Browsing by Subject "Relativity"

Einstein's General Theory of Relativity sits among the pillars of modern physics as the means by which we describe the universe across an enormous range of scales. This theory has furnished our most robust understanding of the origins of the universe, the dynamics of astronomical objects, and the fundamental structure of space and time. For all of general relativity's successes, however, a wide array of deep questions remain. Its sophisticated mathematical structure renders foundational questions surrounding the extent to which the theory is well-posed difficult to answer (and indeed, difficult to ask), and consistent systematic discrepancies between the universe's dynamics and what the theory leads us to expect given our knowledge of the structure of matter leave us puzzling over which of general relativity and particle physics is more incomplete.

This thesis seeks to explore a small cross-section of the fundamental challenges faced by general relativity through two distinct avenues. The first is an investigation of the cosmological properties of scalar field dark matter, often informed by the fact that it may arise through a minor geometric adjustment to the core structure of the theory. The novel cosmological phenomena under consideration primarily include a dark-matter dominated regime in the early universe and a modification to the standard gravitational redshift, and we generally find that (though they are not ruled out) there is little compelling evidence for either amongst the empirical probes considered herein, namely the anisotropies in the cosmic microwave background radiation as measured by the Planck collaboration and a six-year time-domain survey of spectra across many astronomical sources completed by the Anglo-Australian Telescope. The second is a reflection on both the challenge and posing of the Weak Cosmic Censorship Conjecture, the problem of whether singularities in general relativity must generically reside within black holes. We demonstrate that violating singularities are generic within a particular class of spherically symmetric spacetimes, the Vaidya spacetimes, and this reflection leads us to the development of a novel characterization of the phenomenon of black holes, utilized to formulate a more comprehensive rigorous statement of weak cosmic censorship.

In general relativity, the Riemannian Penrose inequality (RPI) provides a lower bound for the ADM mass of an asymptotically flat manifold of nonnegative scalar curvature in terms of the area of the outermost minimal surface, if one exists. In physical terms, an equivalent statement is that the total mass of an asymptotically flat spacetime admitting a time-symmetric spacelike slice is at least the mass of any black holes that are present, assuming nonnegative energy density. The main goal of this thesis is to deduce geometric lower bounds for the ADM mass of manifolds to which neither the RPI nor the famous positive mass theorem (PMT) apply. This is the case, for instance, for manifolds that contain metric singularities or have boundary components that are not minimal surfaces.

The fundamental technique is the use of conformal deformations of a given Riemannian metric to arrive at a new Riemannian manifold to which either the PMT or RPI applies. Along the way we are led to consider the geometry of certain types non-smooth metrics. We prove a result regarding the local structure of area-minimizing hypersurfaces with respect such metrics using geometric measure theory.

One application is to the theory of ``zero area singularities,'' a type of singularity that generalizes the degenerate behavior of the Schwarzschild metric of negative mass. Another application deals with constructing and understanding some new invariants of the harmonic conformal class of an asymptotically flat metric.