Global spectral representations of black hole spacetimes in the complex plane

TL;DR

This paper introduces a global spectral representation of black hole spacetimes in the complex plane, enabling efficient analysis of gravitational waves from binary black hole mergers using Fourier series and complex analysis techniques.

Contribution

It develops a novel spectral representation framework for black hole spacetimes in the complex domain, extending traditional methods and demonstrating its application to initial data and wave equations.

Findings

Spectral representation converges globally in the complex radial coordinate.

Fourier-Legendre expansion effectively models initial data.

Cauchy's integral formula enables accurate signal recovery.

Abstract

Binary black hole coalescence produces a finite burst of gravitational radiation which propagates towards quiescent infinity. These spacetimes are analytic about infinity and contain a dimensionless coupling constant $M/s$, where $M$ denotes the total mass-energy and $s$ an imaginary distance. This introduces globally convergent Fourier series on a complex radial coordinate, allowing spectral representation of black hole spacetimes in all three dimensions. We illustrate this representation theory on a Fourier-Legendre expansion of Boyer-Lindquist initial data and a scalar wave equation with signal recovery by Cauchy's integral formula.