Axially Symmetric Rotating Black Holes, Boyer–Lindquist Coordinates, and Regularity Conditions on Horizons

We consider the metric of an axially symmetric rotating black hole. We do not specify the concrete form of a metric and rely on its behavior near the horizon only. Typically, it is characterized (in the coordinates that generalize the Boyer–Lindquist ones) by two integers \(p\) and \(q\) that enter asymptotic expansions of the time and radial metric coefficients in the main approximation. For given \(p\) and \(q\) we find a general form for which the metric is regular, and how the expansions of the metric coefficients look like. We compare two types of requirement: (i) boundedness of curvature invariants, (ii) boundedness of separate components of the curvature tensor in a freely falling frame. Analysis is done for nonextremal, extremal and ultraextremal horizons separately.