Dynamical black holes and expanding plasmas

Mukund Rangamani; Pau Figueras; Simon F. Ross; Veronika E. Hubeny

REVIEW

Dynamical black holes and expanding plasmas

Pau Figueras, Veronika E. Hubeny, Mukund Rangamani, Simon F. Ross

High Energy Physics - Theory · Submitted 26 Feb 2009 · 31 references

Pith's verdict: Solid, honest paper: the exact conformal-soliton horizon is a genuinely useful example, but the entropy conclusion leans on a choice of entropy the paper never defends.

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CitationMukund Rangamani, Pau Figueras, Simon F. Ross, Veronika E. Hubeny. "Dynamical black holes and expanding plasmas." arXiv:0902.4696 (2009). https://pith.science/paper/XYU22KUC

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Editor's note

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Two things worth knowing. They construct the global event horizon of the Bjorken-flow geometry order by order in the late-time expansion, which closes a real gap left by [22] and speaks to the regularity worry raised in [18,20]. And they give the exact event horizon of the conformal soliton in closed form, showing its area diverges logarithmically while the apparent horizon sits at the BTZ radius with constant area. That second example is the memorable one: a coordinate transformation of a static solution produces two horizons that behave completely differently, and the reason is that the Poincaré patch only sees part of future null infinity. That is a clean, checkable point and it is worth having in the literature. The Bjorken part is careful. The observation that the truncated zeroth-order metric puts the apparent horizon outside the event horizon because the effective stress tensor violates the null energy condition is exactly the kind of honesty you want in a perturbative construction. The CS computation is clean: null geodesics integrated in closed form, the surface verified null by the degeneracy of the induced metric (3.14). Appendix D's argument that an apparent horizon must coincide with a compact Killing horizon for any foliation admitting a complete slice is correct and more general than the example needs. Where it is soft is the jump from 'the event horizon area diverges' to 'entropy tracks the apparent horizon' (§3.3, abstract:s4). The stress-test note is half right. The argument does depend on identifying the relevant entropy with the conformally invariant total thermodynamic entropy; if a Minkowski-patch observer's natural quantity is a patch-dependent entanglement entropy, which the paper itself gestures at via [25], the divergence is not a contradiction. More concretely, the divergence comes from the l→1 generator running out to i0CS, so it looks like a UV divergence, and the paper never checks whether the renormalized area still grows. That is a real gap, not a fatal one. The teleological argument for preferring a quasi-local horizon stands on its own and does not need the divergence. The authors are also upfront in the v2 note that apparent horizons are foliation-dependent and can jump, so the §4 generalization reads as a suggestion, not a theorem. Who should read it: anyone computing horizons in time-dependent duals, or thinking about holographic entropy out of equilibrium. The CS example is a good discussion piece. Recommendation: send it out. The computations are checkable and the interpretive claim is worth a referee's push.

Referee report

Significance

The paper makes a genuinely useful contribution on two fronts. For the Bjorken flow it supplies the explicit global event horizon that the earlier regularity arguments only inferred indirectly, closing a consistency question about the late-time expansion. For the conformal soliton it gives an exact, parameter-free counterexample to the identification of event-horizon area with field-theory entropy: the event horizon is constructed in closed form, its null character is checked by an explicit degeneracy of the induced metric, and the area computation is elementary and reproducible. The distinction drawn between the teleological event horizon and the quasi-local apparent horizon is sharp and physically well motivated, and the paper is appropriately careful to hedge the conclusion as a suggestion rather than a theorem. The Note added in v2, which corrects the horizon terminology to 'dynamical' and 'isolated' horizon, is a welcome clarification. The result is falsifiable in the sense that the geodesic integrals and area formulae can be checked directly, and they are.

Major objections (3)

[§3.3 and abstract (abstract:s3, abstract:s4)] The inference from 'the event horizon area diverges while the apparent horizon area is constant' to 'the entropy of the CFT state is related to the apparent horizon' is drawn for the total thermodynamic entropy of the dual fluid, which the manuscript correctly notes is conformally invariant and therefore constant for a shear-free flow. The manuscript should state this scope explicitly. A Minkowski-patch observer's natural entropy-like quantity could instead be a patch-dependent entanglement entropy, for which a logarithmically growing area is not by itself a contradiction; the manuscript already cites [25] in this connection but does not discuss the alternative. This does not undermine the derivation, but the headline claim would be more robust if §3.3 and the abstract said plainly that the argument concerns total thermodynamic entropy and that a subregion entanglement entropy could behave differently.
[§3.3, Eqs. (3.15)-(3.17), and footnote 14] The abstract's unqualified statement that 'the area of the event horizon diverges' overstates what is slicing-independent. The divergence in Eq. (3.15) arises as l_max approaches 1, i.e. from the generator that reaches the boundary point i0CS, so it is a boundary-touching (UV-type) divergence, and footnote 14 concedes that the location of the divergence at t = 0 is 'really an artifact of our slicing.' The robust statements are that the event-horizon cross-sectional area always exceeds 2 pi r+ and grows without bound along the horizon; the precise time at which the divergence appears is not robust. The manuscript should separate these two levels of statement in §3.3 and in the abstract, so that the sharp event/apparent contrast is not read as a slicing-independent fact about a particular instant.
[§2.4, Eq. (2.19) and Eq. (2.21)] The event horizon is defined as the outermost solution of Eq. (2.19) that does not reach r = infinity at finite tau, but the manuscript then identifies the late-time series solution, Eq. (2.21), with that global object without demonstrating that the series has the requisite boundedness property or that it is the outermost such solution. Since the regularity claim for the Bjorken flow rests on this identification, a short argument (for example, uniqueness of the bounded solution at each order, or a matching of the series to the asymptotic behaviour of the full geodesic equation) would put the conclusion on firmer footing. The parallel discussion of the artificial metric g(0) in the 'curiosity at leading order' paragraph is helpful and could serve as the template.

Minor objections (8)

[Appendix A title] The appendix title reads 'Conformal solitons in higher dimensionss'; the trailing 's' should be removed.
[Fig. 2 caption] 'surperposed' should read 'superposed'.
[§3.2] 'Minksowski' should read 'Minkowski'.
[§2.4] 'co-homogeneity two' should read 'cohomogeneity two'.
[§2.2, Eqs. (2.7) and (2.11)] In Eq. (2.7) the dy^2 coefficient contains the factor (1 + 1/(u tau^{2/3}))^2, whereas in Eq. (2.11) the corresponding factor appears unsquared; please check which is intended.
[§2.3, Eq. (2.15)] The volume element mu = r^3 tau e^b (1 + 1/(tau r)) does not obviously match the metric ansatz of Eq. (2.7), whose natural combination is 1 + 1/(u tau^{2/3}); please verify this expression, since the apparent-horizon coefficients in Eq. (2.18) depend on it.
[Appendix D] The statement that the Killing horizon generators have zero expansion is 'left as an exercise for the reader'; a one-line argument would make the appendix self-contained.
[Throughout] The spelling of 'Poincaré' is inconsistent ('Poincare', 'Poincar´ e'); please standardize, and likewise the notation for future null infinity.

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The paper

The paper claims

For a plasma expanding in a time-dependent spacetime, the entropy is carried by the apparent horizon, not the event horizon.

The reading

The paper asks which geometric surface in a holographic bulk spacetime should be read as the entropy of the dual field theory when the dual state never settles down to equilibrium. It studies two examples: the boost-invariant Bjorken flow, whose bulk dual is known only as a late-proper-time expansion, and the conformal soliton flow, whose bulk dual is an exact Poincaré patch of Schwarzschild-AdS (taken in 2+1 dimensions as BTZ). In the soliton case the two horizons separate dramatically: the event horizon, defined as the boundary of the past of the future infinity visible to a Minkowski-patch observer, has an area that grows without bound at late Poincaré times, while the apparent horizon sits on the global BTZ horizon $r=r_+$ with constant area. Since the dual fluid is shear-free and produces no entropy at all, a diverging horizon area cannot be the field-theory entropy. The authors conclude that entropy should be associated with the apparent (quasi-local) horizon rather than the teleological event horizon, which requires knowing the entire future of the spacetime.

What carries the argument

The load-bearing construction is the definition of the event horizon as the boundary of the causal past of the appropriate future null infinity, computed by tracing the null geodesic congruence that ends on a chosen boundary point. For the conformal soliton the congruence is labelled by the angular momentum parameter $l$ and the horizon is the surface $(r(λ,l), τ(λ,l), φ(λ,l))$; its induced metric is degenerate, as a null surface must be, and its cross-sections are cut off by a caustic line at $φ=π$ where the area integral $A = 2∫_0^{l_{max}} dl/(1-l^2) = 2$ artanh$l_{max}$ runs out. For the Bjorken flow the same logic reduces to a single ordinary differential equation $d r/dτ = r^2 a(r,τ)/2$ for radial null rays in the $(r,τ)$ plane, solved in a $τ^{-2/3}$ expansion. The apparent horizon is located by the vanishing of the invariant product $Θ = θ_+θ_-$ of the expansions of outgoing and ingoing null normals.

Extended reading notes

Core claim

The central claim is that in strongly time-dependent holographic duals the field-theory entropy is captured by the apparent horizon, not the event horizon. The evidence is the conformal soliton: a global Schwarzschild-AdS black hole rewritten in Poincaré coordinates, which describes a finite lump of fluid that collapses and re-expands, and which is related to a static fluid on the Einstein static universe by a conformal transformation. Because the relevant infinity $I^+_{CS}$ is only a subregion of the global future null infinity, the true event horizon of the Poincaré patch lies outside the global horizon and its cross-sectional area diverges logarithmically at late Poincaré times. The apparent horizon, by contrast, coincides with the global BTZ event horizon at $r=r_+$ for any foliation admitting a complete slice of that horizon, so its area is finite and constant. Since the fluid is ideal and dissipation-free, its entropy must be constant, matching the apparent horizon and ruling out the event horizon. The paper also constructs the event horizon of the Bjorken-flow geometry order by order in $1/3$ powers of proper time and confirms it lies outside the apparent horizon at second order, establishing regularity of that spacetime.

Load-bearing premise

The argument assumes the relevant field-theory entropy is the total thermodynamic entropy of the fluid, which is conformally invariant and therefore constant for a shear-free flow, so that a diverging event-horizon area is automatically the wrong answer.

Editorial extensions

If this is right

In any holographic dual that does not relax to a stationary black hole, the event horizon area should not be used as the entropy; the quasi-local horizon is the correct object.
The Bjorken-flow geometry is regular: its global event horizon can be constructed explicitly and cloaks the $r=0$ singularity, confirming the late-time expansion is trustworthy.
For the conformal soliton, the apparent horizon coincides with the global BTZ horizon $r=r_+$ for any foliation that admits a complete slice of that horizon, so the entropy is constant as the ideal-fluid picture demands.
The apparent horizon respects a second-law-type monotonicity of area under the energy conditions, so its pulled-back area form can serve as a local Boltzmann $H$-function on the boundary.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the Minkowski-patch observer's entropy were instead a patch-dependent entanglement entropy, the logarithmic growth of the event horizon area would be a physical effect rather than a disqualifying artifact, and the argument for preferring the apparent horizon would weaken; the paper itself points to the entanglement-entropy literature as the natural place to test this.
The authors note that apparent horizons are foliation dependent and can jump discontinuously, so the prescription 'entropy equals apparent-horizon area' is only clean when the flow is slow enough that all slices ending on the same boundary time give nearly the same area; a sharp formulation probably needs a preferred bulk foliation, which the paper does not supply.
The same diagnostic could be run on numerical far-from-equilibrium collapse geometries, where no exact dual is available: comparing event-horizon and apparent-horizon areas against a separately computed entropy would test whether the two track each other before local equilibrium is reached.

Original abstract

We analyse the global structure of time-dependent geometries dual to expanding plasmas, considering two examples: the boost invariant Bjorken flow, and the conformal soliton flow. While the geometry dual to the Bjorken flow is constructed in a perturbation expansion at late proper time, the conformal soliton flow has an exact dual (which corresponds to a Poincare patch of Schwarzschild-AdS). In particular, we discuss the position and area of event and apparent horizons in the two geometries. The conformal soliton geometry offers a sharp distinction between event and apparent horizon; whereas the area of the event horizon diverges, that of the apparent horizon stays finite and constant. This suggests that the entropy of the corresponding CFT state is related to the apparent horizon rather than the event horizon.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,317 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.41 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.12 · compares with prior results 0.37 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We analyse the global structure of time-dependent geometries dual to expanding plasmas, considering two examples: the boost invariant Bjorken flow, and the conformal soliton flow. While the geometry dual to the Bjorken flow is constructed in a perturbation expansion at late proper time, the conformal soliton flow has an exact dual (which corresponds to a Poincare patch of Schwarzschild-AdS). In particular, we discuss the position and area of event and apparent horizons in the two geometries. The conformal soliton geometry offers a sharp distinction between event and apparent horizon; whereas the area of the event horizon diverges, that of the apparent horizon stays finite and constant. This suggests that the entropy of the corresponding CFT state is related to the apparent horizon rather than the event horizon.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We analyse the global structure of time-dependent geometries dual to expanding plasmas, considering two examples: the boost invariant Bjorken flow, and the conformal soliton flow.	contribution 1.00	0.90	0.04	asserted 0.94
s1	While the geometry dual to the Bjorken flow is constructed in a perturbation expansion at late proper time, the conformal soliton flow has an exact dual (which corresponds to a Poincare patch of Schwarzschild-AdS).	method 0.77	0.29	0.04	asserted 1.00
s2	In particular, we discuss the position and area of event and apparent horizons in the two geometries.	contribution 0.98	0.80	0.04	asserted 0.71
s3	The conformal soliton geometry offers a sharp distinction between event and apparent horizon; whereas the area of the event horizon diverges, that of the apparent horizon stays finite and constant.	result 1.00	0.83	0.31	asserted 1.00
s4	This suggests that the entropy of the corresponding CFT state is related to the apparent horizon rather than the event horizon.	interpretation 1.00	0.59	0.68	hedged possibility 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Illustration of the horizons for the Bjorken flow metrics g (k) at various orders in the perturbation expansion. The event horizons are the solid curves while the apparent horizons are the dashed curves. The locations of the horizons of course should only be trusted at late times as indicated in the figure. behind the event horizon. We illustrate the location of the horizons at various orders in

**Figure 2.** Illustration of Poincar´e coordinates superposed on conformally compactified AdS. The surfaces t = 0, 1, 5 (left), z = 1, 5 (middle), and x = 0, 1, 5 (right) are plotted, with colour-coding blue for 0, green for 1, and red for 5. To guide the eye, surperposed are also the boundary of the I + CS corresponding to t = ±∞ and the t = 0 boundary slice (black curves)

**Figure 3.** Plot of the event horizon of the CS spacetime. For ease of visualization we have also plotted the location of the event horizon of the global Schwarzschild-AdS black hole. It is easy to confirm that the surface described by (3.9) or (3.13) is indeed a null surface. For instance, the induced metric on this 2-surface is simply ds2 ind = d`2 (1 − ` 2 ) 2 , (3.14) which is clearly degenerate, as required of the event horizon

**Figure 4.** Behaviour of the Poincar´e time along the line of caustics tc(`). We have plotted here the situation for a sampling of BTZ horizon size. As explained in the text, as r+ > 1 (recall that we normalize LAdS = 1) the allowed domain in ` shrinks exponentially. However, we can make some general remarks. The slices t = 0 in Poincar´e coordinates and τ = 0 in global coordinates coincide. Hence, the t = 0 cross-section of the horizon is the same as the τ = 0 cross-section, and hence has `max = 1. Further, the slices of t > 0 (< 0) lie entirely in the region τ > 0 (< 0) in the global…

Figures reproduced from arXiv:0902.4696 by the authors.

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The paper derives the complete second-order relativistic viscous fluid equations and the non-equilibrium entropy current, fixing several entropy-current coefficients by requiring that entropy never decreases.
Boost-invariant early time dynamics from AdS/CFT
hep-th 2009-06 conditional novelty 6.0 of 10

Early-time boost-invariant plasma dynamics in AdS/CFT has no scaling solution, depends on bulk initial conditions, and necessarily contains a metric singularity at all times.
New theories of relativistic hydrodynamics in the LHC era
hep-ph 2017-07 accept novelty 5.0 of 10

The hydrodynamic gradient expansion is an asymptotic series whose large-order factorial growth encodes the transient, nonhydrodynamic modes of the underlying plasma.
Holographic thermal field theory on curved spacetimes
hep-th 2013-12 accept novelty 5.0 of 10

The droplet phase of a holographic CFT on a black hole background is confinement in disguise: a conformal transformation turns the near-horizon neck into a confining circle, explaining suppressed heat transport.
Holographic Thermalization with Chemical Potential
hep-th 2012-05 conditional novelty 5.0 of 10

In a holographic model of a charged strongly coupled plasma, the thermalization time is non-monotonic in the chemical potential for small probes: it decreases at small mu/T and increases at large mu/T.
Gravity & Hydrodynamics: Lectures on the fluid-gravity correspondence
hep-th 2009-05 accept novelty 3.0 of 10

Einstein's equations with a negative cosmological constant, expanded in boundary derivatives, reproduce the relativistic Navier-Stokes equations, and every fluid flow lifts to a regular asymptotically AdS black hole.
The AdS/CFT Correspondence
gr-qc 2014-12 accept novelty 2.0 of 10

A self-contained review of the AdS/CFT correspondence: its string theory origin, its dictionary of bulk-to-boundary mappings, and its applications to black holes, fluids, and quantum gravity.
A holographic view on physics out of equilibrium
hep-th 2010-06 accept novelty 2.0 of 10

Holography recasts non-equilibrium dynamics of strongly coupled quantum field theories as classical evolution of black hole geometries, organized by how far the system sits from equilibrium in amplitude and wavelength.
Holographic Entanglement Entropy: An Overview
hep-th 2009-05 conditional novelty 1.0 of 10

A 2009 review argues that entanglement entropy in a conformal field theory is computed by the area of a minimal surface in its dual anti-de Sitter spacetime, and surveys the evidence and applications.

How later papers used it

62 later papers cite it

Evidence: substantial. 54 of the citing papers share no author with it, and 16 of those carry a confident label from Pith's citation classifier.

Pith read 20 of the 106 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

4 later papers by other authors are listed below: 4 used its method or tool.

Uses its method or tool 4 of 8 papers with this label

Followed this paper in associating the non-equilibrium entropy density of the expanding plasma at the cosmological boundary with the Bekenstein-Hawking entropy of the apparent horizon.

Expanding plasmas from Anti de Sitter black holes (2016) · 0.99 · paraphrased: the licence does not allow quoting
Closely followed this paper.

Holographic Entanglement Entropy in Time Dependent Gauss-Bonnet Gravity (2015) · 0.92 · paraphrased: the licence does not allow quoting
Introduced the apparent horizon for the backgrounds following this paper's notations.

Strong Subadditivity, Null Energy Condition and Charged Black Holes (2013) · 0.87 · paraphrased: the licence does not allow quoting
Closely followed this paper.

Holographic Thermalization with Chemical Potential (2012) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

62 arXiv papers cite this in 106 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 76 sentences without a confident label

Uses its method or tool 8 papers

A gravitational geometry [ref] has an apparent horizon located at $r=r_AH$, where [ref] [equation] Following [ref] we associate the non-equilibrium entropy density $s$ of the boundary QFT with the Bekenstein-Hawking entropy density of the apparent horizon [equation] Using the holographic background equations of motion [ref]-[ref] we find [equation] Following [ref] it is easy to prove that the entropy production rate as defined by [ref] is non-negative, [equation] in holographic dynamics governed by [ref]-[ref].

Ringing in de Sitter spacetime · Holographic gravitational dynamics · 0.81
Having determined its location, we now follow [ref] and associate the non-equilibrium entropy density $s$ of the expanding plasma living at the cosmological boundary with the Bekenstein-Hawking entropy of the apparent horizon, namely [equation] From this it follows that, if $R_h=r_h/a$ is the only apparent horizon, then clearly [equation] i.e., there is no entropy production by the plasma during the dynamical process.

Expanding plasmas from Anti de Sitter black holes · Entropy production · 0.99
A gravitational geometry [ref] has an apparent horizon located at $r=r_AH$, where [ref] [equation] Following [ref] we associate the non-equilibrium entropy density $s$ of the boundary QFT with the Bekenstein-Hawking entropy density of the apparent horizon [equation] with $a$ being the QFT background geometry scale factor [ref].

de Sitter Vacua of Strongly Interacting QFT · Holographic evolution of $=2^*$ gauge theory states in $dS_4$ · 0.88

Compares with it 2 papers

Our solution for the difference between the location and the entropy current of the event and apparent horizons is in agreement with the boost invariant Bjorken flow which was done in [ref] .

Relativistic Holographic Hydrodynamics from Black Hole Horizons · Temperature · 0.99
We find strong similarity with the conclusions of [ref] .

The temperature and entropy of CFT on time-dependent backgrounds · Specific cases · 1.00

Cites it as support 7 papers

Currently, there are several evidences that the latter will be more appropriate to define entropy [ref] .

Measuring Black Hole Formations by Entanglement Entropy via Coarse-Graining · An Entropy Puzzle about Time-dependent Holography · 0.97
In fact, as argued e.g. in [ref] , one expects that entropy production is local in time, therefore, associating the far from equilibrium entropy to the area of a dynamic event horizon seems unnatural, since in such a context the event horizon can only be determined by knowing the entire future evolution of the black hole geometry and, consequently, it is a global rather than a local observable.

Holographic entropy production in a Bjorken expanding hot and dense strongly coupled quantum fluid · Introduction · 0.99
The latter identification actually turns out to be more robust, as the example of conformal soliton flow [ref] suggests [ref] .

On the apparent horizon in fluid-gravity duality · Introduction · 0.98

Cites it as background 9 papers

A simple analytic but non-stationary model [ref] of heat flow along horizons constructed by allowing a black string to fall through a Rindler horizon, as well as older models [ref] of localized particles and black holes falling across Rindler or Poincaré horizons.

Holographic thermal field theory on curved spacetimes · Flowing funnels and Hawking radiation · 0.89
Emerging with the path-breaking studies [ref] , themselves motivated by the ideas of [ref] - [ref] , this duality has been since then successfully explored in a series of works [ref] - [ref] ; for a nice review and references see [ref] .

Phase diagram for non-axisymmetric plasma balls · Introduction · 0.98
One of the simplest examples of a secretly static configuration which nevertheless provides an instructive toy model for a time-dependent scenario is given by the conformal soliton geometry originally introduced in [ref] , and then studied extensively in [ref] .

A holographic view on physics out of equilibrium · Mock time-dependence with evolving event horizon · 0.91

What it cites · 61 sentences without a confident label

Uses the method or tool of 3 papers

To translate the variation of the area into a statement of the boundary field theory entropy, we could consider pulling back the area element of the spatial sections of the horizon along radially ingoing null geodesics, as advocated in [ref] .

Local Fluid Dynamical Entropy from Gravity · Event horizon area & field theory entropy · 0.91
We follow [ref] and describe a region in the bulk geometry of the d+1 spacetime in Poincaré coordinates, applying a specific coordinate transformation, one that transforms global d+1 into Poincaré AdS, to the black hole spacetime.

Expanding plasmas and quasinormal modes of anti-de Sitter black holes · The BTZ spacetime as a conformal soliton · 0.89
Let us review the details of this construction: we will follow the conventions of [ref] (except for using $τ$ rather than $τ_+$ to denote the proper time coordinate).

A Holographic Dual of Bjorken Flow · The gravity dual to Bjorken flow · 0.81

Compares with 4 papers

A curiosity at leading order: The location of the event horizon differs from the location of the apparent horizon derived in [ref] at second order in the expansion in $τ^-2/3$.

A Holographic Dual of Bjorken Flow · Event horizon for the BF spacetime · 0.97
Again, the essential difference between the cases we consider here and [ref] is that in that general analysis, it was assumed that the geometry would settle down at late times to a stationary finite-temperature black hole.

Local Fluid Dynamical Entropy from Gravity · Introduction · 0.87
This is different from the result of [ref] because working in Eddington-Finkelstein coordinates imposes regularity on the future event horizon in the bulk geometry, which is the physically correct condition.

On the supergravity description of boost invariant conformal plasma at strong coupling · The gravity dual to Bjorken flow · 0.83

Cites as support 2 papers

This point of view seems natural from studies of entanglement entropy in AdS/CFT [ref] and also in dynamical black hole spacetimes which are out of the hydrodynamic regime [ref] .

A Covariant Holographic Entanglement Entropy Proposal · Event horizon area & field theory entropy · 0.99
This point of view seems natural from studies of entanglement entropy in AdS/CFT [ref] and also in dynamical black hole spacetimes which are out of the hydrodynamic regime [ref] .

Horizon formation and far-from-equilibrium isotropization in supersymmetric Yang-Mills plasma · Event horizon area & field theory entropy · 0.99

Cites as background 8 papers

For the metrics at higher orders, and a comprehensive discussion of the derivation, we refer the reader to [ref] .

A Holographic Dual of Bjorken Flow · The gravity dual to Bjorken flow · 0.89
In [ref] , the authors also constructed the bulk geometry dual to the given fluid dynamical evolution using the AdS/CFT correspondence.[The gravity dual to Bjorken flow in 1+1 dimensions was discussed in [ref] .

Gauge/gravity duality and thermalization of a boost-invariant perfect fluid · The gravity dual to Bjorken flow · 0.80
In [ref] , the authors also constructed the bulk geometry dual to the given fluid dynamical evolution using the AdS/CFT correspondence.[The gravity dual to Bjorken flow in 1+1 dimensions was discussed in [ref] .

Gravity dual of 1+1 dimensional Bjorken expansion · The gravity dual to Bjorken flow · 0.80

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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