A tale of spinors and positive energy theorems

In this thesis, I apply spinorial techniques to develop new positive energy theorems in general relativity, both at the global and quasilocal levels. The new results are mainly in the context of spacetimes with negative cosmological constant. At the global level, I focus on asymptotically, locally AdS spacetimes and give particular attention to spacetimes where conformal infinity has compact, Einstein cross-sections admitting real Killing or parallel spinors. A new positive energy theorem is derived for such spacetimes in terms of geometric data intrinsic to the cross- section. This is followed by the first complete proofs of the BPS inequalities in (the bosonic sectors of) 4D and 5D minimal, gauged supergravity, including with magnetic fields, provided the Maxwell field is exact. The BPS inequalities are proven for asymptotically AdS spacetimes, but also generalised to the aforementioned class of asymptotically, locally AdS spacetimes. Meanwhile, at the quasilocal level, I develop a new notion of quasilocal mass for generic, compact, two dimensional, spacelike surfaces in four dimensional spacetimes with negative cosmological constant. The definition is spinorial and based on work for vanishing cosmological constant by Penrose and Dougan & Mason. Furthermore, this mass is non-negative, equal to the Misner-Sharp mass in spherical symmetry, equal to zero for every generic surface in AdS, has an appropriate form for gravity linearised about AdS and has an appropriate limit for large spheres in asymptotically AdS spacetimes. I finish the thesis with some remarks on natural questions and extensions raised by the results and the methods used to derive them.

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