Accurate mapping of spherically symmetric black holes in a parameterised framework

TL;DR

This paper extends the Rezzolla-Zhidenko framework to accurately parametrize spherically symmetric black-hole spacetimes, enabling precise astrophysical measurements and tests of gravity theories with minimal deviation errors.

Contribution

It introduces an 11-dimensional parametrization space and provides high-accuracy coefficients, significantly improving the precision of black-hole spacetime descriptions.

Findings

Low deviation errors (~10^{-6}) in key observables.

Enhanced accuracy (over 100x) in black-hole parameterization.

Applicability to various compact objects and gravity theories.

Abstract

The Rezzolla-Zhidenko (RZ) framework provides an efficient approach to characterize spherically symmetric black-hole spacetimes in arbitrary metric theories of gravity using a small number of variables [L. Rezzolla and A. Zhidenko, Phys. Rev. D. 90, 084009 (2014)]. These variables can be obtained in principle from near-horizon measurements of various astrophysical processes, thus potentially enabling efficient tests of both black-hole properties and the theory of general relativity in the strong-field regime. Here, we extend this framework to allow for the parametrization of arbitrary asymptotically-flat, spherically symmetric metrics and introduce the notion of a 11-dimensional (11D) parametrization space $\Pi$, on which each solution can be visualised as a curve or surface. An $\mathscr{L}^2$ norm on this space is used to measure the deviation of a particular compact object solution from the Schwarzschild black-hole solution. We calculate various observables, related to particle and photon orbits, within this framework and demonstrate that the relative errors we obtain are low (about $10^{-6}$). In particular, we obtain the innermost stable circular orbit (ISCO) frequency, the unstable photon-orbit impact parameter (shadow radius), the entire orbital angular speed profile for circular Kepler observers and the entire lensing deflection angle curve for various types of compact objects, including non-singular and singular black holes, boson stars and naked singularities, from various theories of gravity. Finally, we provide in a tabular form the first 11 coefficients of the fourth-order RZ parameterization needed to describe a variety of commonly used black-hole spacetimes. When comparing with the first-order RZ parameterization of astrophysical observables such as the ISCO frequency, the coefficients provided here increase the accuracy of two orders of magnitude or more.