglement entropy of a single interval
\be
S(x_1,x_2) = \frac c 3 \log |x_1 - x_2| + \frac c 6 \lsb \phi (x_1) + \phi (x_2)\rsb\,,
\ee
where $x_1, x_2$ are the coordinates of the end points. Let us consider a simple example of the entanglement entropy at $t = \mu$, \ie that of the excited state corresponding to the bulk curve on the $t = 0$ time slice specified by $u = \mu$. The change in the conformal factor from $e^{2\phi(t=\xe, x)} = \frac 1 {\xe^2}$ to $e^{2\phi(t=\mu, x)} = \frac 1 {\mu^2}$ implies that the entanglement entropy is given by $\frac c 3 \log \lb \ell/\mu \rb$ for an interval of length $\ell$, which agrees with the length of a geodesic on the new cutoff surface. We would like to remind the reader that the number density is obtained from the length on geodesics ending on new surface $M[\xS(0)]$ (see \eg Fig.\ref{complexityfig2}(a)), which is guaranteed by the nontrivial fact that the measure in the kinematic space of geodesics always follows from eq.\eqref{generalmeasure} even in a general space without any symmetry. The same conclusion does not necessarily hold for the probes of higher dimensions.

It is very clear that the flux of geodesics depends on the circuit/surface. Graphically, we know that the optimized surface (for vacuum state at $t=0$) receives no contribution from geodesics with both end points in the $t < 0$ region. Instead, every circuit receives contribution from the geodesics connecting the $t>0$ and $t<0$ regions (see Fig.\ref{complexityfig2}(a)). Such a circuit independent contribution serves as the lower bound of the cost, which is saturated by the optimized circuit (corresponding to the $t=0$ slice in the bulk, henceforth $M_0$).

We would like to explore the physical meaning of such a contribution. In \cite{Takayanagi:2018pml}, it is shown that the relevant geodesics come from the entanglement entropy between subsystem $AA'$ and $BB'$, with $A,B$ being subsystems on a slice in the $t<0$ region while $A',B'$ being subsystems on a slice in the $t > 0$ region. The quantum state of the total system is obtained from the mapping given by the path integral with the two slices as the boundaries. More precisely, a mapping like $\st{i}A_{ij}\br j|$ leads to an in general entangled state $\st{i}\st{j} A_{ij}$ by turning bras into kets. Such a practice is common in the study of tensor network. The identity map $\st{i}\br i|$ becomes Bell pairs (more precisely a maximally entangled state) after the move.

\begin{figure}[tbp]
\begin{minipage}[t]{0.5\textwidth}
\includegraphics[scale=0.7]{pic3}
\end{minipage}
\begin{minipage}[t]{0.5\textwidth}
\includegraphics[scale=0.75]{pic2}
\end{minipage}
\caption{(a) A geodesic (pink) with two end points on different sides of $t=0$ always contributes to the cost of the circuit ($M_1, M_2$) while the one (blue) with both end points on the same side does not. We also note that the number density of geodesics can be obtained from the second derivative of the lengths of geodesics between two surfaces $M_1 ,M_2$. (b) A tilted plane is a less optimized circuit and it can be decomposed as the combination of an optimized circuit $s_1$ at time $t=-\xd t$ and a circuit $s_2$ for the time evolution back to $t=0$.}\label{complexityfig2}
\end{figure}

\begin{figure}[htbp]
  \centering
  \includegraphics[width=8cm]{fig3}\\
  \caption{To get a better look at the circuit-dependent flux, we go down to EAdS$_2$. The geometry in the Poincare patch discussed earlier is realized in the region to the right of the dashed line. The optimized circuit is a semi-circle (blue) while the less optimized one is the red curve sharing the same boundary. The green semi-circle (optimized circuit) serves as $s_1$. Geodesics that hit the red but not the blue one are those between $s_1$ and $s_2$.}\label{complexityfig3}
\end{figure}

For better demonstration, let us assume the system is discrete and the state takes the form of a tensor network obtained from RG flow (realized as mappings between various Hilbert spaces with different dimensions, see \eg \cite{vidal2009entanglement} for a review). The PI integral then becomes the mapping between two slices $t_2=-t_1>0$, which is given by $W^\dagger(0, t_1) W(0, t_1)$ with $W(0, t_1)$ being the mapping of RG flow. It is known that such a product should be proportional to identity $W^\dagger(0, t_1) W(0, t_1) = \xa I$ \footnote{$W$ is a map between Hilbert spaces at different energy scales and therefore we need $\xa$ to take into account the difference in dimensions.} and hence there are Bell pairs between the subsystems at $t_1$ ($AB$) and $t_2$ ($A'B'$) whose total number gives the entanglement entropy between the two subsystems.

The next step is to consider states constructed from two arbitrary slices $t_1, t_2$ (assuming $0<t_2 < |t_1|$). The unitary $W(0,t_1)$ can be broken up as $W(0,-t_2)W(-t_2, t_1)$. The mapping from $t_1$ to $t_2$ then becomes
\[W^\dagger(0, -t_2) W(0, t_1) = \xa I_{-t_2} \, W(-t_2, t_1)\,.\]
The reduced density matrix for slice $t_1$ is obtained by tracing out the system at $-t_2$
\[\rho_{t_1} \sim W^\dagger (-t_2, t_1) W(-t_2, t_1)  \sim I_{t_1}\,,\]
which is equal to the identity operator again after appropriate normalization and hence the total entanglement entropy is determined by the size of the Hilbert space at the scale of $t_1$. The interesting part is that the specific forms of $W(0, t_1)$'s, which lead to new quantum circuits do not affect the total number of Bell pairs \ie the entanglement entropy. To see that we first rewrite the new circuit $W'(0, t_1)$ as the product of (see Fig.~\ref{complexityfig2}(b))
\[W'(0, t_1) = U(\xd t) U^\dagger(\xd t) W(0, t_1) U(\xd t)\,,\]
where $U(\xd t)$ is the time translation by the amount $\xd t$. The mapping to consider is
\[W^\dagger(0, -t_2) W'(0, t_1) = W(-t_2, t_1) U(\xd t)\,,\]
(or $W^\dagger(t_1, -t_2) U(\xd t)$ if $t_2 > |t_1|$) and one can then use the same argument to show the invariance of the entanglement between $t_1$ and $t_2$. The slice $t_2$ is on the surface $M_0$ while the slice $t_1$ is on $M$. In the heuristic picture discussed above a Bell pair turns into a geodesic in the continuous limit and the entanglement is measured by the geodesics with one point on $M_0$ and the other on $M_1$. In fact since $M_1$ is closed and shares the boundary with $M_0$, every geodesic going through $M_0$ must also hit $M_1$. What we learn from the circuit point of view is that the flux of these geodesics corresponds to the circuit-independent entanglement. For comparison, we can also take a look at the circuit-dependent contribution. As we can see from Fig.~\ref{complexityfig3}, the extra flux follows from the geodesics between $s_2$ and $s_1$, which is the consequence of the additional piece $s_2$ corresponding to the circuit $U(\xd t)$.

Despite only a hand-waving argument, it does give a reasonable picture in which the flux of geodesics going through $M_0$ measures the entanglement between the different Euclidean times (or energy scales in the RG sense) and provides the circuit-independent contribution to the cost. This lower bound is saturated when other circuit-dependent contributions all drop out \ie when surface is $M_0$.

\section*{Acknowledgments}
XH is supported by the NWU Starting Grant No.0115/338050048 and the Double First-class University Construction Project of Northwest University.