Time-Dependent Geometries and Holographic Thermalization

A collapsing asymptotically AdS geometry can model a specified boundary energy injection, but “thermalization time” is observable-dependent. Local one-point functions, two-point probes, Wilson loops, and extremal-surface entropies sample different radial regions and can equilibrate at different times. Apparent-horizon formation is a geometric diagnostic on a chosen slicing, not proof that every boundary observable is thermal.

Required background. Schwinger–Keldysh Contours and Real-Time Bulk Geometries fixes the real-time contour, and Holographic Initial States and Euclidean Caps fixes admissible initial data.

Helpful background. Equilibration, Thermalization, and Dephasing distinguishes boundary claims; Causal, Killing, Trapping, and Apparent Horizons: The QFT Interface fixes horizon terminology.

Thin-shell collapse in AdS

In ingoing coordinates, an AdS–Vaidya model has

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

A thin shell takes $m(v)=m_f\Theta(v)$. Its boundary stress tensor changes with $m(v)$, so the construction represents a homogeneous injection rather than unitary relaxation from an arbitrary pure state. The apparent horizon satisfies $f(v,z_{\mathrm{AH}})=0$ in this foliation; the event horizon is defined globally and begins growing before the shell arrives.

The metric is a useful analytic model, while fully dynamical Einstein evolution requires initial and boundary data plus numerical constraint monitoring; see Chesler and Yaffe 2014.

Three probes, three equilibration scales

The first application compares:

1. A local one-point function. Near-boundary coefficients respond directly to $m(v)$ and can settle on the injection time.

2. A heavy-operator two-point function. In a geodesic approximation,

   $$\langle\mathcal O(t,\mathbf x)\mathcal O(t,\mathbf 0)\rangle \sim e^{-\Delta\mathcal L_{\mathrm{ren}}(t,\mathbf x)}.$$

   The geodesic reaches a depth set by the separation and may cross the shell after local quantities have settled.

3. A region entropy. The covariant extremal surface obeys

   $$S_A(t)=\frac{\operatorname{Area}(\gamma_A(t))}{4G_N}, \qquad \partial\gamma_A=\partial A,$$

   at leading large $N$. This is the covariant prescription of Hubeny, Rangamani, and Takayanagi 2007; the surface samples an extended bulk region rather than a single horizon time. Larger regions probe deeper and typically saturate later.

The numerical comparison is reproducible only after the shell profile, region shape, renormalization subtraction, and boundary time are fixed. The hierarchy of nonlocal thermalization scales in Vaidya collapse was demonstrated by Balasubramanian et al. 2011.

Horizon adversary

Declare “thermalization complete” at the first apparent-horizon slice. Now evaluate a geodesic or extremal surface whose turning point remains outside the equilibrated portion of the geometry. Its renormalized value still differs from the final black-brane answer. This directly refutes the observable-independent statement.

The converse also fails: a coarse local observable can look thermal before an apparent horizon appears on a different slicing. Event, apparent, and trapping horizons answer causal or quasi-local geometric questions; none is identical to a boundary operational definition of equilibration.

Causal domain and evidence ceiling

The classical saddle controls leading large-$N$, strong-coupling observables for the prepared state and the time interval before finite-$N$ or quantum-gravity corrections accumulate. It does not show exact thermalization in a finite system, determine recurrences, or establish that the Vaidya stress tensor arises from a microscopic quench in every CFT.

Thermal and Nonequilibrium QFT owns equilibration concepts, curved-spacetime QFT owns dynamical horizons, and the next page isolates entanglement growth.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Balasubramanian, Vijay; Bernamonti, Alice; de Boer, Jan; Copland, Neil; Craps, Ben; Keski-Vakkuri, Esko; Müller, Bernd; Schäfer, Andreas; Shigemori, Masaki; and Staessens, Wieland. “Holographic Thermalization.” Physical Review D 84, 026010 (2011). doi:10.1103/PhysRevD.84.026010.
• Chesler, Paul M., and Laurence G. Yaffe. “Numerical Solution of Gravitational Dynamics in Asymptotically Anti-de Sitter Spacetimes.” Journal of High Energy Physics 2014, 086 (2014). doi:10.1007/JHEP07(2014)086.
• Hubeny, Veronika E.; Rangamani, Mukund; and Takayanagi, Tadashi. “A Covariant Holographic Entanglement Entropy Proposal.” Journal of High Energy Physics 2007, 062 (2007). doi:10.1088/1126-6708/2007/07/062.