A pseudo-Riemannian manifold is called CSI if all scalar polynomial invariants constructed from the curvature tensor and its covariant derivatives are constant. In the Lorentzian case, the CSI spacetimes have been studied extensively due to their app
lication to gravity theories. It is conjectured that a CSI spacetime is either locally homogeneous or belongs to the subclass of degenerate Kundt metrics. Independent of this conjecture, any CSI spacetime can be related to a particular locally homogeneous degenerate Kundt metric sharing the same scalar polynomial curvature invariants. In this paper we will invariantly classify the entire subclass of locally homogeneous CSI Kundt spacetimes which are of alignment type {bf D} to all orders and show that any other CSI Kundt metric can be constructed from them.
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 particul
ar 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.
The locally rotationally symmetric tilted perfect fluid Bianchi type V cosmological model provides examples of future geodesically complete spacetimes that admit a `kinematic singularity at which the fluid congruence is inextendible but all frame com
ponents of the Weyl and Ricci tensors remain bounded. We show that for any positive integer n there are examples of Bianchi type V spacetimes admitting a kinematic singularity such that the covariant derivatives of the Weyl and Ricci tensors up to the n-th order also stay bounded. We briefly discuss singularities in classical spacetimes.
Kundt spacetimes are of great importance in general relativity in 4 dimensions and have a number of topical applications in higher dimensions in the context of string theory. The degenerate Kundt spacetimes have many special and unique mathematical p
roperties, including their invariant curvature structure and their holonomy structure. We provide a rigorous geometrical kinematical definition of the general Kundt spacetime in 4 dimensions; essentially a Kundt spacetime is defined as one admitting a null vector that is geodesic, expansion-free, shear-free and twist-free. A Kundt spacetime is said to be degenerate if the preferred kinematic and curvature null frames are all aligned. The degenerate Kundt spacetimes are the only spacetimes in 4 dimensions that are not $mathcal{I}$-non-degenerate, so that they are not determined by their scalar polynomial curvature invariants. We first discuss the non-aligned Kundt spacetimes, and then turn our attention to the degenerate Kundt spacetimes. The degenerate Kundt spacetimes are classified algebraically by the Riemann tensor and its covariant derivatives in the aligned kinematic frame; as an example, we classify Riemann type D degenerate Kundt spacetimes in which $ abla(Riem), abla^{(2)}(Riem)$ are also of type D. We discuss other local characteristics of the degenerate Kundt spacetimes. Finally, we discuss degenerate Kundt spacetimes in higher dimensions.
We study a class of constant scalar invariant (CSI) spacetimes, which belong to the higher-dimensional Kundt class, that are solutions of supergravity. We review the known CSI supergravity solutions in this class and we explicitly present a number of
new exact CSI supergravity solutions, some of which are Einstein.
We investigate solutions of the classical Einstein or supergravity equations that solve any set of quantum corrected Einstein equations in which the Einstein tensor plus a multiple of the metric is equated to a symmetric conserved tensor $T_{mu u}$
constructed from sums of terms the involving contractions of the metric and powers of arbitrary covariant derivatives of the curvature tensor. A classical solution, such as an Einstein metric, is called {it universal} if, when evaluated on that Einstein metric, $T_{mu u}$ is a multiple of the metric. A Ricci flat classical solution is called {it strongly universal} if, when evaluated on that Ricci flat metric, $T_{mu u}$ vanishes. It is well known that pp-waves in four spacetime dimensions are strongly universal. We focus attention on a natural generalisation; Einstein metrics with holonomy ${rm Sim} (n-2)$ in which all scalar invariants are zero or constant. In four dimensions we demonstrate that the generalised Ghanam-Thompson metric is weakly universal and that the Goldberg-Kerr metric is strongly universal; indeed, we show that universality extends to all 4-dimensional ${rm Sim}(2)$ Einstein metrics. We also discuss generalizations to higher dimensions.
We study the late-time behaviour of tilted perfect fluid Bianchi type III models using a dynamical systems approach. We consider models with dust, and perfect fluids stiffer than dust, and eludicate the late-time behaviour by studying the centre mani
fold which dominates the behaviour of the model at late times. In the dust case, this centre manifold is 3-dimensional and can be considered as a double bifurcation as the 2 parameters ($h$ and $gamma$) of the type VI$_h$ model are varied. We calculate the decay rates and show that for dust or stiffer the models approach a vacuum spacetime, however, it does so rather slowly: $rho/H^2sim 1/ln t$.
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 admi
t 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.
