I will report on joint work with Emil Have, Stefan Prohazka and Jakob Salzer in which we present a seemingly novel projective compactification of Minkowski space which reveals a rich asymptotic geometry including the blow-ups at spacelike and timelike infinity and a novel four-dimensional space fibering over null infinity. All these spaces are homogeneous under the Poincaré group and moreover...
Universal spacetimes simultaneously solve vacuum field equations of all metric theories of gravity with the Lagrangian constructed from the Riemann tensor and its covariant derivatives of arbitrary order. For spacetimes that are sufficiently “close” to universal spacetimes, the field equations of any theory are also considerably simplified. We introduce one such class of spacetimes - almost...
A Lorentzian manifold $(M,g)$ is said to be $I$-non-degenerate if any deformation of the metric which leaves the collection of scalar curvature invariants unchanged results in a spacetime which is locally isometric to $(M,g)$. In this talk we characterize $I$-non-degeneracy in terms of algebraic type and present progress towards resolving the Kundt conjecture, which states that if $(M,g)$ is...
The classification of higher dimensional black holes is a major open problem in general relativity. In this talk I will consider smooth, asymptotically flat, supersymmetric black holes of five dimensional minimal supergravity. I present a classification of black hole space-times which admit an axial Killing field that `commutes’ with the remaining supersymmetry. I show that these solutions...
We introduce a conformally invariant connection that charaterizes integrable complex structures (and their Lorentzian analogues) in terms of parallel spinors, and we show how this can be used to: reduce the conformal Einstein equations for algebraically special spaces to a single scalar equation, reconstruct the conformal structure, and understand the geometry underlying the perturbation...
The induced geometry at black hole horizons satisfies equations that have consequences similar to those known from global black hole theory: the preferred spherical topology of horizon cross sections - "topological censorship", the existence of axial symmetry - "rigidity", the solution space parametrized by few parameters - "no hair". One of these equations, which knows the secrets of black...
The question in the title was initiated by H. Weyl at the beginning of the 20th century and later was discussed e.g. in an often cited paper of Ehlers, Pirani and Schild from 1972 or in the book of A.Z. Petrov 1969. I will explain different approaches to interesting versions of this question and will in most cases concentrate on the existence and uniqueness of such a reconstruction.
The...
The spacetimes with the NUT parameter are Petrov type D spacetimes commonly associated with a problematic singularity of their axis of symmetry.
One of the possible solutions is to compactify the orbits of the symmetries, which imposes the Hopf fibration structure onto the spacetime. In the simplest case of the Taub-NUT solution this has been done already by Misner, via a so called Misner's...
The teleological nature of black hole horizons, e.g. for finding their location it is necessary to track the focussing properties of a bundle of light rays in the full spacetime although general relativity is a local theory, has been tamed a few years ago. In fact, it has been proved that the location of the horizon corresponds to the algebraic zeroes of some appropriate curvature invariants....
Powers of Laplacians are weakly hyperbolic in the Lorentzian signature, which is not enough for well posed Cauchy problem. However, the conformal GJMS operators have better properties due to their construction via the ambient metric. Similar properties are shared by the obstruction tensor in the harmonic gauge. This fact can be used to prove stability of asymptotically de Sitter spaces (a...
We build on the ideas of Robert Low and show that null congruences correspond to Legendrian submanifolds in the space of all light rays of a spacetime. In particular this allows one to show that the null cone is a null hypersurface even past the conjugate locus. For the part of the null cone before the conjugate locus this was first observed by Roger Penrose.
In general relativity, the invariant classification of spacetimes is typically formulated in terms of a pseudo-algorithm proposed by Anders Karlhede in 1980. At first glance, this algorithm and its subsequent refinements do not bear much resemblence to Cartan's method for the equivalence of G-structures. Indeed, even if one limits the scope of the equivalence method to that of Riemannian...
We study conformal transformations of indecomposable Lorentzian symmetric spaces of non-constant sectional curvature, the so-called Cahen-Wallach spaces. We will present the following results: When a Cahen-Wallach space is conformally curved, its conformal transformations are homotheties. Using this we show that a conformal transformation of a conformally curved Cahen-Wallach space is...
We study exact non-vacuum solutions of infinite derivative gravity (IDG). IDG is a ghost-free non-local theory of gravity, quadratic in curvature, whose field equations are highly convoluted. First, we employ the so-called almost universal spacetimes as an ansatz reducing IDG field equations to a single non-local but linear equation which is exactly solvable. This procedure allows us to obtain...
Any four-dimensional conformal manifold of Lorentzian signature equipped with a twisting non-shearing congruence of null geodesics can locally be constructed on a circle bundle over a three-dimensional contact Cauchy-Riemann (CR) manifold. We describe families of such conformal structures that admit metrics with prescribed Ricci tensor. In particular, we show how the existence of an Einstein...
In general, field equations of quadratic gravity are too complicated to attempt to find exact solutions. However, an additional assumption that the Ricci scalar is constant, which has both a mathematical and physical motivation, leads to some simplifications of the field equations. In particular, all corrections to the Einstein equations are then proportional to the Bach tensor. Since the...
The relation between black hole event horizons and trapping horizons is investigated in a number of different situations. A notion of constant change is introduced that in certain situations allows the location of the event horizon to be identified. A modified definition is introduced to invariantly define the location of the trapping horizon under a conformal transformation. In this case the...
An asymptotically flat gravitational instanton is a four-dimensional, Ricci flat, complete Riemannian manifold that approaches $S^1 \times \mathbb{R}^3$ at infinity. This includes the notable example of the euclidean Kerr instanton, which was conjectured by Gibbons, Hawking and Lapedes to be the unique solution in this class. Remarkably, over 30 years later, Chen and Teo constructed an...
Certain classes of Einstein spacetimes are known to be "immune" to higher-order corrections, i.e., they are simultaneous solutions of (virtually) any modification of Einstein's gravity in vacuum. In this talk we report on recent progress for the case of Einstein-Maxwell solutions, for which one has to consider also the backreaction of the electromagnetic field as well as corrections to the...
We compute generators for the algebra of rational scalar differential invariants of general and degenerate Kundt spacetimes. Special attention is given to dimensions 3 and 4 since in those dimensions the degenerate Kundt metrics are known to be exactly the Lorentzian metrics that can not be distinguished by polynomial curvature invariants constructed from the Riemann tensor and its covariant...
Given a compact differentiable manifold M, the critical points of the total scalar curvature functional Sc on the space of all unit volume Riemannian metrics on M are called Einstein metrics and play a fundamental role in Differential Geometry and Physics. Among Einstein metrics with positive scalar curvature, those which are stable as critical points of Sc (i.e., negative definite Hessian)...
General Relativity (GR) is formulated using Lorentzian manifolds with a connection which is required to be metric-compatible and the resulting torsion tensor vanishes. Interestingly, GR can be reformulated on Lorentzian manifolds with a different connection which is still metric-compatible but now the curvature tensor must vanish. This alternative gravity theory is known as the teleparallel...
Critical metrics for the Hilbert-Einstein functional in dimension three are very rigid since they are necessarily of constant sectional curvature. Substituting the scalar curvature by a quadratic curvature invariant, one is leaded to the consideration of the functionals
$$
\mathcal{S}: g\mapsto\mathcal{S}(g)=\int_M\tau_g...
The stability of MOTS embedded in perfect fluid spacetimes is studied and an upper bound on the area of stable MOTS is obtained. Aspects of the topology of the MOTS, as well as the case when an extension is made to imperfect fluids, are discussed. and the ``growth" of certain matter and curvature quantities on stable MOTS are provided.
