Definition
Redshift is often denoted with the dimensionless variable, defined as the fractional change of the wavelength
Where is the wavelength of the electromagnetic radiation (photon) as measured by the observer. is the wavelength of the electromagnetic radiation (photon) when measured at the source of emission.
The gravitational redshift of a photon can be calculated in the framework of General Relativity (using the Schwarzschild metric) as
with the Schwarzschild radius
,
where denotes Newton's gravitational constant, the mass of the gravitating body, the speed of light, and the distance between the center of mass of the gravitating body and the point at which the photon is emitted. The redshift is not defined for photons emitted inside the Scharzschild radius, the distance from the body where the escape velocity is greater than the speed of light. Therefore this formula only applies when is at least as large as . When the photon is emitted at a distance equal to the Schwarzschild radius, the redshift will be infinitely large. When the photon is emitted at an infinitely large distance, there is no redshift.
In the Newtonian limit, i.e. when is sufficiently large compared to the Schwarzschild radius, the redshift can be approximated by a binomial expansion to become
Read more about this topic: Gravitational Redshift
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