Correction to: Cartan invariants and event horizon detection

Abstract

The following sentences at the end of section 2: "There is a fundamental difficulty with applying the alignment classification, as it may depend on solving degree five polynomials to determine the WANDs. The solutions to these polynomials may not be expressible in terms of algebraic functions, and instead require transcendental functions, which are often too complex to implement in practice. Thus, unlike the case in 4D, the ability to explicitly determine the WANDs is not guaranteed." should be replaced by:

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.…”

“…As for the analysis dealing with the detection of the horizon, this finding would confirm that the physical quantities characterizing the black hole are indeed local, but it would also provide a more convenient procedure for estimating them in numerical computations. Thus, the seminal result of [39] has been extended to the case of the Kerr-Newman-NUT-(Anti)-de Sitter black hole in [51] by using both scalar polynomial curvature invariants and Cartan curvature invariants separately, and to the case of the massive Banados-Teitelboim-Zanelli black hole in [40] by using a mixture of these two types of curvature quantities.…”

Understanding Gravitational Entropy of Black Holes: A New Proposal via Curvature Invariants

“…It should be appreciated that our procedure is not sensitive to the matter content of the spacetime, its asymptotic flatness properties, or to the fact that f (r) should be found by integrating the Einstein equations for having a physically relevant spacetime -all we require is the existence of a horizon. We also note that the vanishing of the Cartan invariant DW on the horizon, which made it an appropriate quantity for taming the teleological nature of black hole spacetimes [48], is cured by the same property of the function 1/ f (r) entering the hyperspace volume element dV 3 = (r 2 sin θ/ f (r)) dr dθ dφ.…”

“…for an explicit solution which -for the Λ < 0 case -is of interest in string theory and for applications of the AdS/CFT correspondence [57]. By following [48] [Eqs. (4.1.2)-(4.1.…”

“…This is because some of the frame components of the Weyl tensor and of its first derivative are algebraically dependent among each other. By the theorems due to Cartan these are scalars (because they have been computed in the frame constructed in [48]) and thus they are well-defined integrands. This means that we have some freedom in tuning the constant appearing in front of the integral (if it has some relevant physical meaning) depending on which choices we make.…”