The Schwarzschild Model 95The importance of the de Sitter model will be illustrated later when we deal with
exponential expansion at very early times in inflationary scenarios.
5.3 The Schwarzschild Model
If the Einstein Equations (3.29) were difficult to derive, it was even more difficult
to find solutions to this system of ten coupled nonlinear differential equations. A
particularly simple case, however, is a single spherically symmetric star of mass푀
surrounded by a vacuum universe, or in any case far away from the gravitational
influence of other bodies.
The Schwarzschild Metric. Let the metric near the star be described by a time
coordinate푡and a radial elevation푟measured from the center of the star. We assume
stable conditions so that the field is stationary, but it varies with elevation. The metric
is then not flat, but the 00 time–time component and the 11 space–space component
must be modified by some functions of푟. Thus it can be written in the form
d푠^2 =퐵(푟)푐^2 d푡^2 −퐴(푟)d푟^2 , (5.66)where퐵(푟)and퐴(푟)have to be found by solving the Einstein equations.
Far away from the star the space-time may be taken to be flat. This gives us the
asymptotic conditions
푟lim→∞퐴(푟) = lim푟→∞퐵(푟)=^1. (5.67)
From Equation (3.41) the Newtonian limit of푔 00 is known. Here퐵(푟)plays the role
of푔 00 ; thus we have
퐵(푟)= 1 −2 GM
푐^2 푟
. (5.68)
To obtain퐴(푟)from the Einstein equations is more difficult, and we shall not go to
the trouble of deriving it. The exact solution found byKarl Schwarzschild(1873–1916)
in 1916 preceded any solution found by Einstein himself. The result is simply
퐴(푟)=퐵(푟)−^1. (5.69)These functions clearly satisfy the asymptotic conditions [Equation (5.67)].
Let us introduce the concept ofSchwarzschild radius푟cfor a star of mass푀,defined
by퐵(푟c)=0. It follows that
푟c≡2 GM
푐^2
. (5.70)
The physical meaning of푟cis the following. Consider a test body of mass푚and radial
velocity푣attempting to escape from the gravitational field of the star. To succeed, its
kinetic energy must overcome the gravitational potential. In the nonrelativistic case
the condition for this is
1
2
푚푣^2 ⩾GMm∕푟. (5.71)