Black Hole Collapse in the 1/c Expansion

Abstract

We present a first-principles CFT calculation corresponding to the spherical collapse of a shell of matter in three dimensional quantum gravity. In field theory terms, we describe the equilibration process, from early times to thermalization, of a CFT following a sudden injection of energy at time t=0. By formulating a continuum version of Zamolodchikov's monodromy method to calculate conformal blocks at large central charge c, we give a framework to compute a general class of probe observables in the collapse state, incorporating the full backreaction of matter fields on the dual geometry. This is illustrated by calculating a scalar field two-point function at time-like separation and the time-dependent entanglement entropy of an interval, both showing thermalization at late times. The results are in perfect agreement with previous gravity calculations in the AdS$_3$-Vaidya geometry. Information loss appears in the CFT as an explicit violation of unitarity in the 1/c expansion, restored by nonperturbative corrections.

Figures (7)

• Figure 2.1: A shell made up of individual null dust particles collapses to form a BTZ black hole. We have labelled the particles by their dual operator insertion on the boundary in anticipation of our CFT construction in this paper.
• Figure 3.1: Two different OPE channels for a given four-point function. These two channels have the same trivalent graph, but correspond to two distinct conformal block expansions. They differ by moving one insertion point around another.
• Figure 3.2: Two different OPE channels contributing to the correlator (\ref{['eq.expectationValueVaidya']}). The differential equation (\ref{['eq.monodromicODE']}) is required to have trivial monodromy around each cycle indicated in red. The dashed circle is at $|z|=1$.
• Figure 3.3: Two different cycles ${\color{red} \gamma_I}$ and ${\color{red} \gamma_{II}}$ with trivial monodromy for Eq. (\ref{['eq.monodromicODE']}) using the expression (\ref{['eq.ContinuumStressTensor']}) for the stress tensor. In fact any loop straddling the annulus in this fashion has trivial monodromy for the stress tensor (\ref{['eq.ContinuumStressTensor']}).
• Figure 4.1: Path $\gamma$ defining the channel of our correlation function. The black solid lines are the shockwave insertions at $|z| = 1 \pm \sigma$. The path $\gamma$ actually crosses the shockwave twice at the same point, but the crossings are separated in the figure for clarity.