Scalar invariants and theoretical cosmology

I am interested in scalar polynomial invariants (SPI) in general relativity (GR). I proved the important theorem that a 4D Lorentzian spacetime is either completely characterised by its SPI or is a so-called "degenerate Kundt spacetime". There are many important applications of this work in string theory and quantum gravity. I have shown that subclasses of the degenerate Kundt spacetimes are exact classical solutions in supergravity and string theory and that a class of Einstein metrics are "universal". I introduced the concept of a geometric horizon (GH), which is a geometrically defined unique, smooth quasi-local surface on which the curvature tensor is algebraically special (and that can be identified by SPI), and motivated its use for the detection of a quasi-local horizon of a black hole (BH). In recent numerical work on binary BH mergers the behaviour of curvature invariants was studied and it was found that the numerical results provide evidence that a (unique) smooth GH can be identified throughout all stages of the binary BH merger. I propose to further study methods (within a frame approach) to simplify computations (both quasi-analytically and numerically) and, in particular, to study the possible role of SPI in more general dynamical BH simulations. I intend to study the mathematical properties of teleparallel gravity and its extensions, in which the frame basis and connection replace the metric tensor as the primary object of study. This includes further investigating their (affine) symmetries, their algebraic classification and the equivalence problem. We have recently determined the correct field equations in a number of physically important situations, and in future work I intend to correct many of the models used in applications in the literature. In addition, I have argued that teleparallel geometry is the more natural framework within which to formulate a mathematically precise, fully covariant, and exact averaging procedure for tensor fields on a manifold. I propose to investigate the applications of this averaging approach in cosmology. I also propose to work on a number of projects within theoretical cosmology, some of which utilize SPI. For example, classically BH can persist in a Universe which collapses to a minimum radius before returning to expansion. I have helped develop a dynamical cosmology near the time of the bounce and, using the notion of a GH (characterized by SPI), I have shown that the individual BH do not merge before nor at the bounce. In future work I aim to focus on the possible cosmological implications of this work, since persistent BH could contribute to the dark matter, provide seeds for galaxies, generate entropy and even drive the bounce itself. A number of more technical projects are also described in the Proposal.