Cartan invariants and event horizon detection

Abstract

We show that it is possible to locate the event horizon of a black hole (in arbitrary dimensions) by the zeros of certain Cartan invariants. This approach accounts for the recent results on the detection of stationary horizons using scalar polynomial curvature invariants, and improves upon them since the proposed method is computationally less expensive. As an application, we produce Cartan invariants that locate the event horizons for various exact four-dimensional and five-dimensional stationary, asymptotically flat (or (anti) de Sitter), black hole solutions and compare the Cartan invariants with the corresponding scalar curvature invariants that detect the event horizon.

Curvature invariants and lower dimensional black hole horizons

“…In this section we want to establish whether the argument proposed in [39] for locating the horizon applies also to lower dimensional solutions, and/or if it requires any modification due to the different number of independent components of the curvature tensor. In fact, it is well known that the Weyl tensor identically vanishes for any spacetime in (2+1)-dimensional gravity, implying that the Riemann tensor R µνρσ can be fully written in terms of the Ricci tensor R µν , the Ricci scalar R and the metric tensor g µν as follows [43]:…”

“…Therefore a hypothetical spaceship crossing the horizon should in principle be able to recognize it from a change in the behavior of the measured tidal force. We propose that this behavior of the tidal forces can be a valuable technique for distinguishing black holes from other star-like objects which do not have an horizon and therefore according to [39] do not exhibit this property of the Cartan invariants (that is having a spacetime point in which a Cartan invariant connected to the tidal tensor vanishes). Interests in lower-dimensional black holes is motivated from the existence of physical systems whose motion is known to be confined in lower dimensions, like cosmic strings and domain walls which arise in the Polyakov model [73,74] (see also [75] for a review about applications of BTZ black holes in the light of this connection).…”

“…Other possible techniques require approximating the horizon as a minimal Riemannian surface or as a marginally outer trapped surface [36]. Recently it has been proved that it can be located by solving the zeroes of some invariants of the Riemann tensor [37,38], and later the same argument has been extended to the use of the Cartan curvature invariants whose utility has been explicitly demonstrated for (3+1)-and (4+1)-dimensional black holes [39] leading to the geometric horizon conjecture [40,41], which has been tested as well in multi-black holes configurations such as the Kastor-Traschen metric [42]. This literature has also shown that those curvature invariants can be used for extracting locally the values of the mass, electric charge and angular velocity of a black hole, whose mutual relationships can then be compared with the ones predicted by the previously mentioned mechanical laws governing a black hole, and in particular the scaling relation between mass and temperature, which for our lower dimensional case is going to zero when the former is vanishing contrary to the case of ordinary (3+1)-dimensional black holes [43].…”

Geometric horizons in the Kastor-Traschen multi-black-hole solutions

“…For a stationary black hole spacetime, if we know the Killing vector field which acts as the null generator on the event horizon then the horizon is defined locally. This is reflected in the curvature invariants as there is a general procedure to produce scalar polynomial curvature invariants (SPIs) which will vanish on the stationary horizon [9,10] or by employing Cartan invariants [11,12]. This can be generalized to the concept of an isolated horizon (IH) which arises as a non-expanding horizon (NEH) where a class of null normals, { }, exist for which the Lie derivative of { } and the induced covariant derivative on the NEH commute [13,14,15].…”

“…Motivated by the fact that the event horizons of the Reisner-Nordström-(anti) de Sitter solution are detected by SPIs or Cartan invariants [12], we will investigate the existence of GHs in the multi-black hole four-dimensional (4D) KT solutions using the frame approach and utilizing Cartan invariants. We will compare our results with the results of [29] in the case of two black holes, and examine the upper bound on black holes with area larger than 4π/Λ [32].…”