Regularizing Parameterized Kerr Spacetime

Probing gravity in extreme environments with strong and dynamic gravitational fields, such as those around rotating black holes, can allow for small deviations from General Relativity (GR) to be amplified and detected. Various parameterized Kerr spacetimes have been proposed to perform strong-field tests with black hole observations in a theory-agnostic way. These 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 treat the non-GR deviations as small perturbations in one of 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 the new expansions of the functions have remedied the fictitious divergences seen with the original expansion. Additionally, we find overall the parameterized metric does fairly well at recovering the beyond-GR metrics, the root-mean-square error is smaller than about 0.1 in most cases, and can even recover some non-GR solutions exactly.