Curvature invariants and lower dimensional black hole horizons

Abstract

It is known that the event horizon of a black hole can often be identified from the zeroes of some curvature invariants. The situation in lower dimensions has not been thoroughly clarified. In this work we investigate both (2+1)-and (1+1)-dimensional black hole horizons of static, stationary and dynamical black holes, identified with the zeroes of scalar polynomial and Cartan curvature invariants, with the purpose of discriminating the different roles played by the Weyl and Riemann curvature tensors. The situations and applicability of the methods are found to be quite different from that in 4-dimensional spacetime. The suitable Cartan invariants employed for detecting the horizon can be interpreted as a local extremum of the tidal force suggesting that the horizon of a black hole is a genuine special hypersurface within the full manifold, contrary to the usual claim that there is nothing special at the horizon, which is said to be a consequence of the equivalence principle.

A Critical Assessment of Black Hole Solutions With a Linear Term in Their Redshift Function

“…its behavior at null spatial infinity, for locating its horizon(s) which can therefore be observed by experimental devices of finite sizes, taming its teleological nature pointed out in [43,44]. Therefore, we have explicitly shown that the horizon constitutes a local property of the manifold (in agreement with the core principles of any relativistic field theory), and that a curvature invariant can be constructed for its detection once the geometrical symmetries of the spacetime are known regardless the gravitational theory behind it: this is consistent with the geometric horizon conjecture [40,[45][46][47][48][49][50][51][52][53]. On the other hand, our results are important also from the practical point of view in light of the so-called excision technique in numerical relativity: the black hole horizon constitutes a causal boundary separating the evolutions of phenomena occurring outside it from what it may happen inside; thus, the spacetime region delimited by the horizon must be removed (or excised) when performing numerical simulations of the evolution of a black hole.…”

A critical assessment of black hole solutions with a linear term in their redshift function

“…its behavior at null spatial infinity, for locating its horizon(s) which can therefore be observed by experimental devices of finite sizes, taming its teleological nature pointed out in [43,44]. Therefore, we have explicitly shown that the horizon constitutes a local property of the manifold (in agreement with the core principles of any relativistic field theory), and that a curvature invariant can be constructed for its detection once the geometrical symmetries of the spacetime are known regardless the gravitational theory behind it: this is consistent with the geometric horizon conjecture [40,[45][46][47][48][49][50][51][52][53]. On the other hand, our results are important also from the practical point of view in light of the so-called excision technique in numerical relativity: the black hole horizon constitutes a causal boundary separating the evolutions of phenomena occurring outside it from what it may happen inside; thus, the spacetime region delimited by the horizon must be removed (or excised) when performing numerical simulations of the evolution of a black hole.…”

Event horizon detecting invariants

“…Beyond five dimensions, just like the method of [4], generically with our method, we need more than one horizon detecting invariant because usually the cohomogeneity is more than two. For more on the construction of horizon detecting invariants see [15] where Cartan invariants are suggested and see [16] where they were employed for lower dimensional nonvacuum black holes such as the charged BTZ metric.…”