Quasi-local energy of rotating black hole spacetimes and isometric embeddings of 2-surfaces in Euclidean 3-space

One of the most fundamental problems in classical general relativity is the
measure of e↵ective mass of a pure gravitational field. The principle of equivalence
prohibits a purely local measure of this mass. This thesis critically examines the most
recent quasi-local measure by Wang and Yau for a maximally rotating black hole
spacetime. In particular, it examines a family of spacelike 2-surfaces with constant
radii in Boyer-Lindquist coordinates. There exists a critical radius r* below which, the
Wang and Yau quasi-local energy has yet to be explored. In this region, the results of
this thesis indicate that the Wang and Yau quasi-local energy yields complex values
and is essentially equivalent to the previously defined Brown and York quasi-local
energy. However, an application of their quasi-local mass is suggested in a dynamical
setting, which can potentially give new and meaningful measures. In supporting this
thesis, the development of a novel adiabatic isometric mapping algorithm is included.
Its purpose is to provide the isometric embedding of convex 2-surfaces with spherical
topology into Euclidean 3-space necessary for completing the calculation of quasilocal
energy in numerical relativity codes. The innovation of this algorithm is the
guided adiabatic pull- back routine. This uses Ricci flow and Newtons method to give isometric embeddings of piecewise simplicial 2-manifolds, which allows the algorithm
to provide accuracy of the edge lengths up to a user set tolerance.