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

Abstract: We compute analytically differential invariants for accelerating, rotating and charged black holes with a cosmological constant $\Lambda$. In particular, we compute in closed form novel explicit algebraic expressions for curvature invariants constructed from covariant derivatives of the Riemann and Weyl tensors, such as the Karlhede and the Abdelqader-Lake invariants, for the Kerr-Newman-(anti-)de Sitter and accelerating Kerr-Newman-(anti-)de Sitter black hole spacetimes. We explicitly show that some of the computed curvature invariants are vanishing at the event and Cauchy horizons or the ergosurface of the accelerating, charged and rotating black holes with a non-zero cosmological constant. Using a particular generalised null-tetrad and the Bianchi identities we compute in the Newman-Penrose formalism in closed-analytic form the Page-Shoom curvature invariant for the accelerating Kerr-Newman black hole in (anti-)de Sitter spacetime and prove that is vanishing at the black hole event and Cauchy horizon radii. Therefore such invariants can serve as possible detectors of the event horizon and ergosurface for such black hole metrics which belong to the most general type D solution of the Einstein-Maxwell equations with a cosmological constant. Also the norms associated with the gradients of the first two Weyl invariants in the Zakhary-McIntosh classification, were studied 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 angular momentum and charge.