Regularizing Parameterized Kerr Spacetime

Deviations from General Relativity (GR) may become large and detectable in extreme environments with strong and dynamic gravitational fields, such as those around rotating black holes. Various parameterized Kerr spacetimes have been proposed to perform strong-field tests with black hole observations in a theory-agnostic way. Some of these parameterized Kerr spacetimes are constructed such that the modified black hole spacetimes still possess certain symmetries of the Kerr black hole. Such symmetry-preserving spacetimes consist of arbitrary functions of the radial coordinate that capture deviations from Kerr in GR. Practically, one expands these functions about infinity and truncates to extract a finite number of deviation parameters. We find this truncation can introduce pathologies such as nonphysical divergences. To overcome this, we take two different attempts: (i) rescale the arbitrary functions, and (ii) treat the non-GR deviations as small perturbations in the parameterized Kerr spacetime, expand, and keep to linear order in the deviation. We then map black hole solutions in several example non-GR theories to the refined parameterized metric and quantify how well the latter can recover the former with a root-mean-square error analysis. We find that both the rescaling and small deviation approximation attempts can remedy the fictitious divergences seen with the original expansion in most cases. Additionally, we find overall the parameterized metric does fairly well at recovering the existing beyond-GR metrics in many cases and can even recover some non-GR solutions exactly.