Schwarzschild Metric - Part 2

We will now take the spherically symmetric general form for the metric tensor and let the Einstein Field Equations determine the exact functions  and  that appear in that metric.

The covariant form of the spherically symmetric metric is

(18.1)

and the contravariant form is

(18.2)

We will use the following notation for derivatives with respect to time and derivatives with respect to space.

We will also adopt units where .

The Einstein Tensor is now calculated. Its nonzero components are as follows.

 (18.4)

(18.6)

We now impose the Einstein Equations for the vacuum.

 (18.7)

It turns out that the Bianchi Identity Equation

 (18.8)

relates equations (18.3) - (18.6) in such a way that if the equations (18.3)-(18.5) are zero then (18.6) is forced to be zero automatically. Hence (18.6) is not needed. There are only 3 distinct coupled differential equations to be solved arising from (18.3)-(18.5). They are

 (18.10)

 (18.11)

We add the first two of these equations together to get

 (18.12)

This in turn integrates directly to give

 (18.13)

The third differential equation implies that

 (18.14)

We now manipulate the first differential equation (18.9) to give

 (18.15)

We choose the integration constant to have the value

 (18.16)

We take m to be the mass of the gravitational field source. Note that in the units we are using m has units of length. With this choice for the constant, we get the following

 (18.17)

Now continuing with the calculation of the other exponential we do the following

We therefore get that

 (18.18)

We now know that the metric has the following functional form

(18.19)

At this stage the spherically symmetric gravitational field is time dependent. The metric has a form due to spherical symmetry however that allows us to do a transformation that completely removes this time dependence.

Introduce a new time coordinate given by

(18.20)

The only part of the metric that changes with this conversion is the  component.

We now drop all primes from the symbols. The new metric tensor solution of the Einstein Equations is given as

(18.21)

This solution is called the Schwarzschild Solution of the Vacuum Einstein Equations.

The line element is called the Schwarzschild Line Element and it takes the form

(18.22)

We now note two important attributes of the line element solution.

(1) The line element is time translation invariant:  doesn't change the 4-D length. The gravitational field doesn't change in time as long as the spherical symmetry is maintained. Metrics like this are called stationary.

(2) The line element is time symmetric: the time reversal  doesn't change the 4-D distance. Metrics like this, which are also stationary, are called static. Nonstatic metrics have cross-term differentials in their line elements.

We have shown that the most general spherically symmetric metric is such that all time dependence of the gravitational field can be transformed away. Formally this is stated in the following theorem.

Birkhoff’s Theorem: A spherically symmetric vacuum solution is necessarily static.

In Newton’s gravitation theory there is no such connection between the time dependence of the gravitational field and the symmetry of the gravitational field. This theorem implies that a pulsating spherically symmetric star cannot propagate any disturbances into the environment of the star. Thus no gravitational waves can arise in this way.

From taking the flat space limit of the line element it is easy to show that we get consistency with Newtonian physics is the line element has the following form with normal units.

(18.23)

Note that the gravitational field has singularities at  in these coordinates. The special case

(18.24)