Editing Gravitational redshift (section)

== Prediction by the equivalence principle and general relativity ==
=== Uniform gravitational field or acceleration ===
Einstein's theory of general relativity incorporates the [[equivalence principle]], which can be stated in various different ways. One such statement is that gravitational effects are locally undetectable for a free-falling observer. Therefore, in a laboratory experiment at the surface of the Earth, all gravitational effects should be equivalent to the effects that would have been observed if the laboratory had been accelerating through outer space at ''g''. One consequence is a gravitational [[Doppler effect]]. If a light pulse is emitted at the floor of the laboratory, then a free-falling observer says that by the time it reaches the ceiling, the ceiling has accelerated away from it, and therefore when observed by a detector fixed to the ceiling, it will be observed to have been Doppler shifted toward the red end of the spectrum. This shift, which the free-falling observer considers to be a kinematical Doppler shift, is thought of by the laboratory observer as a gravitational redshift. Such an effect was verified in the 1959 [[Pound–Rebka experiment]]. In a case such as this, where the gravitational field is uniform, the change in wavelength is given by
: <math>z = \frac{\Delta\lambda}{\lambda}\approx \frac{g\Delta y}{c^2},</math>
where <math>\Delta y</math> is the change in height. Since this prediction arises directly from the equivalence principle, it does not require any of the mathematical apparatus of general relativity, and its verification does not specifically support general relativity over any other theory that incorporates the equivalence principle.

On Earth's surface (or in a spaceship accelerating at 1&nbsp;''g''), the gravitational redshift is approximately {{val|1.1|e=−16}}, the equivalent of a {{val|3.3|e=−8|u=m/s}} Doppler shift for every 1&nbsp;m of altitude.

=== Spherically symmetric gravitational field ===
When the field is not uniform, the simplest and most useful case to consider is that of a spherically symmetric field. By [[Birkhoff's theorem (relativity)|Birkhoff's theorem]], such a field is described in general relativity by the [[Schwarzschild metric]], <math>d\tau^2 = \left(1 - r_\text{S}/R\right)dt^2 + \ldots</math>, where <math>d\tau</math> is the clock time of an observer at distance ''R'' from the center, <math>dt</math> is the time measured by an observer at infinity, <math>r_\text{S}</math> is the Schwarzschild radius <math>2GM/c^2</math>, "..." represents terms that vanish if the observer is at rest, <math>G</math> is the [[Newtonian constant of gravitation]], <math>M</math> the [[mass]] of the gravitating body, and <math>c</math> the [[speed of light]]. The result is that frequencies and wavelengths are shifted according to the ratio
: <math>1 + z = \frac{\lambda_\infty}{\lambda_\text{e}} = \left(1 - \frac{r_\text{S}}{R_\text{e}}\right)^{-\frac{1}{2}}</math>
where
* <math>\lambda_\infty\,</math>is the wavelength of the light as measured by the observer at infinity,
* <math>\lambda_\text{e}\,</math> is the wavelength measured at the source of emission, and
* <math>R_\text{e}</math> is the radius at which the photon is emitted.

This can be related to the [[Redshift|redshift parameter]] conventionally defined as <math>z = \lambda_\infty/\lambda_\text{e} - 1</math>.

In the case where neither the emitter nor the observer is at infinity, the [[transitive relation|transitivity]] of Doppler shifts allows us to generalize the result to <math>\lambda_1/\lambda_2 = \left[\left(1 - r_\text{S}/R_1\right)/\left(1 - r_\text{S}/R_2\right)\right]^{1/2}</math>. The redshift formula for the frequency <math>\nu = c/\lambda</math> is <math>\nu_o/\nu_\text{e} = \lambda_\text{e}/\lambda_o</math>. When <math>R_1 - R_2</math> is small, these results are consistent with the equation given above based on the equivalence principle.

The redshift ratio may also be expressed in terms of a (Newtonian) escape velocity <math>v_\text{e}</math> at <math>R_\text{e} = 2GM/v_\text{e}^2</math>, resulting in the corresponding [[Lorentz factor]]:
: <math>1 + z = \gamma_\text{e} = \frac{1}{\sqrt{1 - (v_\text{e}/c)^2}}</math>.

For an object compact enough to have an [[event horizon]], the redshift is not defined for photons emitted inside the Schwarzschild radius, both because signals cannot escape from inside the horizon and because an object such as the emitter cannot be stationary inside the horizon, as was assumed above. Therefore, this formula only applies when <math>R_\text{e}</math> is larger than <math>r_\text{S}</math>. When the photon is emitted at a distance equal to the Schwarzschild radius, the redshift will be ''infinitely'' large, and it will not escape to ''any'' finite distance from the Schwarzschild sphere. When the photon is emitted at an infinitely large distance, there is no redshift.

==== Newtonian limit ====
In the Newtonian limit, i.e. when <math>R_\text{e}</math> is sufficiently large compared to the Schwarzschild radius <math>r_\text{S}</math>, the redshift can be approximated as
: <math>z = \frac{\Delta\lambda}{\lambda} \approx \frac{1}{2}\frac{r_\text{S}}{R_\text{e}} = \frac{GM}{R_\text{e} c^2} = \frac{g R_\text{e}}{c^2}</math>
where <math>g</math> is the [[gravitational acceleration]] at <math>R_\text{e}</math>. For Earth's surface with respect to infinity, ''z'' is approximately {{val|7|e=−10}} (the equivalent of a 0.2&nbsp;m/s radial Doppler shift); for the Moon it is approximately {{val|3|e=−11}} (about 1&nbsp;cm/s). The value for the surface of the Sun is about {{val|2|e=−6}}, corresponding to 0.64&nbsp;km/s. (For non-relativistic velocities, the radial [[Relativistic Doppler effect|Doppler equivalent velocity]] can be approximated by multiplying ''z'' with the speed of light.)

The z-value can be expressed succinctly in terms of the [[escape velocity]] at <math>R_\text{e}</math>, since the [[gravitational potential]] is equal to half the square of the [[escape velocity]], thus:
: <math>z \approx \frac{1}{2}\left( \frac{v_\text{e}}{c} \right)^2</math>
where <math>v_\text{e}</math> is the escape velocity at <math>R_\text{e}</math>.

It can also be related to the circular orbit velocity <math>v_\text{o}</math> at <math>R_\text{e}</math>, which equals <math>v_\text{e}/\sqrt{2}</math>, thus
: <math>z \approx \left( \frac{v_\text{o}}{c} \right)^2</math>.

For example, the gravitational blueshift of distant starlight due to the Sun's gravity, which the Earth is orbiting at about 30&nbsp;km/s, would be approximately 1 × 10<sup>−8</sup> or the equivalent of a 3&nbsp;m/s radial Doppler shift.

For an object in a (circular) orbit, the gravitational redshift is of comparable magnitude as the [[transverse Doppler effect]], <math>z \approx \tfrac{1}{2} \beta^2</math> where {{nowrap|1=''β'' = ''v''/''c''}}, while both are much smaller than the [[Relativistic Doppler effect|radial Doppler effect]], for which <math>z \approx \beta</math>.

==== 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.