108 Theory of Relativity(2.4.25).- The operator R in (2.4.22) is denned as follows:
RB=aij(*)Rijg- (2-4.28)
EXERCISE 2.4.6. (a)c With the summation convention now in force, write
all the missing sums in (24.26), (2.4.27), and (2.4.28). (b)B Verify that,
for all x and all i,j,k,m, TJ™- = V%, and so there are at most 40 distinct
values of T™y Hint: all you need is that flij(x) = Qji(x). (c)A Verify that
Rijg — Rjig. Hint: find a function f so that r^ = df/dxj.
(d)B Verify that
8 y(x)^-[fl](x) - \gi\x)R[g\{x)gij{x) = -R[g](x). (2.4.29)- The collection of operators Ty, i,j = 1,...,4, in (2.4.22) is called
the stress-energy tensor and is determined by a given distribution of
the gravitational matter in a region of space. This tensor is symmetric:
Ty = Tji (it has to be, since the left-hand side of (2.4.22) is symmetric). At
this level of generality, no specific representation for Ty can be provided:
the particular form of Ty depends on the particular problem. In what
follows, we will concentrate on the vacuum solutions of the field equations,
that is, the solutions of (2.4.22) corresponding to Ty = 0; accordingly, we
will not discuss the stress-energy tensor any further. - The remaining letters in (2.4.22) have the following meaning: G «
6.67 • 10 -11 m^3 /(kg- sec^2 ) is the universal gravitational constant, c « 3 • 108
m/sec is the speed of light, n « 3.14.
EXERCISE 2.4.7.C Verify that (2.4-22) is a system of ten equations for the
ten components gy, 1 < i < j < 4 of the metric tensor.
The system of equations (2.4.22) does not depend on the particular
choice of the coordinate system (ar^4 ) in space-time; in fact, the
functions g^, T™fc, Ri3 [g], and R[g] can be defined in a coordinate-free way,
and the formulas (2.4.26)-(2.4.28) turn out the same in every coordinate
system (of course, the particular form of fly will depend on the coordinate
system). The resulting independence of the field equations (2.4.22) of the
coordinate system is the very foundation of general relativity.
Equations (2.4.22) are too complicated to justify as a postulate or as
a generalization of experimental facts. Instead, the main postulate of gen-