Editing Redshift (section)

=== Summary table ===
Several important special-case formulae for redshift in certain special spacetime geometries, as summarized in the following table. In all cases the magnitude of the shift (the value of {{math|''z''}}) is independent of the wavelength.<ref name="basicastronomy">See Binney and Merrifeld (1998), Carroll and Ostlie (1996), Kutner (2003) for applications in astronomy.</ref>

{| class="wikitable" style="max-width:1000px;"
|+ Redshift summary
! Redshift type !! Geometry !! Formulae<ref>Where z = redshift; v<sub>||</sub> = [[velocity]] parallel to line-of-sight (positive if moving away from receiver); c = [[speed of light]]; ''γ'' = [[Lorentz factor]]; ''a'' = [[scale factor (Universe)|scale factor]]; G = [[gravitational constant]]; M = object [[mass]]; r = [[Schwarzschild coordinates|radial Schwarzschild coordinate]], g<sub>tt</sub> = t,t component of the [[metric tensor]]</ref>
|-
| [[Relativistic Doppler effect|Relativistic Doppler]]|| [[Minkowski space]]<br />(flat spacetime) ||
For motion completely in the radial or<br />line-of-sight direction:

:<big><math>1 + z = \gamma \left(1 + \frac{v_{\parallel}}{c}\right) = \sqrt{\frac{1+\frac{v_{\parallel}}{c}}{1-\frac{v_{\parallel}}{c}}}</math></big>
:<math>z \approx \frac{v_{\parallel}}{c}</math>  for&nbsp;small <math>v_{\parallel}</math>

<br />
For motion completely in the transverse direction:

:<math>1 + z=\frac{1}{\sqrt{1-\frac{v_\perp^2}{c^2}}}</math>
:<math>z \approx \frac{1}{2} \left( \frac{v_{\perp}}{c} \right)^2</math>  for&nbsp;small <math>v_{\perp}</math>

|-
| [[Cosmological redshift]]|| [[Friedmann–Lemaître–Robertson–Walker|FLRW spacetime]]<br />(expanding Big Bang universe) || 
:<math>1 + z = \frac{a_{\mathrm{now}}}{a_{\mathrm{then}}}</math>

[[Hubble's law]]:

:<math>z \approx \frac{H_0 D}{c}</math>  for <math>D \ll \frac{c}{H_0}</math>

|-
| [[Gravitational redshift]]|| Any [[stationary spacetime]] || 
:<math>1 + z = \sqrt{\frac{g_{tt}(\text{receiver})}{g_{tt}(\text{source})}}</math>
For the [[Schwarzschild geometry]]:

:<big><math>1 + z = \sqrt{\frac{1 - \frac{r_S}{r_{\text{receiver}}}}{1 - \frac{r_S}{r_{\text{source} }}}} = \sqrt{\frac{1 - \frac{2GM}{ c^2  r_{\text{receiver}}}}{1 - \frac{2GM}{ c^2 r_{\text{source} }}}} </math></big>

:<big><math>z \approx \frac{1}{2} \left( \frac{r_S}{r_\text{source}} - \frac{r_S}{r_\text{receiver}} \right)</math>  for <math>r \gg r_S</math></big>

In terms of [[escape velocity]]:

:<math>z \approx \frac{1}{2} \left(\frac{v_\text{e}}{c}\right)_\text{source}^2 - \frac{1}{2} \left(\frac{v_\text{e}}{c}\right)_\text{receiver}^2 </math> 
for <math>v_\text{e} \ll c</math>

|}