In general relativity, the nature of mass is non-local. However, an appropriate def-inition of mass at a quasi-local level could give a more detailed characterization ofthe gravitational field around massive bodies. Several attempts have been made tofind such a definition. One of the candidates is the Hawking mass. This thesispresents a method for calculating the spin coefficients used in the expression for theHawking mass, and gives a closed-form expression for the Hawking mass of ellipsoidal2-surfaces in Minkowski spacetime. Furthermore, the Hawking mass is shown to havethe correct limits, both in Minkowski and Schwarzschild, along particular foliationsof leaves approaching a metric 2-sphere. Numerical results for Schwarzschild are alsopresented.

<p>In general relativity, the nature of mass is non-local. However, an appropriate def-inition of mass at a quasi-local level could give a more detailed characterization ofthe gravitational field around massive bodies. Several attempts have been made tofind such a definition. One of the candidates is the Hawking mass. This thesispresents a method for calculating the spin coefficients used in the expression for theHawking mass, and gives a closed-form expression for the Hawking mass of ellipsoidal2-surfaces in Minkowski spacetime. Furthermore, the Hawking mass is shown to havethe correct limits, both in Minkowski and Schwarzschild, along particular foliationsof leaves approaching a metric 2-sphere. Numerical results for Schwarzschild are alsopresented.</p>

<p>In this thesis we calculate the Hawking mass numerically for surfaces in Kerr spacetime. The Hawking mass is a useful tool for proving the Penrose inequality and the result does not contradict the inequality. It also does not contradict the assumption that the Hawking mass should be monotonic for surfaces in Kerr spacetime. The Hawking mass is quasi-local and defined by the spin coefficents of Newman and Penrose, so first we give a discussion about quasi-local quantities and then a short description of the Newman-Penrose formalism.</p>

In this thesis we calculate the Hawking mass numerically for surfaces in Kerr spacetime. The Hawking mass is a useful tool for proving the Penrose inequality and the result does not contradict the inequality. It also does not contradict the assumption that the Hawking mass should be monotonic for surfaces in Kerr spacetime. The Hawking mass is quasi-local and defined by the spin coefficents of Newman and Penrose, so first we give a discussion about quasi-local quantities and then a short description of the Newman-Penrose formalism.

<p>The central object of study of this thesis is inverse mean curvature vector flow of two-dimensional surfaces in four-dimensional spacetimes. Being a system of forward-backward parabolic PDEs, inverse mean curvature vector flow equation lacks a general existence theory. Our main contribution is proving that there exist infinitely many spacetimes, not necessarily spherically symmetric or static, that admit smooth global solutions to inverse mean curvature vector flow. Prior to our work, such solutions were only known in spherically symmetric and static spacetimes. The technique used in this thesis might be important to prove the Spacetime Penrose Conjecture, which remains open today. </p><p>Given a spacetime $(N^{4}, \gbar)$ and a spacelike hypersurface $M$. For any closed surface $\Sigma$ embedded in $M$ satisfying some natural conditions, one can ``steer'' the spacetime metric $\gbar$ such that the mean curvature vector field of $\Sigma$ becomes tangential to $M$ while keeping the induced metric on $M$. This can be used to construct more examples of smooth solutions to inverse mean curvature vector flow from smooth solutions to inverse mean curvature flow in a spacelike hypersurface.</p> / Dissertation

<p>Penrose's inequality which relates the total mass of a space-time containing a black hole with the area of the event horizon, is a yet unproven condition that is required for the cosmic censorship hypothesis. It is believed that the inequality could be proved by using properties of the Hawking mass. This thesis gives analytical expressions for the Hawking mass in slowly rotating Kerr and Kerr-Newman space-times. It is also shown that the expressions are monotonically increasing, a result that does not contradict Penrose's inequality.</p>

Penrose's inequality which relates the total mass of a space-time containing a black hole with the area of the event horizon, is a yet unproven condition that is required for the cosmic censorship hypothesis. It is believed that the inequality could be proved by using properties of the Hawking mass. This thesis gives analytical expressions for the Hawking mass in slowly rotating Kerr and Kerr-Newman space-times. It is also shown that the expressions are monotonically increasing, a result that does not contradict Penrose's inequality.