Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 8; Problem 8.5.
This is an example of the geodesic equation in a 2-d space-time (with one time and one space dimension). The metric is given in a general way as
where is the generalized spatial coordinate, and is an arbitrary function. The metric tensor is then
Using the time component of the geodesic equation, we set in:
From this we conclude that
for some constant .
Using the condition we get
We can write this in terms of :
That is, the geodesic is the solution of this differential equation.
If we define a new coordinate system in which and is the antiderivative (integral) of then we can transform the metric tensor to this new coordinate system using the standard transformation formula
By implicit differentiation:
The only other non-zero derivative is
The new metric is therefore
This is the metric of flat space-time in rectangular coordinates. Thus any metric with represents flat space-time, since is just a transformation of the flat metric using different coordinates.