Second order gauge invariant measure of a tidally deformed black hole

TL;DR

This paper develops a second order gauge-invariant measure of tidal perturbations on black hole horizons using Lagrangian perturbation theory, extending previous first order analyses and establishing invariance of the expansion rate as a physical observable.

Contribution

It introduces a second order gauge-invariant measure of tidal perturbations on black hole horizons, advancing the understanding of horizon dynamics in general relativity.

Findings

The horizon expansion rate is gauge-invariant at second order.

A general expression for the tidal perturbation measure is derived.

The measure can be used to analyze black hole responses to external disturbances.

Abstract

In this paper, a Lagrangian perturbation theory for the second order treatment of small disturbances of the event horizon in Schwarzchild black holes is introduced. The issue of gauge invariance in the context of general relativistic theory is also discussed. The developments of this paper is a logical continuation of the calculations presented in \cite{Vega+Poisson}, in which the first order coordinate dependance of the intrinsic and exterinsic geometry of the horizon is examined and the first order gauge invariance of the intrinsic geometry of the horizon is shown. In context of second order perturbation theory, It is shown that the rate of the expansion of the congruence of the horizon generators is invariant under a second order reparametrization; so it can be considered as a measure of tidal perturbation. A general expression for this observable, which accomodates tidal perturbations, is also presented.