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.

Singular space-times with bounded algebraic curvature scalars

“…Zakhary and McIntosh introduced the first (algebraically) complete set, K, of algebraic curvature invariants of the Riemann tensor for a general class of metrics [37]. Completeness stands for: (i) any other invariant can be expressed in an algebraic relation which determines that invariant for all Petrov or Segre types, with some or all of the elements of K and their complex conjugates; (ii) no invariant I i ∈ K can, for all Petrov or Segre types, be expressed in an algebraic relation with the remaining invariants in K. The ZM set corresponds to 17 scalars, being 4 Weyl invariants, 4 Ricci invariants and 9 mixed invariants, namely [37,[64][65][66]:…”

Shadow of a charged black hole with scalar hair

“…So, r = r B is the end of the spacetime. In order to check whether a spacetime is singular, one can compute the curvature invariants, such as Zakhary-McIntosh invariant, Kretschmann scalar and Euler-Poincare invariant [86]. We give the expression of the Kretschmann scalar of the fourdimensional effective spacetime considered in this paper…”