Parameterizations of black-hole spacetimes beyond circularity

We discuss parameterizations of black-hole spacetimes in and beyond general relativity in view of their symmetry constraints: within the class of axisymmetric, stationary spacetimes, we propose a parameterization that includes non-circular spacetimes, both in Boyer-Lindquist as well as in horizon-penetrating coordinates. We show how existing parameterizations, which make additional symmetry assumptions (first, circularity; second, a hidden constant of motion), are included in the new parameterization. Further, we explain why horizon-penetrating coordinates may be more suitable to parameterize non-circular deviations from the Kerr geometry. Our investigation is motivated by our result that the regular, spinning black-hole spacetimes proposed in Eichhorn and Held (2021 Eur. Phys. J. C 81 933); Eichhorn and Held (2021 J. Cosmol. Astropart. Phys. 5 73) are non-circular. This particular deviation from circularity can result in cusps, a dent and an asymmetry in the photon rings surrounding the black-hole shadow. Finally, we explore a new class of non-circular deviations from Kerr black holes, which promote the spin parameter to a function, and find indications that regularity cannot be achieved in this setting. This result strengthens the case for regular black holes based on a promotion of the mass parameter to a function.

Automated summary

We discuss the parameterizations of black-hole spacetimes in and beyond general relativity, considering their symmetry constraints. A new parameterization method is proposed that includes non-circular spacetimes and horizon-penetrating coordinates. We show how existing parameterizations, which assume circularity and a hidden constant of motion, are included in the new method. Our investigation is motivated by the finding that regular, spinning black-hole spacetimes can have non-circular deviations, resulting in cusps, a dent, and an asymmetry in the photon rings surrounding the black-hole shadow. Furthermore, we explore a new class of non-circular deviations that promote the spin parameter to a function, and find indications that regularity cannot be achieved in this setting.