Editing Gravitational redshift (section)

==== Prediction of the Newtonian limit using the properties of photons ====
The formula for the gravitational red shift in the Newtonian limit can also be derived using the properties of a photon:<ref>A. Malcherek: ''Elektromagnetismus und Gravitation'', Vereinheitlichung und Erweiterung der klassischen Physik. 2. Edition, Springer-Vieweg, Wiesbaden, 2023, ISBN 978-3-658-42701-6. [[doi:10.1007/978-3-658-42702-3]]</ref>

In a gravitational field <math>\vec{g}</math> a particle of mass <math>m</math> and velocity  <math>\vec{v}</math> changes it's  energy <math>E</math> according to:
: <math>\frac{\mathrm dE}{\mathrm dt} = m \vec{g}\cdot \vec{v} = \vec{g}\cdot\vec{p}</math>.

For a massless photon described by its energy <math>E = h \nu = \hbar \omega</math> and momentum   <math>\vec{p} = \hbar\vec{k}</math> this equation becomes after dividing by the Planck constant <math>\hbar</math>:
: <math>\frac{\mathrm d \omega}{\mathrm dt} = \vec{g}\cdot \vec{k}</math>

Inserting the gravitational field of a spherical body of mass <math>M</math> within the distance  <math>\vec{r}</math>
: <math>\vec{g} = -G M \frac{\vec{r}}{r^3}</math>
and the wave vector of a photon leaving the gravitational field in radial direction
: <math>\vec{k} = \frac{\omega}{c} \frac{\vec{r}}{r}</math>
the energy equation becomes
: <math>\frac{\mathrm d \omega}{\mathrm dt} = -\frac{G M}{c} \frac{\omega}{r^2}.</math>
Using <math>\mathrm dr = c \,\mathrm dt</math> an ordinary differential equation which is only dependent on the radial distance <math>r</math> is obtained:
: <math>\frac{\mathrm d \omega}{\mathrm dr} = -\frac{G M}{c^2} \frac{\omega}{r^2} </math>

For a photon starting at the surface of a spherical body with a Radius <math>R_e</math> with a frequency <math>\omega_0 = 2 \pi \nu_0</math> the analytical solution is:
: <math>\frac{\mathrm d \omega}{\mathrm dr} = -\frac{G M}{c^2} \frac{\omega}{r^2} \quad \Rightarrow \quad \omega(r) = \omega_0 \exp \left ( -\frac{G M}{c^2} \left( \frac{1}{R_e} - \frac{1}{r} \right) \right) </math>

In a large distance from the body <math>r \rightarrow \infty</math> an observer  measures the frequency :
: <math>\omega_\text{obs} = \omega_0 \exp \left ( -\frac{G M}{c^2} \left( \frac{1}{R_e} \right) \right) \simeq \omega_0 \left( 1 - \frac{G M}{R_e c^2} + \frac{1}{2} \frac{G^2 M^2}{R_e^2 c^4} - \ldots \right). </math>

Therefore, the red shift is:
: <math> z = \frac{\omega_0 - \omega_\text{obs}}{\omega_\text{obs}}
= \frac{1 - \exp \left( -\frac{G M}{R_e c^2} \right)}{\exp \left( -\frac{G M}{R_e c^2} \right)} = \frac{1 - \exp \left( -\frac{r_S}{2 R_e} \right)}{\exp \left( -\frac{r_S}{2 R_e} \right)} </math>

In the linear approximation 
: <math>z = \frac{ \frac{G M}{R_e c^2} - \frac{1}{2} \frac{G^2 M^2}{R_e^2 c^4} + \dots}{ 1 - \frac{G M}{R_e c^2} + \frac{1}{2} \frac{G^2 M^2}{R_e^2 c^4} - \ldots } \simeq \frac{ \frac{G M}{R_e c^2} }{ 1 - \frac{G M}{R_e c^2} + \frac{1}{2} \frac{G^2 M^2}{R_e^2 c^4} - \dots} \simeq \frac{G M}{c^2 R_e} </math>

the Newtonian limit for the gravitational red shift of General Relativity is obtained.