Light frequency change is proportional to gravitational potential change

When light is propagating in a gravitational field, its frequency changes proportionally to the change in the potential of the gravitational field.

Let us consider an infinitesimally small section \(d \vec r\) of the light’s path, such that the frequency of light is constant within that section. In that case we can obtain a dependency between the change in light’s frequency and the change in the gravitational potential.

Notes:

1. The gravitational potential \(\varphi\) is defined as a scalar quantity such that the equation \(\vec g = - \nabla \varphi\) holds where \(\vec g\) is the vector of acceleration due to gravity and \(\nabla\) is the nabla operator.

2. \(d \varphi = - \left( \vec g, d \vec r \right)\) where \(\left( \vec a_1, \vec a_2 \right)\) is the dot product between \(\vec a_1\) and \(\vec a_2\).

Links:

1. Formula 72.4 on p. 378 of “General Course of Physics” (Obschiy kurs fiziki), vol. 1 by Sivukhin D.V. (1979).

frequency_change

The infinitesimal change in temporal_frequency after passing an infinitesimal section \(d \vec r\).

Symbol:

d(f)

Latex:

\(df\)

Dimension:

frequency

frequency

The temporal_frequency of light within an infinitesimal section \(d \vec r\).

Symbol:

f

Latex:

\(f\)

Dimension:

frequency

gravitational_potential_change

The infinitesimal change in gravitational potential after passing an infinitesimal section \(d \vec r\).

Symbol:

d(phi)

Latex:

\(d \phi\)

Dimension:

velocity**2

law

d(f) / f = -d(phi) / c^2

Latex:

\[\frac{df}{f} = - \frac{d \phi}{c^{2}}\]