Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 10; Problem 10.3.
Another example of using the Schwarzschild metric to calculate some properties of a particle’s trajectory. This time we start off with two masses at infinity, with one at rest (so ) and another moving radially inwards towards the central mass with . What are the speeds of these two particles when they pass the point ?
We can start by looking at the invariant equation
where is the particle’s momentum as measured by an observer at rest at , and is the four-velocity of the observer. In the Schwarzschild metric an observer at rest has four-velocity
The relation above then becomes
where is the particle’s mass.
We know that the conserved quantity is given by
so we can substitute for above to get
The energy of a particle moving at speed is (from special relativity)
so combining the two results we get
For and :