Curvature invariants for accelerating Kerr–Newman black holes in (anti-)de Sitter spacetime

Abstract

The curvature scalar invariants of the Riemann tensor are important in general relativity because they allow a manifestly coordinate invariant characterisation of certain geometrical properties of spacetimes such as, among others, curvature singularities, gravitomagnetism. We calculate explicit analytic expressions for the set of Zakhary–McIntosh curvature invariants for accelerating Kerr–Newman black holes in (anti-)de Sitter spacetime as well as for the Kerr–Newman–(anti-)de Sitter black hole. These black hole metrics belong to the most general type D solution of the Einstein–Maxwell equations with a cosmological constant. Explicit analytic expressions for the Euler–Poincare density invariant, which is relevant for the computation of the Euler–Poincare characteristic χ(M), and the Kretschmann scalar are also provided for both cases. We perform a detailed plotting of the curvature invariants that reveal a rich structure of the spacetime geometry surrounding the singularity of a rotating, electrically charged and accelerating black hole. These graphs also help us in an exact mathematical way to explore the interior of these black holes. Our explicit closed form expressions show that the above gravitational backgrounds possess a non-trivial Hirzebruch signature density. Possible physical applications of this property for the electromagnetic duality anomaly in curved spacetimes that can spoil helicity conservation are briefly discussed.