Geometric inequalities for initial data with symmetries

We consider a class of initial data sets (Σ,h,K) for the Einstein constraint equations which we define to be generalized Brill (GB) data. This class of data is simply connected, U(1)²-invariant, maximal, and four-dimensional with two asymptotic ends. We study the properties of GB data and in particular the topology of Σ. The GB initial data sets have applications in geometric inequalities in general relativity. We construct a mass functional M for GB initial data sets and we show:(i) the mass of any GB data is greater than or equals M, (ii) it is a non-negative functional for a broad subclass of GB data, (iii) it evaluates to the ADM mass of reduced t − φi symmetric data set, (iv) its critical points are stationary U(1)²-invariant vacuum solutions to the Einstein equations. Then we use this mass functional and prove two geometric inequalities: (1) a positive mass theorem for subclass of GB initial data which includes Myers-Perry black holes, (2) a class of local mass-angular momenta inequalities for U(1)²-invariant black holes. Finally, we construct a one-parameter family of initial data sets which we show can be seen as small deformations of the extreme Myers- Perry black hole which preserve the horizon geometry and angular momenta but have strictly greater energy.

Some near horizon studies onWeyl and Vaidya spacetimes

This thesis comprises some studies on the Weyl, Vaidya and Weyl distorted Schwarzschild (WDS) spacetimes. The main focal areas are : a) construction of near horizon metric(NHM) for WDS spacetime and subsequently a "stretched horizon" prescribed by the membrane formalism for black holes, b) application of membrane formalism and construction of stretched horizons for Vaidya spacetime and c) using the thin shell formalism to construct an asymptotically flat spacetime with a Weyl interior where the construction does not violate energy conditions. For a), a standard formalism developed in [1] has been used wherein the metric is expanded as a Taylor series in ingoing Gaussian null coordinates with the affine parameter as the expansion parameter. This expansion is used to construct a timelike "stretched horizon" just outside the true horizon to facilitate some membrane formalism studies, the theory for which was first introduced in [2]. b) applies the membrane formalism to Vaidya spacetime and also extends a part of the work done in [1] in which event horizon candidates were located perturbatively. Here, we locate stretched horizons in close proximity to every event horizon candidate located in [1]. c) is an attempt to induce Weyl distortions with a thin shell of matter in an asymptotically flat spacetime without violating energy conditions.