Gravitational Observatories

TL;DR

This paper analyzes linearized gravity in four-dimensional general relativity with boundaries, exploring mode stability, boundary conditions, and implications for black hole thermodynamics.

Contribution

It introduces a method to control exponential growth modes in linearized gravity by varying boundary extrinsic curvature and extends black hole thermodynamics to conformal boundary conditions.

Findings

Growing modes can be controlled by adjusting boundary extrinsic curvature.

Dirichlet problem for spherical boundary likely has unique solutions at linear level.

Black hole entropy formula remains valid under conformal boundary conditions.

Abstract

We consider four-dimensional general relativity with vanishing cosmological constant defined on a manifold with a boundary. In Lorentzian signature, the timelike boundary is of the form $\boldsymbol{\sigma} \times \mathbb{R}$, with $\boldsymbol{\sigma}$ a spatial two-manifold that we take to be either flat or $S^2$. In Euclidean signature, we take the boundary to be $S^2\times S^1$. We consider conformal boundary conditions, whereby the conformal class of the induced metric and trace $K$ of the extrinsic curvature are fixed at the timelike boundary. The problem of linearised gravity is analysed using the Kodama-Ishibashi formalism. It is shown that for a round metric on $S^2$ with constant $K$, there are modes that grow exponentially in time. We discuss a method to control the growing modes by varying $K$. The growing modes are absent for a conformally flat induced metric on the timelike boundary. We provide evidence that the Dirichlet problem for a spherical boundary does not suffer from non-uniqueness issues at the linearised level. We consider the extension of black hole thermodynamics to the case of conformal boundary conditions, and show that the form of the Bekenstein-Hawking entropy is retained.