Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapters 9; Problem 9.2.
The gravitational redshift due to a spherical mass measured between the Schwarzschild emission radius and reception radius is
For , the formula reduces to
We can test this formula with some real data, obtained from the Hubble telescope on Sirius B, the white dwarf companion of Sirius A, the brightest star in the night sky. Sirius B is believed to have a mass 1.02 times that of the sun and a radius of km. The fractional redshift of its light is with an uncertainty of around 6%, so we can see if this is consistent with the predicted value.
Since we’re making measurements from the Earth, we can take and thus use the simplified formula. In relativistic units we have
The fractional redshift is
The agreement is thus within the error limits of the measurement.