Systematic study of event horizons and pathologies of parametrically deformed Kerr spacetimes

In general relativity, all black holes in vacuum are described by the Kerr metric, which has only two independent parameters: the mass and the spin. The unique dependence on these two parameters is known as the no-hair theorem. This theorem may be tested observationally by using electromagnetic or gravitational-wave observations to map the spacetime around a candidate black hole and measure potential deviations from the Kerr metric. Several parametric frameworks have been constructed for tests of the no-hair theorem. Due to the uniqueness of the Kerr metric, any such parametric framework must violate at least one of the assumptions of the no-hair theorem. This can lead to pathologies in the spacetime, such as closed timelike curves or singularities, which may hamper using the metric in the strong-field regime. In this paper, I analyze in detail several parametric frameworks and show explicitly the manner in which they differ from the Kerr metric. I calculate the coordinate locations of event horizons in these metrics, if any exist, using methods adapted from the numerical relativity literature. I identify the regions where each parametric deviation is unphysical as well as the range of coordinates and parameters for which each spacetime remains a regular extension of the Kerr metric and is, therefore, suitable for observational tests of the no-hair theorem.