Physics. Basic General Relativity

BASIC GENERAL RELATIVITY BENJAMIN MCKAY Contents 1. Introduction 2. Special relativity 3. Notation 4. The action of a relativistic particle 5. The metric on the world line 6. Wave equations 7. General relativity 8. Notation 9. Examples of fields and field equations 10. Parallel transport and covariant derivatives 11. Conservation of the energy-momentum tensor 12. Curvature 13. The field equations of Einstein References 1 2 2 2 5 5 8 10 10 12 17 20 22 24 “Either the well was very deep, or she fell very slowly, for she had plenty of time as she went down to look about her, and to wonder what was going to happen next” Lewis Carroll, Alice’s Adventures in Wonderland 1. Introduction Weinberg [2] is a beautiful explanation of general relativity. Hawking & Ellis [1] present the most influential examples of spacetime models, and prove the necessity of gravitational collapse under mild physical hypotheses. Unfortunately, in any approach to this subject we have to use some messy tensor calculations. We will assume at least one course in differential geometry. Ultimately we want to consider quantum field theories using path integrals, so we are forced to set up all of our classical physical theories in terms of Lagrangians and principles of least action. Date: November 19, 2001. 1 2 BENJAMIN MCKAY 2. Special relativity Following Jim’s discussion of classical electrodynamics, we see that electrodynamics leads us to believe that spacetime is Minkowski space R1+3 , and we will write it as R1+n to make generalizations easier. How does a material particle move in this space in the absence of any electromagnetic field? 3. Notation Let us establish notation for Minkowski space. The speed of light is c. Write coordinates of points as
(t, ~x) = x0 , x1 , . . . , xn . Write xi for components with i = 1, . . . , n and xµ for components with µ = 0, . . . , n. In th