Pseudo-Riemannian Geometry and invariants in General Relativity

30 May 2022, 08:00 → 3 Jun 2022, 17:00 Europe/Oslo

AR V-101 (University of Stavanger)

Monday 30 May

1 Poincaré-homogeneous geometries at conformal infinity

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 they coexist harmoniously embedded in a pseudo-euclidean space as codimension-two orbits of a Poincaré subgroup acting linearly in the ambient space.

Speaker: José Figueroa-O'Farrill (School of Mathematics, The University of Edinburgh)

2 Universal and almost universal spacetimes

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 universal spacetimes and study the (theory-dependent) form of the resulting field equations and their solutions.

Speaker: Vojtech Pravda (Institute of Mathematics of the Czech Academy of Sciences)

3 Progress on the Kundt conjecture.

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 not $I$-non-degenerate, then there exists some open subset $U\subset M$ which admits a degenerate Kundt null-congruence.

Speaker: Matthew Terje Aadne (UiS TN IMF)

4 Supersymmetric black holes with a single axial symmetry in five dimensions.

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 have a Gibbons-Hawking base of multi-centred type, and the associated harmonic functions have simple poles at the centres on $\mathbb{R}^3$. Smoothness and asymptotic flatness requires these harmonic functions to satisfy certain algebraic constraints, however, these can be satisfied even when the solution does not have any additional symmetry. This gives the first examples of five-dimensional stationary black holes that admit only a single axial symmetry, in contrast to the previously known cases with toroidal isometry.

Speaker: Dávid Katona (University of Edinburgh)

5 Generalized parallel spinors and reductions of Einstein's equations

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 theory of black holes and the relation to the twistor programme of Roger Penrose.

Speaker: Bernardo Araneda (Max Planck Institute for Gravitational Physics)

Tuesday 31 May

6 Black hole horizon equations and the Petrov type D - NUT spacetimes.

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 holes, follows from the Petrov type D of the Weyl tensor on the horizon. For extreme horizons, these equations take a stronger form. There are still important open problems related to the existence of solutions. The class of Petrov type D spacetimes, which are solutions of the Einstein equations, is characterized by a nonzero NUT parameter. In this case, a certain pair of Killing vectors is distinguished by properties of the geometry of the space of orbits. This observation leads to the construction of a globally defined Kerr-NUT-(A)-de-Sitter spacetimes free of the conical singularity. It may be also considered in the acceperated Kerr-NUT-(A)-de-Sitter case.

Speaker: Jerzy Lewandowski (Uniwersytet Warszawski)

7 Can we reconstruct a space-time by a projective and/or conformal structure?

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 talk is based on mutually unrelated papers with A. Trautman, E. Scholz, M. Eastwood, B. Kruglikov and V. Kiosak.

Speaker: Vladimir Matveev (University of Jena)

8 Non-singular spacetimes with the NUT parameter

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 interpretation.

We provide a recipe for extension of this method to the general type D spacetimes (including mass, rotation, acceleration, NUT parameter and the cosmological constant) - it is achieved by imposing the U(1)-principle bundle structure onto the space of the orbits of suitably chosen Killing vector fields, from which the non-singular spacetime can be constructed.

I will discuss the relation to the local theory of horizons, as well as possible applications to the cosmology and regularising the curvature singularities of more widely studied black-hole spacetimes.

Speaker: Maciej Ossowski (University of Warsaw)

9 Locating black hole horizons via curvature invariants: foundation and applications

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. In my talk, I will elaborate on the mathematical foundation of this technique and also point out some motivations from the physical side for trying to further extend it. Specifically, I will discuss the applicability of this method in lower-dimensional gravitational theory in which the Weyl part of the curvature is trivial mentioning some potential physical applications in light of the AdS/CFT correspondence. Next, I will get back to astrophysical settings discussing the evolution of the horizon of the McVittie spacetime, and of another black hole spacetime which arises as a solution in a number of different gravitational theories. I will also explain why it is relevant to know the location of the dynamical apparent horizon in some spacetimes in light of the cosmological holographic principle. Finally, I will show that a certain combination of curvature invariants may serve as an appropriate notion of density of gravitational entropy consistent with the Weyl curvature conjecture.

Speaker: Daniele Gregoris (Jiangsu University of Science and Technology)

10 Propagation properties of the ambient metric equations

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 method proposed by Anderson and Chrusciel).

Speaker: Wojciech Kaminski (University of Warsaw)

11 Null Congruences and Legendrian submanifolds

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.

Speaker: Prof. Vladimir Chernov (Dartmouth College, USA)

12 The Karlhede algorithm and the Cartan equivalence method.

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 geometries, it is difficult to perceive the relation between the two approaches. To wit, Karlhede's algorithm does not make use of the bundle of orthogonal frames and relies instead on iterated normalizations of the curvature tensor. My aim will be to explain the relativity approach to an audience familiar with the Cartan formalism and to highlight some computational advantages of this way of doing equivalence problems.

Speaker: Robert Milson (Dalhousie University)

Wednesday 1 June

13 Conformal transformations of Cahen-Wallach spaces.

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 essential if and only if it has a fixed point. Then we explore the possibility of properly discontinuous groups of conformal transformations acting with a compact orbit space on a conformally curved Cahen-Wallach space. Here our result is that any such group cannot centralise an essential homothety and that for Cahen-Wallach spaces of imaginary type must be contained within the isometries.
This is joint work with Stuart Teisseire (University of Auckland).

Speaker: Thomas Leistner (University of Adelaide)

14 Algebraically special Einstein spacetimes and CR manifolds.

Speaker: Paweł Nurowski (Center for Theoretical Physics Polish Academy of Sciences)

15 Exact radiative solutions of nonlocal gravity

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 non-local analogues of Aichelburg–Sexl and Hotta–Tanaka solutions which represent gravitational waves generated by null sources propagating in Minkowski, de Sitter or anti-de Sitter backgrounds. Then, we step out the class of almost universal spacetimes and focus on axially symmetric type III pp-waves. This ansatz admits gyratonic sources and reduces IDG field equations to a partly linear and decoupled set of two ordinary differential equations. It turns out that with a Gaussian beam of the spinning null matter, this system is still completely solvable and provides an exact gyratonic solution of IDG.

Speaker: Tomáš Málek (Institute of Mathematics, Czech Academy of Sciences)

16 Conformal Lorentzian manifolds from three-dimensional CR structures, and their Einstein metrics

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 metric contained in such a conformal class can be characterised in terms of CR data.

Speaker: Arman Taghavi-Chabert (University of Warsaw)

17 Black holes and other solutions to quadratic gravity

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 Bach tensor is conformally well-behaved, it enables us to construct vacuum solutions to quadratic gravity using conformal transformations. We present various examples of such solutions, including the non-Schwarzschild spherically symmetric black hole obtained by a conformal transformation of a Kundt spacetime.

Speaker: Alena Pravdová (Institute of Mathematics of the Czech Academy of Sciences)

18 Geometric invariants for tracking the horizons of black holes

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 trapping horizon is not always a marginally outer trapped surface. A model is introduced that shows how trapping horizons can be expected to appear outside the event horizon before the black hole starts to evaporate.

Speaker: Alex Nielsen (University of Stavanger)

Thursday 2 June

19 Toric gravitational instantons

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 explicit counterexample to this Riemannian version of the no-hair conjecture. I will discuss recent uniqueness and existence theorems for instantons in this class that possess a torus symmetry, which includes the aforementioned examples.

Speaker: Dr James Lucietti (University of Edinburgh)

20 Universal Einstein-Maxwell solutions

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 Maxwell equations. After giving a characterization of (at least some) universal Einstein-Maxwell solutions, we exemplify the obtained results in the case of particular theories such as ModMax or Horndeski electrodynamics coupled to Einstein gravity.

Speaker: Marcello Ortaggio (Institute of Mathematics of the Czech Academy of Sciences)

21 Differential invariants of Kundt spacetimes

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 derivatives. The talk is based on joint work with Boris Kruglikov.

Speaker: Eivind Schneider (UiT The Arctic University of Norway)

Friday 3 June

22 On the stability of homogeneous Einstein manifolds

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) on the subspace of all constant scalar curvature metrics, and in particular local maxima, seems to be extremely rare.

In this talk, after some general preliminaries, we will focus on the case when the metrics and the variations are G-invariant for some compact Lie group G acting transitively on M. The standard metrics (i.e., defined by minus the Killing form of G) which are Einstein will be specially treated.

This is joint work with Emilio Lauret (Universidad Nacional del Sur and INMABB (CONICET), Argentina) and Cynthia Will (Universidad Nacional de Córdoba and CIEM (CONICET), Argentina).

Speaker: Jorge Lauret (Universidad Nacional de Cordoba)

23 When can a teleparallel geometry be classified by its scalar polynomial torsion invariants

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 equivalent of general relativity (TEGR) and any geometry with this choice of connection will be called a teleparallel geometry.

For a given solution of GR, it is possible to generate a corresponding solution of TEGR, however, this approach is not one-to-one, and we can generate many inequivalent solutions of TEGR. To determine if two solutions of TEGR are equivalent, we could compare their respective scalar polynomial torsion invariants. However, it is expected that the set of all scalar polynomial torsion invariants will not uniquely characterize all teleparallel geometries, so it would be helpful to know when we can use these torsion invariants. In this talk I will discuss the problem in four dimensions and introduce the class of teleparallel geometries which are not characterized by their scalar polynomial torsion invariants.

Speaker: David McNutt (University of Stavanger)

24 Three-dimensional Lorentz critical metrics

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
$$ \mathcal{S}: g\mapsto\mathcal{S}(g)=\int_M\tau_g dvol_g\,, \qquad \mathcal{F}_t:g\mapsto\mathcal{F}_t(g)=\int\{\|\rho_g\|^2+t\tau_g^2\}dvol_g, $$ where $\rho$ and $\tau$ denote the Ricci tensor and the scalar curvature of the metric $g$.
These functionals have been extensively studied in mathematics and physics (see, for example [1,2,3]).

The purpose of this lecture is to present some new results on the classification of critical metrics both in the homogeneous and curvature-homogeneous settings. A specific feature of the Lorentzian situation is the existence of non-Einstein metrics which are critical for all quadratic curvature functionals.

References:

1. M. Berger, Quelques formules de variation pour une structure riemannienne,
   \emph{Ann. Sci. École Norm. Sup.} \textbf{3} (1970), no. 4, 285--294.
2. E. A. Bergshoeff, O. Hohm, and P. K. Townsend, Massive gravity in three dimensions, \emph{Phys. Rev. Lett.} \textbf{102} (20) (2009), 201301, 4 pp.
3. M. J. Gursky and J. A. Viaclovsky, A new variational characterization of three-dimensional space forms, \emph{Invent. Math.} \textbf{145} (2001), 251--278.

Speaker: Eduardo Garcia-Rio (University of Santiago de Compostela)