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

TL;DR

This paper derives explicit analytic expressions for curvature invariants of accelerating Kerr-Newman black holes in (anti-)de Sitter spacetime, revealing complex geometric structures and potential physical implications like electromagnetic duality anomalies.

Contribution

It provides the first detailed analytic formulas for Zakhary-McIntosh invariants and related curvature scalars for these general black hole solutions.

Findings

Rich structure of spacetime geometry near singularities

Explicit formulas for Euler-Poincare and Kretschmann scalars

Identification of non-trivial Hirzebruch signature density

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 $\chi(M)$, 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.