Hyperboloidal framework for the Kerr spacetime

TL;DR

This paper develops a hyperboloidal coordinate framework for Kerr spacetime, enabling accurate wave calculations for gravitational waveforms from extreme-mass-ratio inspirals, and introduces a minimal gauge with simplified equations and extremal limits.

Contribution

It introduces a new hyperboloidal formalism and minimal gauge for Kerr spacetime, facilitating wave equation solutions and connecting with existing formalisms like Leaver's.

Findings

Derived a wave-like equation in 2+1 dimensions with unique solutions from initial data.

Extended hyperboloidal formulation into the frequency domain for Kerr.

Introduced the minimal gauge with simple expressions and extremal limits.

Abstract

Motivated by the need of a robust geometrical framework for the calculation of long, and highly accurate waveforms for extreme-mass-ratio inspirals, this work presents an extensive study of the hyperboloidal formalism for the Kerr spacetime and the Teukolsky equation. In a first step, we introduce a generic coordinate system foliating the Kerr spacetime into hypersurfaces of constant time extending between the black-hole horizon and future null infinity, while keeping track of the underlying degrees of freedom. Then, we express the Teukolsky equation in terms of these generic coordinates with focus on applications in both the time and frequency domains. Specifically, we derive a wave-like equation in $2+1$ dimensions, whose unique solution follows directly from the prescription of initial data (no external boundary conditions). Moreover, we extend the hyperboloidal formulation into the frequency domain. A comparison with the standard form of the Teukolsky equations allows us to express the regularisation factors in terms of the hyperboloidal degrees of freedom. In the second part, we discuss several hyperboloidal gauges for the Kerr solution. Of particular importance, this paper introduces the minimal gauge. The resulting expressions for the Kerr metric and underlying equations are simple enough for eventual (semi)-analytical studies. Despite the simplicity, the gauge has a very rich structure as it naturally leads to two possible limits to extremality, namely the standard extremal Kerr spacetime and its near-horizon geometry. When applied to the Teukolsky equation in the frequency domain, we show that the minimal gauge actually provides the spacetime counterpart of the well-known Leaver's formalism. Finally, we recast the hyperboloidal gauges for the Kerr spacetime available in the literature within the framework introduced here.