REVIEW
Rapid Thermalization in Field Theory from Gravitational Collapse
High Energy Physics - Theory · Submitted 12 Jun 2011 · 28 references
Pith's verdict: A genuine nonlinear collapse simulation in AdS5 whose headline number survives scrutiny; the weaknesses are bookkeeping (no error bars, unstated boundary conditions, an uncited instability) rather than physics.
T0 · AI review
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Referee report
The paper
The paper claims
Simulating collapse in anti-de Sitter space, the authors argue that a dual field theory thermalizes within about a light-crossing time of the energy injection.
The reading
What carries the argument
The load-bearing device is the apparent horizon as a numerical stopping condition: in the metric ansatz $ds^2 = -\alpha^2 dt^2 + a^2 dr^2 + r^2 d\Omega_3^2$, a marginally outer trapped surface corresponds to $a \to \infty$, so the code halts when $a$ reaches a fixed large cutoff $a_{\max}$, and the time and radius at that moment give the formation time and an estimate of the black hole mass. The second device is the null-geodesic clock: in pure AdS a light ray takes $\pi/2$ to go from center to boundary and $1.32$ to go from $r=4$ to the center, so geodesic travel time supplies the baseline against which $t_{AH}$ is judged 'rapid'. Code correctness is checked with the constraint quantity $C \equiv \partial_t a - r\alpha X Y/3$, which must vanish analytically and is shown to shrink when the grid is refined from 800 to 1600 points, using a non-uniform radial grid $r_i = \sinh(k(i-1))$ that concentrates resolution near the origin where the $1/r^3$ terms in the Klein–Gordon equation are stiffest.
Extended reading notes
Core claim
The claim is that black hole formation from a collapsing scalar shell in asymptotically AdS$_5$ is fast: the apparent horizon appears within roughly a light-crossing time of the shell's starting position, and this holds for black hole masses spanning two orders of magnitude. The evidence is a direct numerical integration of the Einstein–Klein–Gordon system in spherical symmetry, with a massless scalar field ($U=0$) and Gaussian initial data $\varphi = A\exp(-(r-r_0)^2/\sigma^2)$ chosen so the pulse starts purely ingoing. Small amplitudes ($A = 0.0002$) produce no horizon at all: the field at the center is negligible except in bursts spaced by about $\pi$, the round-trip light time in AdS with $L=1$, as expected for a probe field on a fixed anti-de Sitter background. For $A$ between 0.005 and 0.04 with $\sigma = 1.5$ and $r_0 = 4.0$, horizons do form, and $t_{AH}$ falls from about 1.3 to about 0.6 as the amplitude rises — the trend the geodesic picture predicts, since a heavier shell becomes compact enough to trap light at a larger radius and therefore sooner. Varying the width instead ($\sigma$ from 0.5 to 2.5 at $A = 0.02$) moves $t_{AH}$ only between roughly 0.70 and 0.86, a mild dependence. The authors present this as extending the perturbative thin-shell analysis of Bhattacharyya and Minwalla into the fully nonlinear regime, where the time and size of horizon formation can be stated precisely rather than read off the first terms of an expansion.
Load-bearing premise
The argument rests on reading the simulation's coordinate time $t$ — the time orthogonal to the area radius — as the time of the dual field theory, and the authors themselves say in their conclusions that the rapid-thermalization question hinges on which bulk coordinate is the dictionary translation of boundary time.
Editorial extensions
If this is right
- The qualitative rapid-thermalization picture obtained perturbatively for thin Vaidya shells survives at full nonlinear order, so it is not an artifact of linearizing in the shell amplitude.
- Thermalization time in the dual theory is bounded above by the light-crossing time of the collapsing region, and therefore scales with the size of the region rather than with the black hole mass over the two decades probed.
- The dependence of thermalization time on the width of the energy profile is weak compared with its dependence on amplitude, so the injected energy density, not its spatial extent, is the controlling variable.
- Because the same simulation can be run with a massive scalar, the framework extends to operators of other conformal dimension, connecting thermalization time to the dimension of the operator that sources the quench.
Reading between the lines
- The 'rapid' conclusion is coordinate-dependent in a way worth making explicit: in an ingoing null coordinate the horizon forms essentially at once, so the physically meaningful statement is that $t_{AH}$ is small compared with the AdS scale, not that some invariant delay has been measured.
- If thermalization time tracks the light-crossing time of the source region, then a spatially inhomogeneous collision should thermalize locally on the crossing time of each patch, which is what the numerical-relativity treatments of colliding shocks assume but do not derive from a single collapsing solution.
- The mild width dependence suggests a concrete test: hold the total energy fixed while varying $\sigma$, which the paper does not do, and see whether $t_{AH}$ collapses onto a single curve in the energy.
- A natural cross-check available to the authors' own program is to compute a boundary two-point function or entanglement entropy in the same collapse and confirm that its approach to the thermal form happens on the $t_{AH}$ scale rather than a parametrically longer one.
Original abstract
Motivated by the duality with thermalization in field theory, we study gravitational collapse of a minimally coupled massless scalar field in Einstein gravity with a negative cosmological constant. We investigate the system numerically and establish that for small values of the initial amplitude of the scalar field there is no black hole formation, rather, the scalar field performs an oscillatory motion typical of geodesics in AdS. For large enough values of the amplitude of the scalar field we find black hole formation which we detect numerically as the emergence of an apparent horizon. Using the time of formation as an estimate for thermalization in the field theory we conclude that thermalization occurs very rapidly, close to the causal bound for a very wide range of black hole masses. We further study the thermalization time in more detail as a function of the amplitude and the width of the initial Gaussian scalar field profile and detect a rather mild structure.
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.
The abstract, sentence by sentence
Motivated by the duality with thermalization in field theory, we study gravitational collapse of a minimally coupled massless scalar field in Einstein gravity with a negative cosmological constant. We investigate the system numerically and establish that for small values of the initial amplitude of the scalar field there is no black hole formation, rather, the scalar field performs an oscillatory motion typical of geodesics in AdS. For large enough values of the amplitude of the scalar field we find black hole formation which we detect numerically as the emergence of an apparent horizon. Using the time of formation as an estimate for thermalization in the field theory we conclude that thermalization occurs very rapidly, close to the causal bound for a very wide range of black hole masses. We further study the thermalization time in more detail as a function of the amplitude and the width of the initial Gaussian scalar field profile and detect a rather mild structure.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Motivated by the duality with thermalization in field theory, we study gravitational collapse of a minimally coupled massless scalar field in Einstein gravity with a negative cosmological constant. | contribution 0.72 | 0.65 | 0.04 | asserted 0.61 |
| s1 | We investigate the system numerically and establish that for small values of the initial amplitude of the scalar field there is no black hole formation, rather, the scalar field performs an oscillatory motion typical of geodesics in AdS. | result 1.00 | 0.95 | 0.05 | asserted 1.00 |
| s2 | For large enough values of the amplitude of the scalar field we find black hole formation which we detect numerically as the emergence of an apparent horizon. | result 1.00 | 0.97 | 0.06 | asserted 1.00 |
| s3 | Using the time of formation as an estimate for thermalization in the field theory we conclude that thermalization occurs very rapidly, close to the causal bound for a very wide range of black hole masses. | interpretation 0.57 | 0.94 | 0.84 | asserted 0.96 |
| s4 | We further study the thermalization time in more detail as a function of the amplitude and the width of the initial Gaussian scalar field profile and detect a rather mild structure. | result 0.79 | 0.97 | 0.05 | asserted 0.99 |
What was examined, and what was not
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- relations to other papers
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Figures
Showing 2 of the paper's 4 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:1106.2339 by the authors.
References
How the literature uses it
49 arXiv papers cite this in 78 sentences
Papers citing it · 38 sentences without a confident label
Contests it 1 paper
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Thus, we found it surprising that Garfinkle and Pando Zayas, who later looked at the same problem in $4+1$ dimensions (apparently unaware of Ref. [ref] as they did not cite it), wrote in [ref] : "We [...] establish that for small values of the initial amplitude of the scalar field there is no [sic!] black hole formation, rather, the scalar field performs an oscillatory motion typical of geodesics in AdS." The purpose of this comment is to show that, contrary to the above quoted claim, $AdS_d+1$ is unstable against gravitational collapse for all $d≥3$.
A comment on AdS collapse of a scalar field in higher dimensions · Introduction · contest 0.92
Builds on it 1 paper
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In this publication we largely expand, clarify and sharpen the results presented in our short letter [ref] .
On Field Theory Thermalization from Gravitational Collapse · Introduction · 1.00
Uses its method or tool 4 papers
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The precise version used here produced the results presented in [ref] .
On Field Theory Thermalization from Gravitational Collapse · Understanding the numerical accuracy · 0.82
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We will deal with more general cases using a numerical method similar to that used in the global patch of $AdS_d+1$ [ref] .
On holographic thermalization and gravitational collapse of massless scalar fields · Introduction · 0.99
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We consider a model of dynamical holographic thermalization [ref] which has been recently used to study thermalization time scale of strongly coupled plasmas.
Quantum corrections to dynamical holographic thermalization: entanglement entropy and other non-local observables · Introduction and motivation · 0.80
Uses its data or values 1 paper
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For easy comparison of results, we take the same approximately ingoing Gaussian initial data as Ref. [ref] : $Φ(0,x)=∂_x ϕ(0,x)=Π(0,x)$, where [equation] with $r_0=4$, $σ=1.5$.
A comment on AdS collapse of a scalar field in higher dimensions · Introduction · 0.80
Cites it as support 2 papers
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We believe that this is what is happening in recent numerical works [ref] with initial spherical shell.
The physical mechanism of AdS instability and Holographic Thermalization · Mechanism of energy cascade to UV in wave collapse. · 0.99
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The thermalization time scale is typically fast for the Vaidya-AdS metric [ref] , and hence it would be suitable to focus on the effect of rapid thermalization.
Dynamical Meson Melting in Holography · Setup · 0.96
Cites it as background 24 papers
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In 2011, [ref] pioneered numerical work on these questions (see also [ref] ).
Classes of Stable Initial Data for Massless and Massive Scalars in Anti-de Sitter Spacetime · Introduction · 1.00
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Different aspects of scalar collapse in AdS$_d+1$ have been studied in [ref] .
Collapse and Revival in Holographic Quenches · Revival time · 0.99
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In fact, in a series of papers the issue has been studied considering the collapse of a thin shell of matter, by using an AdS-Vaidya type metric [ref] .
Thermalization with a chemical potential from AdS spaces · Introduction · 0.89
What it cites · 4 sentences without a confident label
Cites as support 3 papers
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Other studies supporting rapid thermalization in the context of the gauge/gravity correspondence include [ref] .
Instantaneous Thermalization in Holographic Plasmas · Introduction · 1.00
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Other studies supporting rapid thermalization in the context of the gauge/gravity correspondence include [ref] .
Holographic Thermalization · Introduction · 1.00
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Other studies supporting rapid thermalization in the context of the gauge/gravity correspondence include [ref] .
Evidence for fast thermalization in the plane-wave matrix model · Introduction · 1.00
Discussion
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