Date of Completion
Cosmology, Relativity, and Gravity | Geometry and Topology
One hundred years ago, Albert Einstein revolutionized our understanding of gravity, and thus the large-scale structure of spacetime, by implementing differential geometry as the pri- mary medium of its description, thereby condensing the relationship between mass, energy and curvature of spacetime manifolds with the Einstein field equations (EFE), the primary compo- nent of his theory of General Relativity. In this paper, we use the language of Semi-Riemannian Geometry to examine the Schwarzschild and the Friedmann-Lemaˆıtre-Robertson-Walker met- rics, which represent some of the most well-known solutions to the EFE. Our investigation of these metrics will lead us to the problem of singularities arising within them, which have mathematical meaning, but whose physical meaning at first seems dubious, due to the highly symmetric nature of the metrics. We then use techniques of causal structure on Lorentz man- ifolds to see how theorems due to Roger Penrose and Stephen Hawking justify that physical singularities do, in fact, occur where we guessed they would.
Dul, Filip, "The Geometry of Spacetime and its Singular Nature" (2016). Honors Scholar Theses. 497.