Affiliation:
1. Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, Copenhagen 2100, Denmark
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–Nordström–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.
This article is part of a discussion meeting issue ‘At the interface of asymptotics, conformal methods and analysis in general relativity’.
Funder
Danmarks Grundforskningsfond
H2020 European Research Council
Villum Fonden
Subject
General Physics and Astronomy,General Engineering,General Mathematics
Cited by
1 articles.
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1. At the interface of asymptotics, conformal methods and analysis in general relativity;Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences;2024-01-15