Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapters 9; Problem 9.6.
This is a first example of the use of the time component of the Schwarzschild metric. This metric is, for a spherical mass :
Suppose we have an observer at a Schwarzschild radius from the centre of a star of mass , and this observer watches a photon move radially outward. The observer measures the energy of the photon to be . We can use this to calculate the four-momentum of the photon.
In special relativity, for an observer at rest the observer’s four-velocity is so the scalar product of the observer’s four-velocity with another object’s momentum (as measured by the observer) is
since the time component of an object’s four-momentum is its energy. Since this is a tensor equation, it should be true in curved space-time as well. In the Schwarzschild metric, an observer at rest has
Therefore, we get
For a photon, , and for a photon moving in the radial direction so
Thus the photon’s four-momentum in the Schwarzschild basis is