Planar Black holes and Entanglement Entropy in Analog Gravity Models

TL;DR

This paper constructs explicit analog gravity models for planar black holes and introduces a holographic entanglement entropy concept applicable to a wide class of these models, expanding the known examples of such metrics.

Contribution

It provides a method to realize planar black holes as analog metrics and extends the holographic entanglement entropy framework to more general conformal black-hole spacetimes.

Findings

Explicit Lagrangian for analog black hole metrics

Holographic entanglement entropy for planar black holes

Broader class of known analog metrics

Abstract

Via constructing an explicit Lagrangian for which the perturbation equations are analogues of a scalar field propagating in a planar black hole space-time, it is found that all planar black holes conformal to a Painlev\'e--Gullstrand type line element can be realized as analogue metrics. We also introduce the concept of holographic entanglement entropy for planar black-hole space-times. This is valid for an arbitrary choice of conformal and blackening factor, thereby vastly extending the number of known examples of explicitly known analogue metrics.

Equations

$$\displaystyle ds^{2}=G_{\mu\nu}dx^{\mu}dx^{\nu}=\frac{\Omega(t,x,z)^{2}}{\sqrt{1-\gamma(z)}}\left[-\gamma(z)dt^{2}+\frac{dz^{2}}{\gamma(z)}+d\mbox{\boldmath$x$}^{2}\right],$$

$$\square \phi -m_{\mathrm{eff}}^2\phi \equiv \frac{1}{\sqrt{|G|}}{\partial }_{\mu }\left(\sqrt{|G|}{G}^{\mu \nu }{\partial }_{\nu }\phi \right)-m_{\mathrm{eff}}^2\phi =0.$$

$$\tilde{\square }\tilde{\phi }-{\tilde{m}}_{\mathrm{eff}}^2(t,z,x)\tilde{\phi }=0,$$

$$\displaystyle d\widetilde{s}^{2}=\Omega(t,x,z)^{-2}ds^{2}=\widetilde{G}_{\mu\nu}dx^{\mu}dx^{\nu}=\frac{1}{\sqrt{1-\gamma(z)}}\left[-\gamma(z)dt^{2}+\frac{dz^{2}}{\gamma(z)}+d\mbox{\boldmath$x$}^{2}\right],$$

$${\tilde{m}}_{\mathrm{eff}}^2={\Omega }^2{m}_{\mathrm{eff}}^2+{\Omega }^{1/2-n/2}\tilde{\square }{\Omega }^{n/2-1/2}.$$

$$\displaystyle{\cal L}=F(\chi)-V(\theta,t,z,\mbox{\boldmath$x$}),$$

$$\chi =-g^{\mu \nu }{\theta }_{,\mu }{\theta }_{,\nu }.$$

$$T_{\mu \nu }=2{\mathcal{L}}_{\chi }{\theta }_{,\mu }{\theta }_{,\nu }+\mathcal{L}{g}_{\mu \nu },$$

$$p=\mathcal{L},$$

$$\rho =2\chi {\mathcal{L}}_{\chi }-\mathcal{L},$$

$$u_{\mu }=\frac{{\theta }_{,\mu }}{\sqrt{\chi }}.$$

$${\omega }_{\mu \nu }=h_{\mu }^{\rho }{h}_{\nu }^{\sigma }{u}_{[\rho ;\sigma ]},$$

$$h_{\nu }^{\mu }={\delta }_{\nu }^{\mu }+u^{\mu }{u}_{\nu }$$

$$w{u}_{\mu }={\theta }_{,\mu }$$

$$s_{,\mu }=w^{-1}{u}^{\nu }{s}_{,\nu }{\theta }_{,\mu }.$$

$$w\equiv \frac{p+\rho }{n}=\sqrt{\chi }.$$

$$n=2\sqrt{\chi }{\mathcal{L}}_{\chi }.$$

$$dp=ndw-nTds={\mathcal{L}}_{\chi }d\chi +{\mathcal{L}}_{\theta }d\theta ,$$

$$(2{\mathcal{L}}_{\chi }{g}^{\mu \nu }{\theta }_{,\nu }{)}_{;\mu }+\frac{\partial \mathcal{L}}{\partial \theta }=0.$$

$$(n{u}^{\mu }{)}_{;\mu }$$

$$p\to p+\delta p,n\to n+\delta n,u^{\mu }\to u^{\mu }+\delta u^{\mu },$$

$$\delta w=-u^{\mu }\delta {\theta }_{,\mu },$$

$$w\delta u^{\mu }=(g^{\mu \nu }+u^{\mu }{u}^{\nu })\delta {\theta }_{,\nu }.$$

$${\left(f^{\mu \nu }\delta {\theta }_{,\nu }\right)}_{;\mu }+\left[{\left(\frac{\partial n}{\partial \theta }{u}^{\mu }\right)}_{;\mu }-\left(\frac{{\partial }^{2}V}{\partial {\theta }^2}\right)\right]\delta \theta =0,$$

$$f^{\mu \nu }=\frac{n}{w}\left[g^{\mu \nu }+\left(1-\frac{w}{n}\frac{\partial n}{\partial w}\right)u^{\mu }{u}^{\nu }\right].$$

$$|\tilde{G}{|}^{-1/2}{\left(|\tilde{G}{|}^{1/2}{\tilde{G}}^{\mu \nu }\delta {\theta }_{,\nu }\right)}_{;\mu }-m_{\mathrm{eff}}^2\delta \theta =0,$$

$${\tilde{G}}_{\mu \nu }=\frac{n}{m^{2}wc}\left[g_{\mu \nu }+(1-c^2)u_{\mu }{u}_{\nu }\right]$$

$$\displaystyle c^{2}\equiv\frac{\partial p}{\partial\rho}\Biggr{|}_{s}=\frac{n}{w}\frac{\partial w}{\partial n}\Biggr{|}_{s},$$

$$m^2\sqrt{|\tilde{G}|}{m}_{\mathrm{eff}}^2={\frac{{\partial }^{2}V}{\partial {\theta }^2}|}_{{\theta }_0}.$$

$$m^2\sqrt{|\tilde{G}|}{\tilde{G}}^{\mu \nu }={-\frac{{\partial }^{2}F}{\partial {\theta }_{,\nu }\partial {\theta }_{,\mu }}|}_{{\theta }_0},$$

Full text

Analogue Gravity Models for Planar Black Holes

N.Bilić1, T. Zingg2,3*

1 Division of Theoretical Physics, Rudjer Bošković Institute, 10002 Zagreb, Croatia

2 Nordita, Stockholm University and KTH Royal Institute of Technology

Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden

3 Department of Physics and Helsinki Institute of Physics

P.O.Box 64, FIN-00014 University of Helsinki, Finland

• zingg@nordita.org

March 1, 2024

Abstract

**Via constructing an explicit Lagrangian for which the perturbation equations are analogues of a scalar field propagating in a planar black hole space-time, it is found that all planar black holes conformal to a Painlevé–Gullstrand type line element can be realized as analogue metrics. This is valid for an arbitrary choice of conformal and blackening factor, thereby vastly extending the number of known examples of explicitly known analogue metrics. **

Contents

1. 1 Introduction

2. 2 Conformal Rescaling

3. 3 The Lagrangian

4. 4 Relativistic Acoustic Metric

5. 4.1 The potential

6. 5 Conclusions

1 Introduction

Certain condensed matter systems, respectively the Lagrangians describing these, have a property that small perturbations around a given background are described by the equations of motion of a field propagating in curved space-time. Thus, these systems may serve as ‘analogues’ of phenomena in gravitational physics and could, in principle, be employed to simulate gravity in tabletop experiments. Though already known in theory since the 1980s [1, 2], it is only in recent years that this approach in simulating gravity has attracted more attention, mostly due to new technologies – in particular in dealing with Bose-Einstein condensates (BECs) or cold atom systems – having been developed and making these kinds of experiments more accessible [3, 4, 5, 6, 7, 8, 9].

However, it should be noted that, a priori, not all interesting geometries can be mimicked by analogue geometries. For example, counting the available degrees of freedom, in general relativity (GR) in $3+1$ dimensions we have four degrees of freedom per point in space-time: $10$ independent components of a general metric minus $6$ owing to the $6$ independent Einstein equations. Whereas an analogue metric basically depends on two independent functions, which are the scalar potential $\theta$ that generates the flow velocities and the speed of sound $c$. Thus, the basic analogue gravity setup involving a single scalar field, which is also what we will be considering in the following, can not reproduce all possible metrics that could be derived from GR. However, as additional degrees of freedom enter by coupling to an external potential, which is assumed to be freely tunable, the setup considered is actually more than sufficient to mimic the most important phenomena such as black holes, FRW cosmology, and even some aspects of semiclassical quantum gravity like Hawking radiation.

As these phenomena are all of central importance in gravity physics, it is desirable to extend the class of analog gravity systems to as many metrics as possible. Besides astronomic observations, analog gravity provides the only way to experimentally test such predictions in a lab environment. In this paper, we follow the formalism developed in [10] where it was restricted to a planar AdS5 black hole, and extend the applicability of analog gravity by demonstrating that it can potentially capture all phenomena described by a field propagating in any space-time that is conformal to a rather generic stationary planar black hole, for an arbitrary choice of blackening factor. Our paper provides a substantial generalization and thereby extends the list of examples of planar space-times that have already been found to have an analog dual [11, 12, 13].

The reasons for primarily considering planar black holes is two-fold. First and foremost, we provide a proof of concept on how analogue Lagrangians for a general class of space-times, not just individual, single metrics, can be constructed, thereby tremendously extending the zoo of known analog black hole metrics. The case of generic black hole space-times, not only for specific blackening factors, provides a suitable starting point due to its relevance for phenomena involving an event horizon, which is one of the main research points in analog gravity experiments, and due to previous work on which can be build upon. Secondly, the effects of the curvature of the horizon, e.g. spherical or hyperbolic, is secondary in most situations of practical interest and could, effectively, at any rate be absorbed into a redefinition of the effective mass of the scalar perturbation. Furthermore, experiments that simulate horizon-related phenomena such as the Hawking effect in analogue systems involving the flow of water in a basin, BECs or cold atoms in a trap often employ a linear setup, making planar black holes a more suitable choice for practical purposes.

The paper is organized as follows. In section 2 we define the geometry and its conformally rescaled metric. In section 3 we outline a field theory description of a fluid and derive the propagation equation for acoustic perturbations. The main result follows in section 4, where we show how a generic planar black hole metric can be mapped to the effective geometry of a fluid in which acoustic perturbations propagate. Concluding remarks are given in section 5.

We adopt a convention in which the speed of light and Planck constant $\hbar$ are set to unity, $c$ denotes the speed of sound, and the metric signature is ‘mostly plus’, i.e., $\{-,+,\ldots,+\}$.

2 Conformal Rescaling

For the purpose of being self-contained, we summarize a result from [14], which shows how an additional degree of freedom in the form of a conformal factor can be introduced into an analog metric.

Consider a space-time in $n+1$ dimensions conformal to a rather generic stationary planar black hole metric, which for later convenience we take to be parameterized as

[TABLE]

where the function $\gamma$ is referred to as the ‘blackening factor’. If there is a horizon located at $z=z_{0}$, where $\gamma(z_{0})=0$, the outside region is characterized by $\gamma>0$. A canonical scalar field ${\varphi}$ propagating in this background with effective mass $m_{\rm eff}$ satisfies the equation of motion,

[TABLE]

Via a rescaling111note that we use a slightly different convention than in [14]. $\varphi=\Omega^{\frac{1-n}{2}}\widetilde{\varphi}$, this equation is equivalent to the conformally rescaled equation of motion [15, 14]

[TABLE]

where the rescaled field $\widetilde{\varphi}$ is propagating in the background geometry with conformally rescaled line element

[TABLE]

and effective mass squared

[TABLE]

3 The Lagrangian

We begin this section by introducing the Lagrangian formalism suitable for the general description of nonisentropic fluids. We mostly use notation and definitions established in previous work, see e.g. [10, 16, 17], which is considered fairly standard for this type of systems. Consider a Lagrangian of the form

[TABLE]

where $\theta$ is a dimensionless scalar field, and $F$ is an arbitrary function of the kinetic energy term

[TABLE]

The energy-momentum tensor corresponding to (6) is

[TABLE]

where the subscript $\chi$ denotes a partial derivative with respect to $\chi$. For $\chi>0$, this energy-momentum tensor will describe a perfect fluid if we identify the pressure and energy density as

[TABLE]

[TABLE]

and the fluid velocity vector as

[TABLE]

This equation describes the so called ‘potential flow’. Solutions of this form are the relativistic analogue of potential flow in non-relativistic fluid dynamics [18] and are usually ascribed to isentropic and irrotational flows. Isentropic flow is characterized by vanishing of the gradient $s_{,\mu}=0$, with $s$ being the specific entropy, i.e. the entropy per particle. In general, a flow may be nonisentropic and have a non-vanishing vorticity $\omega_{\mu\nu}$ defined as

[TABLE]

where

[TABLE]

is the projector which projects an arbitrary vector in space-time into its component in the subspace orthogonal to $u^{\mu}$. If the conditions of isentropy and vanishing vorticity are assumed the velocity field may be expressed by

[TABLE]

where $\theta$ is the velocity potential and $w$ is the specific enthalpy. The reverse of the above statement is not true: a potential flow alone implies only vanishing vorticity and implies neither isentropy nor particle number conservation. In a potential flow, as may be easily shown [10], the entropy gradient is proportional to the gradient of the potential, i.e.,

[TABLE]

The assumption (14) is equivalent to (11) if we identify

[TABLE]

Hence, the potential flow is automatically satisfied in the field-theory formalism with a scalar field $\theta$ playing the role of the velocity potential. Furthermore, in view of (16) with (9) and (10) we identify the particle number density as

[TABLE]

This is consistent with the Gibbs relation

[TABLE]

when a functional relationship $s=s(\theta)$ is assumed.

Thus, we have constructed a field theory description of a fluid. Following [10] the ideal irrotational fluid will satisfy the Euler equation – i.e. the energy momentum conservation – if, in addition to the potential flow equation (11), the field satisfies the equation of motion

[TABLE]

Using (11) and (17)) this equation can be written as

[TABLE]

Next, we briefly describe the derivation of the propagation equation for linear perturbations of a nonisentropic flow assuming a fixed background geometry. Given some average bulk motion represented by $p$, $n$, and $u^{\mu}$, following the standard procedure [19] we make a replacement

[TABLE]

where the perturbations $\delta p$, $\delta n$, and $\delta u^{\mu}$ are induced by a small perturbation $\theta=\theta_{0}+\delta\theta$, around the background $\theta_{0}$. From (14) we find

[TABLE]

[TABLE]

Using this and (21) equation (20) at linear order yields

[TABLE]

where

[TABLE]

Then, it may be easily shown that equation (24) can be recast into the form

[TABLE]

where $\widetilde{G}^{\mu\nu}$ is the inverse of the relativistic acoustic metric [16]

[TABLE]

with determinant $\widetilde{G}$. Here $m$ is an arbitrary mass parameter introduced to make the metric dimensionless, $c$ is the speed of sound defined by

[TABLE]

and $m_{\rm eff}$ is the effective mass defined by

[TABLE]

Using (6)-(11) one can derive the relation

[TABLE]

which has also been derived by Babichev et al [17] in a different context.

4 Relativistic Acoustic Metric

Building up on [10] we now proceed to show that an acoustic perturbation in a fluid – the dynamics of which is described by an explicit field theory Lagrangian – can be realized as a scalar field propagating in the background (4). This extends the procedure developed in [10], which was restricted to a planar AdS5 black hole with

[TABLE]

whereas here we will show that the formalism can be generalized to simulate metrics of the form (4) with an arbitrary blackening factor $\gamma(z)$. In particular, we will show that an acoustic perturbation propagating in a fluid described by the Lagrangian of the form (6) represents an analogue dual of a scalar field propagating in the background (4). In other words, if a fluid is described by the Lagrangian (6), the dynamics of acoustic perturbations described by (26-29) will have the form of the Klein-Gordon equation (3) in a curved space-time described by the line element (4).

The first step is to bring the metric (4) to a form that can be compared with the acoustic metric (27). For this purpose, consider a simple coordinate transformation,

[TABLE]

such that the line element from (4) takes the form

[TABLE]

where

[TABLE]

Comparing with (27) allows to read off the non-vanishing components of the 4-velocity

[TABLE]

Next, assuming a potential flow (14) we derive closed expressions for $w$, $n$, and $c$ in terms of the variable $\widetilde{z}$. Since the metric is stationary, the velocity potential must be of the form

[TABLE]

where $m$ is an arbitrary mass and $h(z)$ is a function of $\widetilde{z}$ through $z=g(\widetilde{z})$. The specific enthalpy is then given by

[TABLE]

and the function $h(z)$ is determined through

[TABLE]

Furthermore, in view of (27) and  (33) the particle number density is

[TABLE]

From the definition (28) it follows

[TABLE]

This implies that the sound speed $c$ must satisfy a differential equation

[TABLE]

with solution

[TABLE]

Then we find

[TABLE]

[TABLE]

In principle, $c_{1}$ could be an arbitrary function of $s$. However, since $w$ and $s$ are considered as independent variables, the right hand side of (43) admits no explicit $s$-dependence and hence a consistent choice is $c_{1}\equiv\rm const$. From (43) and (44) it follows

[TABLE]

Then, according to (18) the pressure reads

[TABLE]

where $c_{2}(s)$ is an arbitrary function of $s$. In view of (43) the pressure can also be expressed as

[TABLE]

This expression is precisely of the form (6) in which

[TABLE]

$c_{2}$ is identified with $V$ and the specific enthalpy with $\sqrt{\chi}$ as in (16).

Therefore, we have shown that the Lagrangian (6) with (48) can be used to construct an analogue model for a scalar field propagating in the metric (4) with an arbitrary $\gamma(z)$. It is worth noting that our Lagrangian has the same functional dependence on $\chi$ as the one found in [10] where it was derived from the requirement that the analogue metric correctly reproduces the planar AdS5 black hole. Hence, the functional form (48) is generic. However, the fluid dynamics is not completely determined unless the potential $V(\theta)$ is specified because the flow velocity components are fully determined by the velocity potential $\theta$ which solves the field equations. To find a solution to the field equations we need to specify the potential $V(\theta)$ which will be done in the following section.

4.1 The potential

Recall that we are considering a scalar field $\theta=\theta_{0}+\delta\theta$, i.e. a small acoustic perturbation $\delta\theta$ around a fixed background $\theta_{0}$. If the equation of motion of this perturbation is to consistently be an analog model of a particle propagating in the curved space-time, the potential $V$ has to meet a requirement that its first derivative, when evaluated on the background (36), is determined by the equation of motion (20).

In applications where one wishes to not only simulate a specific metric but also a specific effective mass222such as e.g. in [20], then (29) demands to also impose conditions on the second derivative of $V$. Thus, the potential $V$ has to be chosen such that

[TABLE]

In principle, one could satisfy these conditions in many ways. Quite generally, a suitable potential can be written as

[TABLE]

where $f_{1}(x)$ and $f_{2}(x)$ are arbitrary functions which at $x=1$ (i.e., when $\theta=\theta_{0}$) satisfy

[TABLE]

and $\alpha(\widetilde{z}),\beta(\widetilde{z})$ are chosen to match (49). Therefore, the potential $V$ will generally have to be chosen coordinate dependent. This would present no real obstacle from a practical point of view, as experimental setups for analogue gravity with moving and oscillating horizons are already being conducted, e.g. [5, 6], and time and position dependent external potentials could be simulated with the same setup.

From a theoretical point of view, there could be some caveat that limits the choice of potentials. That comes from the condition that the Gibbs relation (18) must hold. At first sight it may seem a bit odd how the relation containing only two degrees of freedom could be satisfied with a generic potential V(\theta,t,z,\mbox{\boldmathx}). However, one has to keep in mind that the functional identities in section 3 are independent of the specific coordinate dependence of the potential and the crucial point is that the Gibbs relation (18) has to hold as an on-shell functional identity. This is to say that it must be possible to express the pressure $p$ as a functional depending on two variables $w$ and $s$ which are defined on the function space of solutions to the equations of motion. This reduces the effective number of degrees of freedom.333Note that the Gibbs relation need not hold for a generic field that does not satisfy the equations of motion. In practice, however, it could be rather non-trivial to check (18) explicitly, and following construction might be more convenient.

Assume a Lagrangian with no explicit coordinate dependence of the form

[TABLE]

The Gibbs relation (18) is now automatically satisfied and for a solution $\theta_{0}$ of the equations of motion (20) the analogue metric and effective mass for a perturbation follow from (30,29). In this situation one can proceed to construct a potential $V$ that reproduces the desired analog metric in a way similar to [10], where it has been worked out for the case of a planar black hole in AdS space-time. In order to then explicitly match the effective mass to a desired value, consider the Lagrangian (52) changed by an ${\cal O}(\theta-\theta_{0})^{2}$ deformation around the found background solution $\theta_{0}$, e.g.

[TABLE]

By construction, $\theta_{0}$ is still a solution to the equations of motion and all identities from section 3 will hold identically when evaluated for $\theta_{0}$, with the exception of (29), which, as only quantity in the perturbation equations, depends on second order derivatives of the Lagrangian with respect to $\theta$. Thus, the effective mass changes to

[TABLE]

Therefore, by a suitable choice of $a(\theta,t,x,z)$, any $m^{\prime}_{\rm eff}$ can be reproduced without changing the analog metric.

What of course has to be maintained is that (52) has to remain an analog model when considering deviations around $\theta_{0}$, including the Gibbs relation (18), which is the most crucial for the analog gravity construction to work. This however follows directly from the theorem of implicit functions, if (52) is an analog model and $\theta_{0}$ is not a degenerate point in the space of solutions.

5 Conclusions

Using the formalism of analogue gravity for the case of nonisentropic fluids from [10], we have shown that by a suitable transformation of variables and choice of parametrization, a Lagrangian of the form (6) is an analogue model for a scalar field propagating in a space-time that is conformal to a static stationary planar black hole space-time. Furthermore, we have also demonstrated how, with a suitable adjustment of the external potential that couples to the analog Lagrangian, it is possible, for any given analog metric, to simulate an arbitrary effective mass for the perturbation. These results are valid for a generic choice of conformal rescaling and blackening factor of the metric, and for arbitrary effective mass of the scalar perturbation. Therefore, the procedure outlined here allows to vastly extend the class of phenomena in gravity physics that can be simulated in condensed matter systems via the analogue gravity formalism.

This class has now been shown to include most non-rotating planar black hole metrics considered in the literature – as well as several cosmological space-times of particular interest. As our emphasis was put on planar black hole geometries, our result also provides new foundations for the surge of investigations on how analog gravity interlinks with gauge/gravity duality444for reviews see e.g. [21, 22] and condensed matter physics in the last years [23, 24, 25, 26, 27, 20], where the type of space-times considered here also plays a central role.

A generalization to geometries with spherical or axial symmetry is possible and relatively straightforward, but will be left for future work.

Acknowledgement

The work of N.B. has been supported by: the European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (KK.01.1.1.06), the H2020 CSA Twinning project No. 692194, “RBI-T-WINNING”, and the ICTP - SEENET-MTP project NT-03 Cosmology - Classical and Quantum Challenges. T.Z. acknowledges the support of the Swedish Research Council (Vetenskapsrådet) via grant No. 2015-04852.