A Mathematical Derivation of theGeneral Relativistic Schwarzschild MetricAn Honors thesis presentedto the faculty of the Departments of Physics and MathematicsEast Tennessee State UniversityIn partial fulfillment of the requirementsfor the Honors Scholar and Honors-in-Discipline Programs for aBachelor of Science in Physics and MathematicsbyDavid SimpsonApril 2007Robert Gardner, Ph.D.Mark Giroux, Ph.D.Keywords: differential geometry, general relativity, Schwarzschild metric, black holes
ABSTRACTThe Mathematical Derivation of theGeneral Relativistic Schwarzschild MetricbyDavid SimpsonWe brieﬂy discuss some underlying principles of special and general relativity withthe focus on a more geometric interpretation. We outline Einstein’s Equations whichdescribes the geometry of spacetime due to the inﬂuence of mass, and from therederive the Schwarzschild metric. The metric relies on the curvature of spacetime toprovide a means of measuring invariant spacetime intervals around an isolated, static,and spherically symmetric massM, which could represent a star or a black hole. Inthe derivation, we suggest a concise mathematical line of reasoning to evaluate thelarge number of cumbersome equations involved which was not found elsewhere inour survey of the literature.2
1Introduction to RelativityA quantitative comprehensive view of the universe was arguably first initiated withIsaac Newton’s theory of gravity, a little more than three hundred years ago. It wasthis theory that first allowed scientists to describe the motion of the heavenly bodiesand that of objects on earth with the same principles. In Newtonian mechanics, theuniverse was thought to be an unbounded, infinite 3-dimensional space modeled byEuclidean geometry, which describes ﬂat space. Thus, any event in the universe couldbe described by three spatial coordinates and time, generally written as (x, y, z) withthe implied concept of an absolute timet.In 1905, Albert Einstein introduced the Special Theory of Relativity in his paper‘On the Electrodynamics of Moving Bodies.’ Special relativity, as it is usually called,postulated two things. First, any physical law which is valid in one reference frameis also valid for any frame moving uniformly relative to the first. A frame for whichthis holds is referred to as an inertial reference frame. Second, the speed of light invacuum is the same in all inertial reference frames, regardless of how the light sourcemay be moving.