Into the Bulk: A Covariant Approach

TL;DR

This paper introduces a covariant, horizon-independent method to define and compare the depth of bulk regions in holographic geometries, linking bulk depth to boundary correlator singularities and RG flow.

Contribution

It presents a novel, covariant framework for measuring bulk depth that applies to generic geometries and relates to holographic RG flow and correlator singularities.

Findings

Causal wedges probe monotonically deeper regions in the bulk.

The definition aligns with expectations in pure AdS and static black holes.

Bulk depth correlates with boundary correlator singularities and RG flow.

Abstract

I propose a general, covariant way of defining when one region is "deeper in the bulk" than another. This definition is formulated outside of an event horizon (or in the absence thereof) in generic geometries; it may be applied to both points and surfaces, and may be used to compare the depth of bulk points or surfaces relative to a particular boundary subregion or relative to the entire boundary. Using the recently proposed "lightcone cut" formalism, the comparative depth between two bulk points can be determined from the singularity structure of Lorentzian correlators in the dual field theory. I prove that, by this definition, causal wedges of progressively larger regions probe monotonically deeper in the bulk. The definition furthermore matches expectations in pure AdS and in static AdS black holes with isotropic spatial slices, where a well-defined holographic coordinate exists. In terms of holographic RG flow, this new definition of bulk depth makes contact with coarse-graining over both large distances and long time scales.