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A Mathematical Derivation of the General Relativistic Schwarzschild Metric An Honors thesis presented to the faculty of the Departments of Physics and Mathematics East Tennessee State University In partial fulfillment of the requirements for the Honors Scholar and Honors-in-Discipline Programs for a Bachelor of Science in Physics and Mathematics by David Simpson April 2007 Robert Gardner, Ph.D. Mark Giroux, Ph.D. Keywords: differential geometry, general relativity, Schwarzschild metric, black holes
ABSTRACT The Mathematical Derivation of the General Relativistic Schwarzschild Metric by David Simpson We brieﬂy discuss some underlying principles of special and general relativity with the focus on a more geometric interpretation. We outline Einstein’s Equations which describes the geometry of spacetime due to the inﬂuence of mass, and from there derive the Schwarzschild metric. The metric relies on the curvature of spacetime to provide a means of measuring invariant spacetime intervals around an isolated, static, and spherically symmetric mass M , which could represent a star or a black hole. In the derivation, we suggest a concise mathematical line of reasoning to evaluate the large number of cumbersome equations involved which was not found elsewhere in our survey of the literature. 2
CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 Introduction to Relativity . . . . . . . . . . . . . . . . . . . . . . 4 1.1 Minkowski Space . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 What is a black hole? . . . . . . . . . . . . . . . . . . . . . 11 1.3 Geodesics and Christoffel Symbols . . . . . . . . . . . . . 14 2 Einstein’s Field Equations and Requirements for a Solution . 17 2.1 Einstein’s Field Equations . . . . . . . . . . . . . . . . . . 20 3 Derivation of the Schwarzschild Metric . . . . . . . . . . . . . . 21 3.1 Evaluation of the Christoffel Symbols . . . . . . . . . . 25 3.2 Ricci Tensor Components . . . . . . . . . . . . . . . . . . 28 3.3 Solving for the Coeﬃcients . . . . . . . . . . . . . . . . . 36 3.4 Circular Orbits of the Schwarzschild Metric . . . . . . . 40 4 Eddington-Finkelstein Coordinates . . . . . . . . . . . . . . . . 47 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3
1 Introduction to Relativity A quantitative comprehensive view of the universe was arguably first initiated with Isaac Newton’s theory of gravity, a little more than three hundred years ago. It was this theory that first allowed scientists to describe the motion of the heavenly bodies and that of objects on earth with the same principles. In Newtonian mechanics, the universe was thought to be an unbounded, infinite 3-dimensional space modeled by Euclidean geometry, which describes ﬂat space. Thus, any event in the universe could be described by three spatial coordinates and time, generally written as ( x, y, z ) with the implied concept of an absolute time t . 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 frame is also valid for any frame moving uniformly relative to the first. A frame for which this holds is referred to as an inertial reference frame. Second, the speed of light in vacuum is the same in all inertial reference frames, regardless of how the light source may be moving.