Editing Redshift (section)

== Physical origins ==
Redshifts are differences between two wavelength measurements and wavelengths are a property of both the photons and the measuring equipment. Thus redshifts characterize differences between two measurement locations. These differences are 
commonly organized in three groups, attributed to relative motion between the source and the observer, to the expansion of the universe, and to gravity.<ref>{{cite journal |last1=Lewis |first1=Geraint F. |title=On The Relativity of Redshifts: Does Space Really "Expand"? |journal=Australian Physics |date=2016 |volume=53 |page=95 |arxiv=1605.08634 }}</ref> The following sections explain these groups.

===Doppler effect===
{{Main|Doppler effect|Relativistic Doppler effect}}
[[Image:Suzredshift.gif|thumb|[[Doppler effect]], yellow (~575 [[Nanometre|nm]] wavelength) ball appears greenish (blueshift to ~565 nm wavelength) approaching observer, turns [[Orange (colour)|orange]] (redshift to ~585 nm wavelength) as it passes, and returns to yellow when motion stops. To observe such a change in color, the object would have to be traveling at approximately 5,200 [[Metre per second|km/s]], or about 32 times faster than the speed record for the [[Parker Solar Probe|fastest space probe]].]]
[[File:Redshift blueshift.svg|thumb|Redshift and blueshift]]

If a source of the light is moving away from an observer, then redshift ({{math|''z'' > 0}}) occurs; if the source moves towards the observer, then [[blueshift]] ({{math|''z'' < 0}}) occurs. This is true for all electromagnetic waves and is explained by the [[Doppler effect]]. Consequently, this type of redshift is called the ''Doppler redshift''. If the source moves away from the observer with [[velocity]] {{math|''v''}}, which is much less than the speed of light ({{math|''v'' ≪ ''c''}}), the redshift is given by

:<math>z \approx \frac{v}{c}</math>   (since <math>\gamma \approx 1</math>)

where {{math|''c''}} is the [[speed of light]]. In the classical Doppler effect, the frequency of the source is not modified, but the recessional motion causes the illusion of a lower frequency.

A more complete treatment of the Doppler redshift requires considering relativistic effects associated with motion of sources close to the speed of light. A complete derivation of the effect can be found in the article on the [[relativistic Doppler effect]]. In brief, objects moving close to the speed of light will experience deviations from the above formula due to the [[time dilation]] of [[special relativity]] which can be corrected for by introducing the [[Lorentz factor]] {{math|''γ''}} into the classical Doppler formula as follows (for motion solely in the line of sight):

:<math>1 + z = \left(1 + \frac{v}{c}\right) \gamma.</math>

This phenomenon was first observed in a 1938 experiment performed by [[Herbert E. Ives]] and G. R. Stilwell, called the [[Ives–Stilwell experiment]].<ref>{{cite journal | last1 = Ives | first1 = H. | last2 = Stilwell | first2 = G. | year = 1938 | title = An Experimental study of the rate of a moving atomic clock | journal = Journal of the Optical Society of America | volume = 28 | issue = 7| pages = 215–226 | doi=10.1364/josa.28.000215 | bibcode = 1938JOSA...28..215I}}</ref>

Since the Lorentz factor is dependent only on the [[magnitude (mathematics)|magnitude]] of the velocity, this causes the redshift associated with the relativistic correction to be independent of the orientation of the source movement. In contrast, the classical part of the formula is dependent on the [[scalar resolute|projection]] of the movement of the source into the [[Line-of-sight propagation|line-of-sight]] which yields different results for different orientations. If {{math|''θ''}} is the angle between the direction of relative motion and the direction of emission in the observer's frame<ref>{{cite book|last=Freund|first=Jurgen|title=Special Relativity for Beginners|date=2008|publisher=World Scientific|page=120|isbn=978-981-277-160-5}}</ref> (zero angle is directly away from the observer), the full form for the relativistic Doppler effect becomes:

:<math>1+ z = \frac{1 + v \cos (\theta)/c}{\sqrt{1-v^2/c^2}}</math>

and for motion solely in the line of sight ({{math|''θ'' {{=}} 0°}}), this equation reduces to:

:<math>1 + z = \sqrt{\frac{1+v/c}{1-v/c}}</math>

For the special case that the light is moving at [[right angle]] ({{math|''θ'' {{=}} 90°}}) to the direction of relative motion in the observer's frame,<ref>{{cite book|last=Ditchburn|first=R. |title=Light|date=1991|publisher=Dover|page=329|isbn=978-0-12-218101-6}}</ref> the relativistic redshift is known as the [[Relativistic Doppler effect|transverse redshift]], and a redshift:

:<math>1 + z = \frac{1}{\sqrt{1-v^2/c^2}}</math>

is measured, even though the object is not moving away from the observer. Even when the source is moving towards the observer, if there is a transverse component to the motion then there is some speed at which the dilation just cancels the expected blueshift and at higher speed the approaching source will be redshifted.<ref>
See "[http://www.physics.uq.edu.au/people/ross/phys2100/doppler.htm Photons, Relativity, Doppler shift] {{Webarchive|url=https://web.archive.org/web/20060827063802/http://www.physics.uq.edu.au/people/ross/phys2100/doppler.htm |date=2006-08-27 }} " at the University of Queensland
</ref>

===Cosmic expansion===
{{Main|Expansion of the universe}}

The observations of increasing redshifts from more and more distant galaxies can be modeled assuming a [[cosmological principle|homogeneous and isotropic universe]] combined with [[general relativity]].  This cosmological redshift can be written as a function of {{math|''a''}}, the time-dependent cosmic [[Scale factor (cosmology)|scale factor]]:<ref>{{Cite book |last=Peacock |first=J. A. |url=https://www.cambridge.org/core/product/identifier/9780511804533/type/book |title=Cosmological Physics |date=1998-12-28 |publisher=Cambridge University Press |isbn=978-0-521-41072-4 |edition=1 |doi=10.1017/cbo9780511804533}}</ref>{{rp|72}}
:<math>1+z = \frac{a_\mathrm{now}}{a_\mathrm{then}} = \frac{a_0}{a(t)}</math>

The scale factor is [[monotonic function|monotonically increasing]] as time passes. Thus {{math|''z''}} is positive, close to zero for local stars, and increasing for distant galaxies that appear redshifted.

Using a [[Friedmann–Robertson–Walker model]] of the expansion of the universe, redshift can be related to the age of an observed object, the so-called ''[[cosmic time]]–redshift relation''. Denote a density ratio as {{math|Ω<sub>0</sub>}}:

:<math>\Omega_0 = \frac {\rho}{ \rho_\text{crit}} \ , </math>

with {{math|''ρ''<sub>crit</sub>}} the critical density demarcating a universe that eventually crunches from one that simply expands. This density is about three hydrogen atoms per cubic meter of space.<ref Name=Weinberg>{{cite book |first=Steven | last=Weinberg |edition=2nd |title=The First Three Minutes: A Modern View of the Origin of the Universe | page=34 |isbn=9780-465-02437-7 |date=1993 |publisher=Basic Books|title-link=The First Three Minutes: A Modern View of the Origin of the Universe }}</ref> At large redshifts, {{math| ''1 + z'' > Ω<sub>0</sub><sup>−1</sup>}}, one finds:

:<math> t(z) \approx \frac {2}{3 H_0 {\Omega_0}^{1/2} } z^{-3/2}\ , </math>

where {{math|''H''<sub>0</sub>}} is the present-day [[Hubble constant]], and {{math|''z''}} is the redshift.<ref name="Bergström">{{cite book |title=Cosmology and Particle Astrophysics |url=https://books.google.com/books?id=CQYu_sutWAoC&pg=PA77 |page=77, Eq.4.79 |isbn=978-3-540-32924-4 |publisher=Springer |edition=2nd|date=2006|first1 = Lars |last1=Bergström|first2 = Ariel |last2=Goobar|author-link1=Lars Bergström (physicist) |author-link2=Ariel Goobar }}</ref><ref name = Longair>{{cite book |title=Galaxy Formation |first=M. S. |last=Longair |url=https://books.google.com/books?id=2ARuLT-tk5EC&pg=PA161 |page=161 |isbn=978-3-540-63785-1 |publisher=Springer |date=1998}}</ref>

The [[cosmological redshift]] is commonly attributed to stretching of the wavelengths of photons due to the stretching of space. This interpretation can be misleading. 
As required by [[general relativity]], the cosmological expansion of space has no effect on local physics. There is no term related to expansion in [[Maxwell's equations]] that govern light propagation. The cosmological redshift can be interpreted as an accumulation of infinitesimal Doppler shifts along the trajectory of the light.<ref name="Hogg">{{cite journal |author=Bunn |first1=E. F. |last2=Hogg |first2=D. W. |year=2009 |title=The kinematic origin of the cosmological redshift |journal=American Journal of Physics |volume=77 |issue=8 |pages=688–694 |arxiv=0808.1081 |bibcode=2009AmJPh..77..688B |doi=10.1119/1.3129103 |s2cid=1365918}}</ref>

There are several websites for calculating various times and distances from redshift, as the precise calculations require numerical integrals for most values of the parameters.<ref name="UCLA-2018">{{cite web |last=Wright |first=Edward L. |title=UCLA Cosmological Calculator |url=http://www.astro.ucla.edu/~wright/ACC.html |date=2018 |work=[[UCLA]] |access-date=6 August 2022 }} For parameter values as of 2018, H<sub>0</sub>=67.4 and Omega<sub>M</sub>=0.315, see the table at [[Lambda-CDM model#Parameters|Lambda-CDM model § Parameters]].</ref><ref name="ICRAR-2022">{{cite web |author=Staff |title=ICRAR Cosmology Calculator |url=https://cosmocalc.icrar.org/ |date=2022 |work=[[International Centre for Radio Astronomy Research]] |access-date=6 August 2022 }}</ref>

====Distinguishing between cosmological and local effects====

The redshift of a galaxy includes both a component related to [[recessional velocity]] from expansion of the universe, and a component related to the [[peculiar motion]] of the galaxy with respect to its local universe.<ref>{{cite journal
 | title=A comparison between the Doppler and cosmological redshifts
 | last=Bedran | first=M. L. | year=2002
 | journal=American Journal of Physics
 | volume=70 | issue=4 | pages=406–408
 | doi=10.1119/1.1446856 | bibcode=2002AmJPh..70..406B
 | url=http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/cosmo/doppler_redshift.pdf
 | access-date=2023-03-16
}}</ref> The redshift due to expansion of the universe depends upon the recessional velocity in a fashion determined by the cosmological model chosen to describe the expansion of the universe, which is very different from how Doppler redshift depends upon local velocity.<ref name="Harrison2">{{cite journal |last=Harrison |first=Edward |date=1992 |title=The redshift-distance and velocity-distance laws |journal=Astrophysical Journal, Part 1 |volume=403 |pages=28–31 |bibcode=1993ApJ...403...28H |doi=10.1086/172179 |doi-access=free}}. A pdf file can be found here [http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1993ApJ...403...28H&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf].</ref> Describing the cosmological expansion origin of redshift, cosmologist [[Edward Robert Harrison]] said, "Light leaves a galaxy, which is stationary in its local region of space, and is eventually received by observers who are stationary in their own local region of space. Between the galaxy and the observer, light travels through vast regions of expanding space. As a result, all wavelengths of the light are stretched by the expansion of space. It is as simple as that..."<ref>{{Harvnb|Harrison|2000|p=302}}.</ref> [[Steven Weinberg]] clarified, "The increase of wavelength from emission to absorption of light does not depend on the rate of change of {{math|''a''(''t'')}} [the [[Scale factor (cosmology)|scale factor]]] at the times of emission or absorption, but on the increase of {{math|''a''(''t'')}} in the whole period from emission to absorption."<ref name=Weinberg_Cosmology>{{cite book |url=https://books.google.com/books?id=48C-ym2EmZkC&pg=PA11 |first=Steven | last=Weinberg |title=Cosmology |publisher=Oxford University Press |page=11 |date=2008 |isbn=978-0-19-852682-7}}</ref>

If the universe were contracting instead of expanding, we would see distant galaxies blueshifted by an amount proportional to their distance instead of redshifted.<ref>This is only true in a universe where there are no [[peculiar velocity|peculiar velocities]]. Otherwise, redshifts combine as

:<math>1+z=(1+z_{\mathrm{Doppler}})(1+z_{\mathrm{expansion}})</math>
which yields solutions where certain objects that "recede" are blueshifted and other objects that "approach" are redshifted. For more on this bizarre result see: {{cite journal
 | last1=Davis | first1=T. M. | last2=Lineweaver | first2=C. H. | last3=Webb | first3=J. K.
 | title=Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blueshifted objects
 | journal=American Journal of Physics
 | volume=71 | issue=4 | pages=358–364
 | date=April 2003 | doi=10.1119/1.1528916 
 | arxiv=astro-ph/0104349 | bibcode=2003AmJPh..71..358D | s2cid=3219383 }}</ref>

===Gravitational redshift===
{{Main|Gravitational redshift}}
In the theory of [[general relativity]], there is time dilation within a gravitational well. Light emitted within the well will appear to have fewer cycles per second when measured outside of the well, due to differences in the two clocks.<ref>{{Cite book |last=Zee |first=Anthony |title=Einstein Gravity in a Nutshell |date=2013 |publisher=Princeton University Press |isbn=978-0-691-14558-7 |edition=1st |series=In a Nutshell Series |location=Princeton}}</ref>{{rp|284}}  This is known as the [[gravitational redshift]] or ''Einstein Shift''.<ref>{{cite journal | last=Chant | first=C. A. | bibcode = 1930JRASC..24..390C | title = Notes and Queries (Telescopes and Observatory Equipment – The Einstein Shift of Solar Lines) | date = 1930 | journal = [[Journal of the Royal Astronomical Society of Canada]] | volume = 24 | page = 390 }}</ref> The theoretical derivation of this effect follows from the [[Schwarzschild solution]] of the [[Einstein field equations|Einstein equations]] which yields the following formula for redshift associated with a photon traveling in the [[gravitational field]] of an [[Electric charge|uncharged]], [[rotation|nonrotating]], [[spherical symmetry|spherically symmetric]] mass:

:<math>1+z=\frac{1}{\sqrt{1-\frac{2GM}{rc^2}}},</math>

where
* {{math|''G''}} is the [[gravitational constant]],
* {{math|''M''}} is the [[mass]] of the object creating the gravitational field,
* {{math|''r''}} is the radial coordinate of the source (which is analogous to the classical distance from the center of the object, but is actually a [[Schwarzschild coordinates|Schwarzschild coordinate]]), and
* {{math|''c''}} is the [[speed of light]].

This gravitational redshift result can be derived from the assumptions of [[special relativity]] and the [[equivalence principle]]; the full theory of general relativity is not required.<ref>{{cite journal | last = Einstein | first = A. | author-link = Albert Einstein | date = 1907 | title = Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen | journal = Jahrbuch der Radioaktivität und Elektronik | volume = 4 | pages = 411–462 | bibcode=1908JRE.....4..411E}} See p. 458 ''The influence of a gravitational field on clocks''</ref>

The effect is very small but measurable on Earth using the [[Mössbauer effect]] and was first observed in the [[Pound–Rebka experiment]].<ref>{{cite journal | doi = 10.1103/PhysRevLett.4.337 | title = Apparent Weight of Photons | date = 1960 | last1 = Pound | first1 = R. | last2 = Rebka | first2 = G. | journal = Physical Review Letters | volume = 4 | issue = 7 | pages = 337–341 | bibcode=1960PhRvL...4..337P| doi-access = free }}. This paper was the first measurement.</ref> However, it is significant near a [[black hole]], and as an object approaches the [[event horizon]] the red shift becomes infinite. It is also the dominant cause of large angular-scale temperature fluctuations in the [[cosmic microwave background]] radiation (see [[Sachs–Wolfe effect]]).<ref>{{cite journal | last1=Sachs | first1=R. K. | author-link=Rainer K. Sachs | last2=Wolfe | first2=A. M. | author-link2=Arthur M. Wolfe | date=1967 | title=Perturbations of a cosmological model and angular variations of the cosmic microwave background | journal=Astrophysical Journal | volume=147 | issue=73 | doi=10.1086/148982 | page=73 | bibcode=1967ApJ...147...73S }}</ref>

=== Summary table ===
Several important special-case formulae for redshift in certain special spacetime geometries are 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 metric|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>

|}