Gauge invariant formalism for second order perturbations of Schwarzschild spacetimes

TL;DR

This paper introduces a gauge invariant formalism for second order perturbations of Schwarzschild spacetimes, improving the conceptual and computational approach to black hole collision models by avoiding gauge-related issues.

Contribution

It develops a self-contained, gauge invariant framework for second order perturbation calculations in Schwarzschild spacetime, including wave equations, initial data prescription, and gravitational wave power computation.

Findings

Provides gauge invariant wave equations and sources.

Offers a method to derive initial data from fundamental forms.

Enables calculation of gravitational wave power from perturbations.

Abstract

The ``close limit,'' a method based on perturbations of Schwarzschild spacetime, has proved to be a very useful tool for finding approximate solutions to models of black hole collisions. Calculations carried out with second order perturbation theory have been shown to give the limits of applicability of the method without the need for comparison with numerical relativity results. Those second order calculations have been carried out in a fixed coordinate gauge, a method that entails conceptual and computational difficulties. Here we demonstrate a gauge invariant approach to such calculations. For a specific set of models (requiring head on collisions and quadrupole dominance of both the first and second order perturbations), we give a self contained gauge invariant formalism. Specifically, we give (i) wave equations and sources for first and second order gauge invariant wave functions; (ii) the prescription for finding Cauchy data for those equations from initial values of the first and second fundamental forms on an initial hypersurface; (iii) the formula for computing the gravitational wave power from the evolved first and second order wave functions.