Spinors, the Sen-Witten equation, quasilocal energy and time functions in general relativity

This work investigates three uses of spinors in general relativity. First we define a new quasilocal mass based on the Nester-Witten two form and solutions of the Sen-Witten equation. The boundary conditions for the Sen-Witten equation are set so that one of the components of the solution spinor equals the same component of either the holomorphic or anti-holomorphic spinor. We evaluate the mass in flat, Schwarzchild, Tolman and naked black hole spacetimes. In flat spacetime we find that the mass is zero. In the Schwarzchild and Tolman spacetimes we find that mass behaves well in some cases but in others it fails to have properties that we deem desirable. In the naked black hole spacetime we look at the difference between the mass value evaluated in static and infalling reference frames specifically looking for artifacts of the vastly differing experiences of the two sets of observers, none are found. Our results coupled with the computational complexity of the new mass definition provide little reason to prefer it over more established definitions such as Dougan and Mason's or Brown and York's. Next we turn to the question of how the Dougan-Mason mass depends on the timeslice it is evaluated on. We evaluate the mass on the event horizon of a Schwarzchild black hole on a given timeslice and then modify the timeslice using a function of the angular coordinates. We find the mass changes and that the change depends on the ℓ = 0 and ℓ = 1 components of the spherical harmonic expansion of an expression involving the function used to modify the timeslice. Finally we consider the current vectors of solutions to the massless Dirac equation as evolution vectors for numerical relativity. We limit ourselves to spherically symmetric spacetimes where we show that the current vector must always be surface forming. The Dirac equation is solved in flat and Schwarzchild spacetimes with various initial and boundary conditions. We find that the vectors behave well with respect to being used to evolve Einstein's equations for the cases considered.