Mathematical Cosmology and Invariants in General Relativity

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Summary: Mathematical Cosmology and Invariants in General RelativityIn this research program, within mathematical cosmology we shall study inhomogeneous cosmological models using dynamical systems theory and computational methods. We shall study, using both exact solutions and numerics, the effect on the formation of large scale structure of inhomogeneous “spikes” that occur naturally and generically within general relativity and that generate small residual matter perturbations in the early universe. We shall also investigate the possible effects of spatial inhomogeneities, spatial curvature and non-Gaussianities in cosmology through backreaction effects. We propose novel approaches to averaging in cosmology using scalar invariants and within teleparallel gravity. We search for constraints on models of inflation inspired from string theory and higher-dimensional theories. We investigate models which re-collapse and then bounce into a new expansion phase, and determine whether black holes could persist through such a bounce.We have proven in four dimensions (4D) and in higher dimensions that generally a space- time is uniquely characterized by its scalar polynomial (curvature) invariants (SPI). We shall determine a complete (minimal) set of such SPI. We shall obtain a more practical way of determining the algebraic (Weyl) type of a higher dimensional spacetime employing SPI. We shall then algebraically classify higher dimensional black holes and seek new exact black hole solutions and study near horizon geometries. We shall obtain exact solutions in super-gravity and string theory that suffer no quantum corrections to all loop orders in 4D and in higher dimensions. We introduce the “foliation independent” concept of a geometric horizon, which is a surface distinguished by the vanishing of certain SPI (related to algebraical specialization), and motivate its use for the detection of the event horizon for stationary black holes in 4D. We propose to study the application of geometric horizons to more general dynamical black hole scenarios and in higher dimensions. This work is important not only for mathematical relativists, but hopefully also for numerical relativists and astrophysicists.