Conformal description of near-horizon vacuum states

Thomas Banks,Kathryn M Zurek

doi:10.1103/physrevd.104.126026

Thomas Banks, Kathryn M Zurek

Open Access

https://doi.org/10.1103/physrevd.104.126026

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Abstract

Motivated by recent work suggesting observably large spacetime fluctuations in the causal development of an empty region of flat space, we conjecture that these metric fluctuations can be quantitatively described in terms of a conformal field theory of near-horizon vacuum states. One consequence of this conjecture is that fluctuations in the modular Hamiltonian $\Delta K$ of a causal diamond are equal to the entanglement entropy: $\langle \Delta K^2 \rangle = \langle K \rangle = \frac{A(\Sigma_{d-2})}{4 G_d}$, where $A(\Sigma_{d-2})$ is the area of the entangling surface in $d$ dimensions. Our conjecture applies to flat space, the cosmological horizon of dS, and AdS Ryu-Takayanagi diamonds, but not to large finite area diamonds in the bulk of AdS. We focus on three pieces of quantitative evidence, from a Randall-Sundrum II braneworld, from the conformal description of black hole horizons, and from the fluid-gravity correspondence. Our hypothesis also suggests that a broader range of formal results can be brought to bear on observables in flat and dS spaces.

Highlights

Because a detector in any experiment follows a timelike trajectory for a finite proper time, an experimental measurement defines a causal diamond of finite proper time
For a CFT, the general form of the modular Hamiltonian is known inRterms of the stress tensor Tab and the conformal Killing vector (CKV) ζb, K 1⁄4 Hζ 1⁄4 dVad−1Tabζb, where Vd−1, shown as shaded disks in Fig. 1, is a volume element in the CFT
III, we consider that this same result is derived from positing that the near-horizon dynamics of a causal diamond is described by a 2-d CFT with density of states given by the Cardy formula

Summary

INTRODUCTION

Because a detector in any experiment follows a timelike trajectory for a finite proper time, an experimental measurement defines a causal diamond of finite proper time. For a CFT, the general form of the modular Hamiltonian (for any state obtained by acting on the vacuum with a finite number of local operators, spacelike separated from the diamond) is known inRterms of the stress tensor Tab and the CKV ζb, K 1⁄4 Hζ 1⁄4 dVad−1Tabζb, where Vd−1, shown as shaded disks, is a volume element (at fixed time) in the CFT Such a density matrix allows one to compute the thermodynamic quantities hKi and its fluctuations hΔK2i via the free energy. A4NðGBedÞwtionn’as constant Gd. In Sec. III, we consider that this same result is derived from positing that the near-horizon dynamics of a causal diamond is described by a 2-d CFT with density of states given by the Cardy formula. At first sight hΔK2i 1⁄4 hKi for Minkowski diamonds may seem surprising, so we explore why this result could have been expected from the dynamics of conformal field theories in the near-horizon limit

NEAR-HORIZON DYNAMICS

NESTED CAUSAL DIAMONDS AND THE RANDOM WALK

CONCLUSIONS