Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 16; Box 16.2.
In radiation from a black hole, we found an estimate for the energy of a radiated particle as measured in the frame of an observer who is momentarily at rest at the point at which the particle pair is created.
To work out the energy at infinity in the Schwarzschild (S) frame, we can use the particle’s four-momentum in the form
If we work out this equation in S coordinates, we have
so using the S metric, we have
Since the energy per unit mass of a particle is given by
Since is the energy per unit mass at infinity, is the total energy of the particle at infinity, so
For a particle that is created at with energy given by 1, the energy at infinity is
This can be expanded in a series around to get
Thus for very small distances from the event horizon, the energy the particle has at infinity tends to a constant.