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Register a new userAlgebraic classification of five-dimensional spacetimes using scalar invariants
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There are a number of algebraic classifications of spacetimes in higher dimensions utilizing alignment theory, bivectors and discriminants. Previous work gave a set of necessary conditions in terms of discriminants for a spacetime to be of a particular algebraic type. We demonstrate the discriminant approach by applying the techniques to the Sorkin-Gross-Perry soliton, the supersymmetric and doubly-spinning black rings and some other higher dimensional spacetimes. We show that even in the case of some very complicated metrics it is possible to compute the relevant discriminants and extract useful information from them.
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We wish to construct a minimal set of algebraically independent scalar curvature invariants formed by the contraction of the Riemann (Ricci) tensor and its covariant derivatives up to some order of differentiation in three dimensional (3D) Lorentzian spacetimes. In order to do this we utilize the Cartan-Karlhede equivalence algorithm since, in general, all Cartan invariants are related to scalar polynomial curvature invariants. As an example we apply the algorithm to the class of 3D Szekeres cosmological spacetimes with comoving dust and cosmological constant $Lambda$. In this case, we find that there are at most twelve algebraically independent Cartan invariants, including $Lambda$. We present these Cartan invariants, and we relate them to twelve independent scalar polynomial curvature invariants (two, four and six, respectively, zeroth, first, and second order scalar polynomial curvature invariants).
We algebraically classify some higher dimensional spacetimes, including a number of vacuum solutions of the Einstein field equations which can represent higher dimensional black holes. We discuss some consequences of this work.
We introduce the concept of a geometric horizon, which is a surface distinguished by the vanishing of certain curvature invariants which characterize its special algebraic character. We motivate its use for the detection of the event horizon of a stationary black hole by providing a set of appropriate scalar polynomial curvature invariants that vanish on this surface. We extend this result by proving that a non-expanding horizon, which generalizes a Killing horizon, coincides with the geometric horizon. Finally, we consider the imploding spherically symmetric metrics and show that the geometric horizon identifies a unique quasi-local surface corresponding to the unique spherically symmetric marginally trapped tube, implying that the spherically symmetric dynamical black holes admit a geometric horizon. Based on these results, we propose a suite of conjectures concerning the application of geometric horizons to more general dynamical black hole scenarios.
We derive expressions for the general five-dimensional metric for Kerr-(A)dS black holes. The Klein-Gordon equation is explicitly separated and we show that the angular part of the wave equation leads to just one spheroidal wave equation, which is also that for charged five-dimensional Kerr-(A)dS black holes. We present results for the perturbative expansion of the angular eigenvalue in powers of the rotation parameters up to 6th order and compare numerically with the continued fraction method.
We present the explicit metric forms for higher dimensional vanishing scalar invariant (VSI) Lorentzian spacetimes. We note that all of the VSI spacetimes belong to the higher dimensional Kundt class. We determine all of the VSI spacetimes which admit a covariantly constant null vector, and we note that in general in higher dimensions these spacetimes are of Ricci type III and Weyl type III. The Ricci type N subclass is related to the chiral null models and includes the relativistic gyratons and the higher dimensional pp-wave spacetimes. The spacetimes under investigation are of particular interest since they are solutions of supergravity or superstring theory.
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