Hyperboloidal approach for static spherically symmetric spacetimes: a didactical introduction and applications in black-hole physics

TL;DR

This paper introduces a hyperboloidal formalism for static spherically symmetric spacetimes, explaining its geometrical properties, gauge choices, and applications to black-hole physics, including wave equations and various black hole solutions.

Contribution

It provides a didactical framework for hyperboloidal slicing in black-hole spacetimes, including new derivations of the hyperboloidal minimal gauge and applications to diverse black hole models.

Findings

Expression of Trautman-Bondi mass in hyperboloidal coordinates

Derivation of hyperboloidal minimal gauge using in-out and out-in strategies

Application to Schwarzschild, higher-dimensional, and Reissner-Nordström-de Sitter black holes

Abstract

This work offers a didactical introduction to the calculations and geometrical properties of a static, spherically symmetric spacetime foliated by hyperboloidal time surfaces. We discuss the various degrees of freedom involved, namely the height function, responsible for introducing the hyperboloidal time coordinate, and a radial compactification function. A central outcome is the expression of the Trautman-Bondi mass in terms of the hyperboloidal metric functions. Moreover, we apply this formalism to a class of wave equations commonly used in black-hole perturbation theory. Additionally, we provide a comprehensive derivation of the hyperboloidal minimal gauge, introducing two alternative approaches within this conceptual framework: the in-out and out-in strategies. Specifically, we demonstrate that the height function in the in-out strategy follows from the well-known tortoise coordinate by changing the sign of the terms that become singular at future null infinity. Similarly, for the out-in strategy, a sign change also occurs in the tortoise coordinate's regular terms. We apply the methodology to the following spacetimes: Singularity-approaching slices in Schwarzschild, higher-dimensional black holes, black hole with matter halo, and Reissner- Nordstr\"om-de Sitter. From this heuristic study, we conjecture that the out-in strategy is best adapted for black hole geometries that account for environmental or effective quantum effects.