Holographic Quenches and Thermalization

A holographic quench turns a prescribed boundary energy injection into an asymptotically AdS initial-value problem. Different observables sample different bulk depths and therefore need not approach their thermal values together. This page makes that statement quantitative in planar AdS–Vaidya, while separating probe equilibration, hydrodynamization, and the teleological formation of an event horizon.

Required background. Gauge Theory and String-Theory Preparation supplies the large-N saddle expansion and the operator/geometric-probe dictionary used below. Helpful background. Quantum Gravity supplies semiclassical control, Quantum Information supplies entropy diagnostics, Black-Hole Perturbations supplies mode relaxation, and Gravitational Radiation supplies characteristic evolution and constraint checks.

The AdS–Vaidya shell and its observables

In ingoing Eddington–Finkelstein coordinates, a homogeneous null shell can be modeled by

$$ds^2=\frac{1}{z^2}\left[-f(v,z)\,dv^2-2\,dv\,dz+d\mathbf{x}^{2}\right], \qquad f(v,z)=1-m(v)z^d .$$

Here the AdS radius is set to one, $d$ is the boundary spacetime dimension, $z=0$ is the conformal boundary, and $m(v)$ interpolates from zero to a positive constant. Near-boundary data make the energy density proportional to $m(v)$, so the shell width measures the injection time. The benchmark below takes $d=2$, $m(v)=[1+\tanh(v/w)]/2$, $w=0.15$, and a final horizon at $z_h=1$. This is a phenomenological null-dust geometry, not a derivation of a particular microscopic driving protocol.

Each diagnostic poses a different bulk problem. A heavy-operator correlator uses a spacelike geodesic,

$$\langle {\cal O}(t,-\ell/2){\cal O}(t,\ell/2)\rangle \simeq \exp[-\Delta {\cal L}_{\mathrm{ren}}(t,\ell)] ,$$

a Wilson loop uses a string worldsheet, and an interval entropy uses a covariant extremal curve. Their turning depths grow with the boundary scale $\ell$, so larger probes retain memory of the pre-quench region longer. This scale-dependent, top-down pattern was computed explicitly for thin-shell AdS–Vaidya geometries by Balasubramanian et al. 2011, §§ II.1.3–III.3, pp. 6–36. It is a statement about those probes in the specified large-N, strong-coupling saddle regime, not a theorem about every interacting quantum system.

What a thermalization time means

For a chosen observable $X$, tolerance $\epsilon$, and scale $\ell$, define

$$t_X(\ell;\epsilon)=\inf\left\{t_0:\ \frac{|X(t,\ell)-X_{\rm th}(\ell)|}{S_X(\ell)}\lt\epsilon \ \text{for every }t\geq t_0\right\} ,$$

where $S_X$ is a declared nonzero normalization. The “for every later time” clause rejects false plateaus. A one-point function, a two-point function, entanglement entropy, and a hydrodynamic constitutive residual can consequently give four different times. Hydrodynamization asks when conserved stresses are well approximated by a gradient expansion; it does not require isotropy or local thermal equilibrium. The distinction is central in numerical holography, where a small constitutive residual can precede the damping of all nonhydrodynamic structure Chesler and van der Schee 2015, §§ 2–4, pp. 5–24.

An event horizon supplies no competing local clock. For an outgoing radial null curve in the metric above,

$$\frac{dz_{\rm EH}}{dv}=-\frac12\left[1-m(v)z_{\rm EH}^{2}\right] \quad (d=2),$$

and its final boundary condition must be integrated backward. It therefore grows into a region where the boundary source is still exponentially small. That anticipatory behavior is the global definition of the horizon, not acausal boundary response. The apparent horizon $z_{\rm AH}=m(v)^{-1/2}$ is quasilocal but slicing dependent. Neither surface, by itself, establishes that a chosen field-theory observable has equilibrated.

A reproducible bounded benchmark

The downloadable Python program and its committed JSON result implement three checks. The reference environment is Python 3.11.4, NumPy 2.0.2, SciPy 1.13.1, and IEEE-754 binary64 arithmetic. solve_ivp uses DOP853; brentq locates the 99-percent injection time. All dimensionful quantities are measured in units of the final horizon radius. The ODE domain is $-2\leq v\leq4$, the interval has $\ell=1.2$, and the three rows jointly refine relative tolerance, maximum step, and radial cutoff. Use the standalone download-and-run recipe:

curl -fL https://blackhole.org/data/string-holographic-black-holes/holographic-quench-benchmark.py -o holographic-quench-benchmark.py
python3 holographic-quench-benchmark.py --output holographic-quench-benchmark.json

refinement	relative tolerance	$z_{\rm cut}$	$z_{\rm EH}(0)$	change from reference	static-length error	step-integral residual
coarse	1.0e-7	0.020	1.024452656721	1.642e-12	6.208e-5	4.382e-5
medium	1.0e-9	0.010	1.024452656720	8.882e-15	1.553e-5	5.522e-6
fine	1.0e-11	0.005	1.024452656720	0	3.883e-6	6.919e-7

The independent reference uses relative tolerance $5\times10^{-13}$ and maximum step $0.002$. For the static geodesic checks, both endpoints are fixed at $(x,z)=(\pm\ell/2,z_{\rm cut})$. Subtracting the common boundary divergence gives

$${\cal L}_{\rm vac}=2\operatorname{arsinh}\!\left(\frac{\ell}{2z_{\rm cut}}\right) +2\log z_{\rm cut}, \qquad {\cal L}_{\rm BTZ}=2\operatorname{arsinh}\!\left(\frac{\sinh(\ell/2)}{z_{\rm cut}}\right) +2\log z_{\rm cut}.$$

Their difference converges to $2\log[2\sinh(\ell/2)/\ell]=0.1185920506912$. This exact value is the analytic benchmark. The roughly fourfold error decrease under each cutoff halving is the expected quadratic approach. The event-horizon value is separately stable, and the ODE step-integral residual decreases by nearly an order of magnitude per row.

This finite-cutoff prescription differs from the vacuum RT interval calculation, which keeps the boundary-anchored circle fixed and truncates its ends. Here the vacuum circle has radius $\sqrt{(\ell/2)^2+z_{\rm cut}^2}$, so that its endpoints remain separated by $\ell$ at the cutoff. The RT prescription instead gives the bare length $2\operatorname{arcosh}[\ell/(2z_{\rm cut})]$. Both yield the same leading logarithm; their order-$z_{\rm cut}^2$ terms differ because they hold different endpoint data fixed.

The program also intersects a vacuum reference semicircle $z^2+x^2=(\ell/2)^2$, with $v=t-z$, with the surface $v=0$:

$$z_{\times}=t, \qquad x_{\times}=\sqrt{(\ell/2)^2-t^2},$$

This algebraic intersection ends at $t=\ell/2=0.6$. It is not the physical crossing geodesic: a curve joining post-quench endpoints crosses a BTZ region and refracts at the shell, so the vacuum semicircle alone cannot determine its crossing point or length.

The actual thin-shell saturation time follows independently from the final BTZ geodesic. At zero boundary cutoff its turning point is $z_*=\tanh(\ell/2)$, and the ingoing time along a constant-$t$ BTZ slice is $v=t-\operatorname{artanh}z$. Its tip first lies on the shell when

$$v_*=0,\qquad t_{\rm crit}=\operatorname{artanh}z_*=\frac{\ell}{2}.$$

This checks the thin-shell endpoint, without solving the smooth-shell extremal-curve problem. The source reaches 99 percent of its final value at $v=0.3446339888$, while event-horizon formation is already globally encoded; these numbers measure different processes. The saturation result is stated and compared with other dimensions in Balasubramanian et al. 2011, pp. 25–26, Fig. 15.

Interpretation and failure diagnostics

The benchmark supports three deliberately narrow conclusions. First, the subtraction has the correct vacuum and late-BTZ limits. Second, its reference-curve intersection is an elementary geometric check; the physical thin-shell saturation time needs the separate BTZ derivation above. Third, event-horizon convergence is numerically clean but says nothing by itself about boundary equilibration. It does not compute refracted or smooth-shell HRT dynamics, Wilson loops, backreaction of a specified boundary source, finite-coupling corrections, or a microscopic final-state density matrix.

Several failures are made explicit in the program. A terminal horizon value $z(4)=0.9$ produces a $0.1$ late-boundary error and is rejected as the wrong null branch. A forced shell crossing after $t=\ell/2$ makes $x_{\times}$ imaginary and must be rejected rather than continued. A subtraction whose result drifts with $z_{\rm cut}$ has not isolated a finite observable. A pre-quench boundary response would violate causality even though an event horizon may extend there. Missed turning points, stiff shell profiles, and an isolated small residual can each create a false plateau; branch continuity, constraint residuals, cutoff refinement, and late-time analytic limits must be checked together. These controls follow the general real-time holographic workflow reviewed in Rangamani 2009, §§ 2–5, pp. 8–43.

Exercises

Show that the static-length difference is cutoff independent at leading order. Expand the two regulated lengths above through order $z_{\rm cut}^2$ and determine the limiting difference.

Solution

For large $y$, $\operatorname{arsinh}y=\log(2y)+1/(4y^2)+O(y^{-4})$. Substitution gives ${\cal L}_{\rm vac}=2\log\ell+2z_{\rm cut}^2/\ell^2+O(z_{\rm cut}^4)$, whereas ${\cal L}_{\rm BTZ}=2\log[2\sinh(\ell/2)]+z_{\rm cut}^2/[2\sinh^2(\ell/2)]+O(z_{\rm cut}^4)$. Their leading divergences have therefore canceled because both curves approach the same asymptotic AdS boundary. For $\ell=1.2$, the limiting difference is $2\log[2\sinh(0.6)/1.2]=0.1185920506912$. The table’s errors decrease from $6.208\times10^{-5}$ to $1.553\times10^{-5}$ and then $3.883\times10^{-6}$. Each value is approximately one quarter of its predecessor, so the comparison confirms the predicted quadratic cutoff error. It also distinguishes a stable finite difference from two separately regulator-dependent lengths; agreement in only the final row would not establish convergence.

Determine why an event horizon cannot define $t_X$. Use the null equation and its boundary condition to compare it with the observable definition.

Solution

The null equation is first order, but the event horizon is selected by the future condition $z_{\rm EH}(v\to\infty)=z_h$. Integrating backward makes its location depend on the future mass profile. Each boundary observable $X(t)$ instead responds causally to its past preparation. Certifying the all-later-times definition of $t_X$ also needs future observations or analytic late-time control; a finite simulation can establish only a stated finite-window version. The event horizon can move before a local source is appreciable without implying that $X$ responded; its location is not an equilibration observable.

References

• Balasubramanian, Vijay, et al. “Holographic Thermalization.” Physical Review D 84 (2011): 026010. DOI. Open PDF.
• Chesler, Paul M., and Wilke van der Schee. “Early Thermalization, Hydrodynamics and Energy Loss in AdS/CFT.” International Journal of Modern Physics E 24 (2015): 1530011. DOI. Open PDF.
• Rangamani, Mukund. “Gravity and Hydrodynamics: Lectures on the Fluid-Gravity Correspondence.” Classical and Quantum Gravity 26 (2009): 224003. DOI. Open PDF.