106 Theory of Relativityrelativity, the frames can move with variable relative velocities, for exam-
ple, rotate, and there are many admissible coordinate transformations. To
achieve this general invariance, he used tensor analysis and non-Euclidean
geometry. Thus,the mathematical foundation of general relativity is ten-
sor analysis and pseudo-Riemannian geometry, and the main result is a
system of ten nonlinear partial differential equations. This is more than
enough to scare most people away from even trying to understand the sub-
ject. The good news is that, while deriving the equations indeed requires
all this advanced knowledge, solving these equations often requires little
beyond the basic theory of ordinary differential equations. Accordingly,
we start by introducing the equations. We will then describe the mean-
ing of all components of the equations, and finally derive and analyze one
particular solution, known as the Schwarzschild solution. For the sake of
completeness, we summarize the main facts about tensors in Section 8.3 in
Appendix.
At the end of Section 2.4.2, we discussed the metric geometry of rela-
tivistic space-time. This geometry is characterized by the matrix, or the
metric tensor, Q in equation (2.4.20), page 105. In special relativity, the
metric is flat, that is, the matrix g is the same at every point, see (2.4.18)
on page 104.
In general relativity, the metric tensor can be different at different points
and is determined by the gravitational field. The corresponding metric in
space-time is no longer flat, but curved. This equivalence between gravita-
tion and curvature of space-time is the mathematical expression of the main
idea of general relativity. Let us emphasize that general relativity
deals only with gravitational forces.
Einstein's field equations, also known as Einstein's gravitation
equations, or the field equations of general relativity, describe the rela-
tion between the metric tensor g and the gravitational field at every point
of relativistic space-time. To state these equations, we switch from the
(x,y,z,t) variables to (a:^1 ) variables:
x^x^1 , y = x^2 , z = x^3 , t = x*, x = (x^1 ,x^2 ,x^3 ,x^4 ). (2.4.21)Note the use of super-scripts rather than sub-scripts in this definition. The
position of indices is important in tensor calculus; for example, in tensor
calculus, gy is an object very different from Qij. The reader should pay
special attention to position of indices in all formulas in this section.
The equations, introduced by A. Einstein in 1915, are written as follows: