In 1915, the same year as the publication of the Field Equations, the German physicist Karl Schwarzschild produced the first exact solution. He did this while in hospital having developed a fatal skin condition while serving in the German army on the Russian front in World War I.
The Schwarzschild Outer Metric is an exact solution to the
Field Equations for the vacuum of space around a static and
symmetric body. The metric must be independent of time and the
spherical
The metric can be written as:
Where
The metric and its inverse are now defined to be:
Using the comma notation, where a comma subscript means a partial derivative of that variable, the Christoffel symbols are:
There are four sets of Christoffel symbols, one set for each
of
For the
The term
This leaves two symbols:
For the
This gives four symbols:
For the
This gives three symbols:
For the
This gives four symbols:
So, the Christoffel symbols are:
The Ricci tensor is defined to be:
Due to symmetry the Ricci tensor has just the four trace emements:
To calculate
To calculate
To calculate
To calculate
The Ricci tensor is:
The Ricci scalar is:
The Einstein Tensor can now be constructed.
The Einstein tensor is zero because the stress energy tensor
The field equations are:
The first two field equations reduce to:
Subtracting gives:
Integrating gives:
Where
The first equation can be written as:
Integrating gives:
Where
This gives.
Therefore.
As the solution must become the Minkowski metric as
This must agree with Newton’s law of gravity for a small
object of mass
Now substitute the value of
The Schwarzschild radius is:
Now, the Christoffel symbols are:
The Schwarzschild outer solution becomes:
The solution has two singularities. The singularity ar
The singularity at