Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapters 8 & 9; Problem 9.5.
Starting from the geodesic equation we derived earlier, we can transform it into a different form which is sometimes more useful. The starting point is
We can expand the first term to get
We can substitute this back into 1 to get
We can now multiply this by and sum, using
where we’ve relabelled the dummy index to in the second line. Note that the only non-dummy index is so this is actually a set of equations, where is the dimension of the space-time (or just space).
This form of the geodesic equation requires knowing both the contravariant and covariant forms of the metric tensor, but for diagonal metrics . For non-diagonal metrics, we do have to calculate a matrix inverse, but this isn’t usually too hard.
In the Schwarzschild metric, we can use this form of the geodesic equation to get an equation of motion for the radial coordinate. This metric is
Since this metric is diagonal, the inverse metric is easy to calculate.
Now suppose that the object is at rest, so that the spatial components of the four velocity , for . From the condition we therefore get
For the case of an object at rest, the only non-zero terms in the geodesic equation above are those with , so
Since none of the metric components depends on , the term is zero. The term is non-zero only if , in which case we have
The equation of motion then becomes
The second term is non-zero only when , and so we get
If we require this to reduce to the Newtonian theory of gravity for large distances, where then when an object starts at rest a distance from an object of mass , its initial acceleration is
In the Newtonian limit, and so if the correspondence is to work, we must have
This value is known as the Schwarzschild radius.
As an example, suppose we have a neutron star with mass . If we consider a shell around the star with a circumference of , then the radial coordinate is . There is another shell with circumference , and radial coordinate . The radial distance between these shells is then
Note this is greater than the ordinary difference .