Four-dimensional Einstein Spacetimes

Abstract

The primary goal of this thesis lies in the exploration of a class of spacetime manifolds in four dimensions; the Einstein spacetimes. These are spacetimes that contain only the so-called cosmological constant, Λ. The restriction to four dimensions results in unique algebraic properties for these spacetimes that are not possible in any other dimension. Among these is the proof that a spacetime is Einstein if and only if the Riemann curvature tensor commutes with the Hodge dual, with both viewed as operators on the differential 2-forms. Developing such properties and associated techniques in differential geometry and group theory, we demonstrate a number of interesting results regarding these spacetimes. In the first instance uniqueness theorems are proved establishing that a selection of four-dimensional Einstein spacetimes follow directly from this commutator. Further expanding this approach, it is shown that Λ is an integral of motion on this condition. These results are consequently interpreted as a challenge to the conventional cosmological constant problem. The question of the origin of thermodynamic properties of horizons is addressed from the perspective of spacetime topology, and it is established that the Hawking temperature TH results from the interplay between the topological Euler characteristic and the curvature spectrum of spacetime. What freedom remains in the Einstein field equations beyond these algebraic and topological constraints is explored via the reformulations of General Relativity. With emphasis on Cartan geometry, the possibility of a topological formulation of gravity in four-dimensional vacuum is explored. This thesis is therefore an original self-contained exploration of the Nature of gravitational vacuum in four dimensions.

Date of Award	10 May 2025
Original language	American English
Supervisor	FEDOR Kusmartsev (Supervisor)