文档介绍：BASIC GENERAL RELATIVITY
BENJAMIN MCKAY
Contents
1. Introduction 1
2. Special relativity 2
3. Notation 2
4. The action of a relativistic particle 2
5. The metric on the world line 5
6. Wave equations 5
7. General relativity 8
8. Notation 10
9. Examples of fields and field equations 10
10. Parallel transport and covariant derivatives 12
11. Conservation of the energy-momentum tensor 17
12. Curvature 20
13. The field equations of Einstein 22
References 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 electrody-
namics 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 ic 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 ponents with i = 1, . . . , n and xµ ponents with µ = 0, . . . , n.
In these coordinates, our Minkowski metric is
−c2
1
(g ) = .
µν..