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

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LironIf the observer was at the Schwarzschild radius then they would be right on the event horizon and 1-2GM/R = 0 so pt of the photon would have a divide by zero error. What about if they were at another radius r ?