Mathematics of Physics and Engineering

Einstein's Field Equations 113

From now on, we will consider (2.4.34), which is still a system of 10
equations, and call this system the vacuum field equations. Every fiat
metric, that is, a constant non-singular matrix g, is a trivial solution of
(2.4.34). Indeed, for constant g, the numbers T\j are all equal to zero, and
then so are Rij. A relativistic vacuum solution must be invariant under the
Lorentz transformation. By Exercise 2.4.3 on page 104, such a solution g is
a constant multiple of the flat Minkowski metric (2.4.18) on page 104.
EXERCISE 2.4.10.c Verify the above assertion that every trivial solution of
(2.4-34), invariant under the Lorentz transformation, is a constant multiple
of the flat Minkowski metric (2.4-18).
In 1916, the German physicist KARL SCHWARZSCHILD (1873-1916)
found the first non-trivial solution of the vacuum field equations. Vacuum
field equations do not necessarily describe a space without any gravitation
whatsoever; gravitating matter can be present, but in other parts of the
space, where we are not trying to solve the field equations. Accordingly,
in the derivation of the Schwarzschild solution, it is assumed that there is
only a stationary spherical object of mass M somewhere in space.
Introduce spherical coordinates (r, 0, <p) in the space part of space-time
(x,y,z,t): x = rcos9sintp, y = rsmOsimp, z = rcosip. Then we set
x^1 = r, x^2 = 6, x^3 = ip, x^4 = t. Schwarzschild's solution, or
Schwarzschild's metric, g is a diagonal matrix so that

d(s? = X_}R /r)(dr? + r^2 (sinM^)^2 + W') - c^2 (l - {R 0 /r)){dt)\

(Ro/r)

where

(2.4.35)

Ra = ™i (2.4.36)

is called the Schwarzschild radius of the mass M. Schwarzschild found

this metric by looking for a spherically symmetric solution of (2.4.34); see

below for an outline of the corresponding computations.

EXERCISE 2.4.11. c (a) Verify that R 0 indeed has the dimension of the

length, (b) Compute the Schwarzschild radius of the following objects: (i)

The Sun (take M = 2 • 1030 kg), (ii) The Earth, (Hi) yourself, (c) Find the

mass and the Schwarzschild radius of the ball whose density is p = 1000

kg/m^3 and whose radius is R 0. Hint: you have R 0 = 2G(4n/3)Rlp/c^2 ; find

R 0 and then M = (4n/3)Rlp. (d) Repeat part (c) when p = 1018 kg/m^3 , the

density of a typical atomic nucleus, (e) Verify that, as r —> oo, the metric