Outer entropy = Bartnik-Bray inner mass, and the gravitational ant conjecture

TL;DR

This paper establishes a precise mathematical link between outer entropy and the Bartnik-Bray quasilocal mass, revealing new bounds and insights into gravitational energy and entropy in the context of quantum gravity.

Contribution

It demonstrates the exact equivalence between outer entropy and the Bartnik-Bray inner mass for certain surfaces, connecting holographic entropy with quasilocal mass concepts.

Findings

Outer entropy equals the Bartnik-Bray inner mass for outer-minimizing mean-convex surfaces.

Derived new bounds between entropy and gravitational energy.

Computed the small sphere limit of outer entropy proportional to the bulk stress tensor.

Abstract

Entropy and energy are found to be closely tied on our quest for quantum gravity. We point out an interesting connection between the recently proposed outer entropy, a coarse-grained entropy defined for a compact spacetime domain motivated by the holographic duality, and the Bartnik-Bray quasilocal mass long known in the mathematics community. In both scenarios, one seeks an optimal spacetime fill-in of a given closed, connected, spacelike, codimension-two boundary. We show that for an outer-minimizing mean-convex surface, the Bartnik-Bray inner mass matches exactly with the irreducible mass corresponding to the outer entropy. The equivalence implies that the area laws derived from the outer entropy are mathematically equivalent as the monotonicity property of the quasilocal mass. It also gives rise to new bounds between entropy and the gravitational energy, which naturally gives the gravitational counterpart to Wall's ant conjecture. We also observe that the equality can be achieved in a conformal flow of metrics, which is structurally similar to the Ceyhan-Faulkner proof of the ant conjecture. We compute the small sphere limit of the outer entropy and it is proportional to the bulk stress tensor as one would expect for a quasilocal mass. Lastly, we discuss some implications of taking quantum matter into consideration in the semiclassical setting.