Pseudo-Riemannian Geometry and Polynomial Curvature Invariants: Classification, Characterisation and Applications

A hundred years ago Albert Einstein revolutionized how scientists viewed gravity. With Einstein gravity became a geometric theory where the setting was a so-called Lorentzian geometry. The gravitational interaction was essentially reduced to the curvature of space and where matter (including us) travelled in this geometry as geodesics. Much emphasis has been on finding exact solutions of this theory and classifying such solutions. One of the problems is that the same space may look different for different observers. For example, time-dilatation is one such phenomenon; different observers observe different clock-time when travelling relative to each other. In order to avoid such problems, one can compute invariants which are independent of any observer and viewpoint. In this project, we therefore consider invariants that are constructed from the curvature invariants, the so-called polynomial curvature invariants. We know that such invariants cannot entirely distinguish spaces since there are different spaces with identical invariants. For example, plane gravitational wave spacetimes possess this kind of degeneracy: many distinct gravitational wave spaces have identical invariants. The aim of this project is to study these polynomial invariants and precisely classify all those spaces having this degeneracy, i.e., we aim to classify those spaces which cannot be distinguished by considering these polynomial invariants alone. This project considers the Lorentzian case, as well as the more general pseudo-Riemannian case (where there are more "time-directions"). It includes methods from other branches of mathematics, like Lie Group theory, (real) geometric invariant theory, and differential geometry. For example, the project has uncovered a new kind of symmetries playing a vital role for these degenerate spaces, namely those generated by nil-Killing vectorfields. So far, we have established a connection between spaces with identical curvature invariants to so-called Wick-rotations. These Wick-rotations play a vital role for these spaces and connect spaces with different signatures and their properties in a new way. The structure of such Wick-rotated spaces have been investigated, and concrete conditions for when spaces with identical invariants are Wick-rotated have been given. Particularly interesting is the Lorentz case where we have described in full a special class of spaces (so-called type D^k spaces). These have all been determined and can be generalised to provide classes of spacetimes with identical invariants. In addition, we study universal solutions to the field equations; i.e., solutions to any theory of gravity. This concept have also been extended to black hole solutions where the black holes are so-called "universal". This gives insight into the existence of solutions which not necessarily are standard black holes.

-Determined various condition for Wick-rotatable metrics. -Investigated universal metrics. -Formalised the Kundt structure using a GN-structure. -Introduced and investigated the concepts of Nil-Killing vectors and I-preserving diffeomorphisms. -Investigated pseudo-Riemannian metrics being I-degenerate.

The project aims to connect aspects of real Geometric Invariant Theory and pseudo-Riemannian geometry. In particular, the project will investigate the relationship between the scalar polynomial curvature invariants and the pseudo-Riemannian metric. For some metrics, all the (local) information of the metric is given, at least in principle, in the polynomial curvature invariants; i.e., they are characterised by their invariants. However, other metrics are degenerate in the sense that many metrics have the same value of their invariants. This project has a goal to classify all spaces not being characterised by the invariants and will thus provide a better understanding of the interrelation between the set of curvature scalar invariants and the metric for pseudo-Riemannian geometries. Specifically, the project has the following main goals: - Classify the pseudo-Riemannian spaces for which the polynomial curvature invariants do not characterise the metric. - Classify the metrics that have a degenerate structure in the sense that a continuous family of metrics have the same set of invariants. In order to achieve this we will implement tools from real invariant theory by analysing the orbit spaces of linear algebraic groups. These results will then be applied to the curvature tensors of pseudo-Riemannian geometries, leading to algebraic conditions on the curvature tensors when such degeneracy occurs. Next step is to integrate these conditions to obtain canonical forms for the metric in these degenerate cases. We will also use the results for various applications in mathematics and mathematical physics and thereby shed new light on the importance of polynomial invariants in these areas. These applications include holonomy, analytic continuations of metrics, and exact solutions.