Motivation

To give an example of the general methodology one would follow to find a solution to the Einstein Field Equations. The specific example used yields the exterior Schwarzschild metric.

Definitions

Recall the definitions of the basic entities used in the equations.

Einstein Field Equations ( without cosmological constant ) :

(1)

Ricci tensor :

(2)

Christoffel symbols :

(3)

Contracted Christoffel symbols :

(4)

Ansatz

Every solution of the field equations requires an ansatz; in this thread we will look at the simplest possible solution of the equations, which is the vacuum solution of a spherically symmetric gravitational field for a static mass. The solution is called the Schwarzschild Metric. The spherical symmetry and the condition that mass and resulting field are static leads to the following simple ansatz :

(5)

with two as yet unspecified functions A(r) and B(r). Our task will be to find these two functions from the field equations. EDIT : Note that I am using the sign convention (+,-,-,-) for this ansatz.

Field Equations

In a vacuum ( ) the Einstein Field Equations (1) reduce to

(6)

which is a set of partial differential equations for the unknown functions A(r) and B(r).

Calculating the Christoffel Symbols

The elements of the Christoffel symbols which do not vanish are

Calculating the Ricci Tensor

The non-vanishing elements of the Ricci tensor are thus

Solving the Equations

From the above we obtain the system of equations

We now write

and, doing some algebra, we obtain from this

We also know that the gravitational field vanishes at infinity, i.e for we obtain

and therefore

Now we can insert this into the remaining equations :

One can easily verify that these two differential equations are solved by

with an integration constant a. This constant is determined by the condition that the solution of the field equation must reduce the usual Newton's law at infinity; therefore

Putting all this back into the ansatz (5) gives us the solution of the Einstein field equation we were looking for :

This is called the Exterior Schwarzschild Metric, and its form is the simplest possible vacuum solution to the original field equations without cosmological constant.

References

Fliessbach, Prof Torsten : Allgemeine Relativitätstheorie , Mannheim/Wien/Zuerich : BI-Wiss.-Verl. 1990