Intrinsic and extrinsic geometries of a tidally deformed black hole

TL;DR

This paper develops a geometric framework for describing the event horizon of a tidally deformed black hole, highlighting the gauge invariance of intrinsic geometry and applying it to specific tidal scenarios.

Contribution

It extends Gauss-Codazzi theory to null hypersurfaces and analyzes the gauge invariance of intrinsic versus extrinsic horizon geometries in tidal deformations.

Findings

Intrinsic horizon geometry is gauge invariant.

Extrinsic geometry depends on parameterization.

Application to tidally deformed black holes in specific scenarios.

Abstract

A description of the event horizon of a perturbed Schwarzschild black hole is provided in terms of the intrinsic and extrinsic geometries of the null hypersurface. This description relies on a Gauss-Codazzi theory of null hypersurfaces embedded in spacetime, which extends the standard theory of spacelike and timelike hypersurfaces involving the first and second fundamental forms. We show that the intrinsic geometry of the event horizon is invariant under a reparameterization of the null generators, and that the extrinsic geometry depends on the parameterization. Stated differently, we show that while the extrinsic geometry depends on the choice of gauge, the intrinsic geometry is gauge invariant. We apply the formalism to solutions to the vacuum field equations that describe a tidally deformed black hole. In a first instance we consider a slowly-varying, quadrupolar tidal field imposed on the black hole, and in a second instance we examine the tide raised during a close parabolic encounter between the black hole and a small orbiting body.