Holographic Banners

Abstract

This paper is concerned with eternal AdS black holes. The quantum cosmological future and past interior states of the black hole may be placed on an equal footing to the left and right AdS boundary data by considering the on-shell bulk action as a function of the left/right/future/past data: $S[φ^{(0)L},φ^{(0)R},φ^{(0)F},φ^{(0)P}]$. We call this object a holographic banner, and it obeys the Hamilton-Jacobi equation with respect to all four of its arguments. We compute the holographic banner for a scalar field in an AdS black hole background explicitly and use it to construct the semiclassical state in the future interior obtained from a thermofield double state in the past evolved by arbitrary time- and space-dependent boundary sources. When the spacetime itself is dynamical we explain how the holographic banner gives, in principle, a map from boundary data to near-singularity semiclassical quantum cosmology following chaotic BKL dynamics. We obtain the timescale for the BKL dynamics to ergodically mix the future interior quantum state, given a quantum variance in the past state or a classical ensemble of boundary theories.

Figures (3)

• Figure 1: A holographic banner: Boundary couplings $\phi^{(0) L/R}$ evolve the boundary thermofield double state $|\text{tfd}\rangle$ to a late-time boundary state $|\text{tfd}\rangle_\infty$. The boundary couplings source a bulk field $\phi$ which is classically zero in the past interior. In the future interior the bulk field is nonzero, due to the sources, and in a semiclassical quantum state $|\Psi\rangle$ on an interior slice. Near the singularity this state evolves in a simple way under a relational interior time $\tau$.
• Figure 2: The universe near a BKL singularity can be mapped, at each point in space separately, to the trajectory of a particle moving within half of the fundamental domain of $SL(2,{{\mathbb Z}})$ in $\mathbb{H}_2$. The exponential divergence of nearby geodesics causes a semiclassical wavepacket to spread across the entire domain over the mixing time $\tau_\text{mix}$. The figure illustrates the spreading of a collection of geodesics.
• Figure 3: Four regions, each covered by separate $z,t$ coordinates. The direction of increasing Schwarzschild $t$ coordinate in each region is shown. The diagonal lines are the horizons, determined by either $U$ or $V$ vanishing. The direction of increasing $U$ and $V$ is also shown.