Differential curvature invariants and event horizon detection for accelerating Kerr–Newman black holes in (anti-)de Sitter spacetime

Abstract

We compute analytically differential curvature invariants for accelerating, rotating and charged black holes with a cosmological constant \(\varLambda \). Specifically, we compute novel closed-form expressions for the Karlhede and the Abdelqader-Lake invariants, for accelerating Kerr–Newman black holes in (anti-)de Sitter spacetime or subsets thereof with the aim of detecting physically relevant surfaces, like horizons and ergospheres. We explicitly show that some of the computed invariants of the particular class of spacetimes are vanishing at the event, Cauchy and acceleration horizons or ergosurface. Using the Bianchi identities we calculate in the Newman-Penrose tetrad formalism in closed-form the Page-Shoom curvature invariant for the general class of accelerating, rotating and charged Plebański-Demiański black holes with \(\varLambda \not =0\) and we prove that is zero at the relevant surfaces. For the invariants that vanish at horizon radii we show that are non-zero everywhere else, or in the case there are additional roots such roots do not affect their capability to detect the physically relevant surfaces. Such curvature invariants are locally measurable quantities and thus could allow the local experimental detection of the event and acceleration horizons or outer ergosurface. The differential invariants which are norms associated with the gradients of the first two Weyl invariants, are explored in detail. Although both locally single out the horizons, their global behaviour is also intriguing. Both reflect the background angular momentum and electric charge as the volume of space allowing a timelike gradient decreases with increasing spin and charge.