First order perturbations of the Einstein-Straus and Oppenheimer-Snyder models

TL;DR

This paper develops a framework for analyzing how small perturbations affect the matching conditions between Schwarzschild and FLRW spacetimes, revealing constraints on cosmological and gravitational wave perturbations and implications for rotating black holes.

Contribution

It derives the first order perturbed matching conditions between Schwarzschild and FLRW spacetimes, linking rotational and gravitational wave perturbations, and shows that FLRW cannot source Kerr black holes.

Findings

Perturbed matching conditions relate rotational and gravitational wave perturbations.

A perturbed FLRW universe cannot produce a Kerr black hole.

Vacuoles in a perturbed universe are necessarily static and Schwarzschild.

Abstract

We derive the linearly perturbed matching conditions between a Schwarzschild spacetime region with stationary and axially symmetric perturbations and a FLRW spacetime with arbitrary perturbations. The matching hypersurface is also perturbed arbitrarily and, in all cases, the perturbations are decomposed into scalars using the Hodge operator on the sphere. This allows us to write down the matching conditions in a compact way. In particular, we find that the existence of a perturbed (rotating, stationary and vacuum) Schwarzschild cavity in a perturbed FLRW universe forces the cosmological perturbations to satisfy constraints that link rotational and gravitational wave perturbations. We also prove that if the perturbation on the FLRW side vanishes identically, then the vacuole must be perturbatively static and hence Schwarzschild. By the dual nature of the problem, the first result translates into links between rotational and gravitational wave perturbations on a perturbed Oppenheimer-Snyder model, where the perturbed FLRW dust collapses in a perturbed Schwarzschild environment which rotates in equilibrium. The second result implies in particular that no region described by FLRW can be a source of the Kerr metric.