More than a century has passed since Albert Einstein published his general theory of relativity. The theory has been tested many times experimentally, primarily in the relatively weak gravitational fields of the solar system [1,2]. More recently the first experimental results from the strong gravitational fields of two black holes have been measured in the form of gravitational waves, which are another prediction of general relativity. The 2017 Nobel prize in physics was awarded to Kip Thorne, Rainer Weiss, and Barry Barish for their role in the detection of gravitational waves. This year we have seen the first image of a black hole from a team of over 200 scientists, further confirming the predictions of general relativity.
General relativity is one of the pillars of modern physics and explains how mass and energy curve spacetime (a Lorentzian manifold), and how the curvature of spacetime affects matter. As presented in 1915, general relativity is a system of ten nonlinear coupled partial differential equations. This means that it is seldom possible to find analytic solutions to his equations. In fact, Einstein did not believe that an exact solution would ever be found to his theory. Over the course of the last century, numerous exact solutions have been found. The technique used in the derivation of these solutions is to assume certain symmetries and other properties of a solution. These symmetries greatly reduce the complexity of the system and allow for a metric (the solution) to be calculated.
In this manuscript we examine a set of symmetries of a spacetime known as affine collineations, which are transformations of the spacetime that preserve the connection. We show how these symmetries give rise to conserved quantities in the geodesic equation. In the case that the metric possesses certain symmetries such that the metric is ”homogeneous”, we explicitly calculate these symmetries and quantities. By homogeneous, we mean that every point of space has the same geometrical properties as every other point. (Two examples of homogeneous manifolds would be the infinite plane R 2 or the surface of a smooth sphere S 2 ). We finish with a discussion on potential applications of affine collineations in astrophysics and numerical relativity.
Maughan, David, "Physics 4900" (2019). Physics Capstone Project. Paper 72.