Accurate mapping of spherically symmetric black holes in a parametrized framework

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 [Rezzolla and Zhidenko, Phys. Rev. D 90, 084009 (2014), 10.1103/PhysRevD.90.084009]. 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 an 11-dimensional (11D) parametrization space Π , on which each solution can be visualized as a curve or surface. An L² 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 nonsingular 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 parametrization needed to describe a variety of commonly used black hole spacetimes. When comparing with the first-order RZ parametrization of astrophysical observables such as the ISCO frequency, the coefficients provided here increase the accuracy by 2 orders of magnitude or more.