Regular Black Hole Metric with Three Constants of Motion

According to the no-hair theorem, astrophysical black holes are uniquely characterized by their masses and spins and are described by the Kerr metric. Several parametric spacetimes which deviate from the Kerr metric have been proposed in order to test this theorem with observations of black holes in both the electromagnetic and gravitational-wave spectra. Such metrics often contain naked singularities or closed timelike curves in the vicinity of the compact objects that can limit the applicability of the metrics to compact objects that do not spin rapidly, and generally admit only two constants of motion. The existence of a third constant, however, can facilitate the calculation of observables, because the equations of motion can be written in first-order form. In this paper, I design a Kerr-like black hole metric which is regular everywhere outside of the event horizon, possesses three independent constants of motion, and depends nonlinearly on four free functions that parameterize potential deviations from the Kerr metric. This metric is generally not a solution to the field equations of any particular gravity theory, but can be mapped to known four-dimensional black hole solutions of modified theories of gravity for suitable choices of the deviation functions. I derive expressions for the energy, angular momentum, and epicyclic frequencies of a particle on a circular equatorial orbit around the black hole and compute the location of the innermost stable circular orbit. In addition, I write the metric in a Kerr-Schild-like form, which allows for a straightforward implementation of fully relativistic magnetohydrodynamic simulations of accretion flows in this metric. The properties of this metric make it a well-suited spacetime for strong-field tests of the no-hair theorem in the electromagnetic spectrum with black holes of arbitrary spin.