Gauge invariant formalism for second order perturbations of Schwarzschild spacetimes

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, and (iii) the formula for computing the gravitational wave power from the evolved first and second order wave functions.