Holographic probes of collapsing black holes

TL;DR

This work investigates how holographic probes—specifically boundary-anchored spacelike geodesics and codimension-two extremal surfaces—penetrate the interior of a dynamically forming black hole in global AdS, modeled by Vaidya-AdS spacetimes. By analyzing equal-time boundary-anchored geodesics and symmetric extremal surfaces, the authors uncover dimension-dependent penetration depths: geodesics can reach arbitrarily close to the singularity shortly after collapse (though only in higher dimensions), while extremal surfaces remain bounded away from the singularity but sample interior regions up to a finite depth inside the horizon. The study demonstrates a rich structure in the post-quench evolution, including multiple branches of shortest geodesics and a monotonic growth of extremal-surface areas tied to entanglement entropy, with significant finite-volume effects arising from the global, spherical geometry. These results illuminate how bulk locality and interior geometry are encoded in boundary observables during far-from-equilibrium thermalization, and suggest nuanced, dimension-sensitive limits on how deeply holographic probes can access strong-curvature regions. The findings have implications for understanding the bulk emergence of spacetime and the dynamics of quantum quenches in strongly coupled field theories.

Abstract

We continue the programme of exploring the means of holographically decoding the geometry of spacetime inside a black hole using the gauge/gravity correspondence. To this end, we study the behaviour of certain extremal surfaces (focusing on those relevant for equal-time correlators and entanglement entropy in the dual CFT) in a dynamically evolving asymptotically AdS spacetime, specifically examining how deep such probes reach. To highlight the novel effects of putting the system far out of equilibrium and at finite volume, we consider spherically symmetric Vaidya-AdS, describing black hole formation by gravitational collapse of a null shell, which provides a convenient toy model of a quantum quench in the field theory. Extremal surfaces anchored on the boundary exhibit rather rich behaviour, whose features depend on dimension of both the spacetime and the surface, as well as on the anchoring region. The main common feature is that they reach inside the horizon even in the post-collapse part of the geometry. In 3-dimensional spacetime, we find that for sub-AdS-sized black holes, the entire spacetime is accessible by the restricted class of geodesics whereas in larger black holes a small region near the imploding shell cannot be reached by any boundary-anchored geodesic. In higher dimensions, the deepest reach is attained by geodesics which (despite being asymmetric) connect equal time and antipodal boundary points soon after the collapse; these can attain spacetime regions of arbitrarily high curvature and simultaneously have smallest length. Higher-dimensional surfaces can penetrate the horizon while anchored on the boundary at arbitrarily late times, but are bounded away from the singularity. We also study the details of length or area growth during thermalization. While the area of extremal surfaces increases monotonically, geodesic length is neither monotonic nor continuous.

Figures (21)

• Figure 1: Eddington-Finkelstein (left) and Carter-Penrose (right) diagrams for the AdS-Vaidya spacetime, with $d=4$ and $r_+=1$. The vertical black dashed line on the left side in each panel is the origin of spherical coordinates before the shell begins, and the thick dashed curve the singularity. The AdS boundary is the solid thick line on the right. The infalling shell of matter is indicated by the red shading (its width indicating the shell thickness $\delta$ used in our numerical calculations), and the blue dashed line denotes the event horizon.
• Figure 2: Effective potentials for spacelike geodesics in the BTZ (left) and Schwarzschild-AdS$_5$ (right) geometry with horizon radius $r_+=1$, for various values of the angular momenta: $L=0$ (red) to $L=2$ (purple), in increments of 0.2. The two cases are qualitatively different for low-$L$ values.
• Figure 3: A radial geodesic (solid blue curve) with $v_0=-1.4$ and $E_0=12$, in Schwarzschild-AdS$_5$ with $r_+ =1$, plotted on Eddington (left) and Penrose (right) diagrams, as described in Fig. \ref{['f:penrose']}. We have cut off the uninteresting bottom part of the geodesic; its continuation approaches the boundary in a similar manner to the top part. Note that on the Penrose diagram in the right panel, the geodesic looks like it reaches the singularity, but this is misleading effect of the coordinates, as evident in the Eddington diagram on the left panel.
• Figure 4: Contours of $\Delta t$ for radial geodesics. They are parameterised by the value of $v=v_0$ and the energy $E_0$ when they pass through the origin. The green lines give the $\Delta t=0$ contours, corresponding to ETEBA geodesics.
• Figure 5: Radial geodesics passing through the origin at $v_0=-0.3$, with increasing energy, plotted on Eddington diagram (left) and Penrose diagram (right), with $d=4$ and $r_+=1$. The blue curve has zero initial energy, so is symmetric, the purple has initial energy $E_0=0.5$, and the yellow has $E_0=2.7$, close to the energy required to give equal-time endpoints.