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{{dablink|This article is about the physical phenomenon. For the photochemical usage, see [[bathochromic shift]]. For other uses of the phrase "red shift" or "redshift", see [[redshift (disambiguation)]].}}

[[Image:Redshift.png|thumb|200px|[[spectral line|Absorption lines]] in the [[visible spectrum|optical spectrum]] of a supercluster of distant galaxies (right), as compared to absorption lines in the optical spectrum of the Sun (left). Arrows indicate redshift. Wavelength increases up towards the red and beyond (frequency decreases).]]
[[Image:Redshift blueshift.svg|thumb|Redshift and blue shift]]

In [[physics]] and [[astronomy]], '''redshift''' occurs when [[electromagnetic radiation]]—usually [[visible light]]—emitted or reflected by an object is shifted towards the (less energetic) red end of the [[electromagnetic spectrum]] due to the [[Doppler effect]] or other gravitationally-induced effects. More generally, redshift is defined as an ''increase'' in the [[wavelength]] of [[electromagnetic radiation]] received by a detector compared with the wavelength [[Emission (electromagnetic radiation)|emitted]] by the source. This increase in wavelength corresponds to a drop in the [[frequency]] of the [[electromagnetic radiation]]. Conversely, a ''decrease'' in wavelength is called [[blue shift]].

Any increase in wavelength is called "redshift", even if it occurs in electromagnetic radiation of non-optical wavelengths, such as [[gamma ray]]s, [[x-ray]]s and [[ultraviolet]]. This nomenclature might be confusing since, at wavelengths longer than red (e.g., [[infrared]], [[microwave]]s, and [[radio waves]]), redshifts shift the radiation ''away'' from the red wavelengths.

An observed redshift due to the Doppler effect occurs whenever a light source moves away from the observer, corresponding to the Doppler shift that changes the perceived frequency of [[sound|sound waves]]. Although observing such redshifts, or complementary blue shifts, has several terrestrial applications (e.g., [[Doppler radar]] and [[radar gun]]s),<ref>See Feynman, Leighton and Sands (1989) or any introductory undergraduate (and many high school) [[Physics#Undergraduate|physics textbooks]]. See Taylor (1992) for a relativistic discussion.</ref> [[astronomical spectroscopy|spectroscopic]] astrophysics uses Doppler redshifts to determine the movement of distant astronomical objects.<ref name=basicastronomy>See Binney and Merrifeld (1998), Carroll and Ostlie (1996), Kutner (2003) for applications in astronomy.</ref>

A [[special relativity|special relativistic]] [[#Redshift formulas|redshift formula]] (and its [[Newtonian physics|Newtonian approximation]]) is used when [[spacetime]] is [[Minkowski space|flat]]. Where [[gravity|gravitational effects]] are important, redshift must be calculated using [[general relativity]]. Two important special-case formulas are the so-called [[gravitational redshift]] formula which applies to any stationary (that is, unchanging with time) gravitational field, and the [[cosmological redshift]] formula which applies to the [[metric expansion of space|expanding universe]] of [[Big Bang cosmology]].<ref>See Misner, Thorne and Wheeler (1973) and Weinberg (1971) or any of the [[physical cosmology#Textbooks|physical cosmology textbooks]]</ref>

Special relativistic, gravitational, and cosmological redshifts can be understood under the umbrella of [[Frame of reference|frame transformation laws]]. There exist other physical processes that can lead to a shift in the frequency of electromagnetic radiation that are not generally referred to as "redshifts", including [[scattering]] and [[physical optics|optical effects]] (for more see section on [[#Effects due to physical optics or radiative transfer|physical optics and radiative transfer]]).

==History==

The history of the subject began with the development in the 19th century of [[wave mechanics]] and the exploration of phenomena associated with the [[Doppler effect]]. The effect is named after [[Christian Andreas Doppler]], who offered the first known physical explanation for the phenomenon in 1842.<ref>Doppler, Christian, (1846), ''Prag, Druck von G. Haase sohne''</ref> The hypothesis was tested and confirmed for [[sound|sound waves]] by the [[Netherlands|Dutch]] scientist [[C.H.D. Buys Ballot|Christoph Hendrik Diederik Buys Ballot]] in 1845.<ref>Dev Maulik, "[http://books.google.com/books?id=HedeGJms0n4C&vid=ISBN3540230882&dq=%22Buys+Ballot%22&pg=PA3&lpg=PA3&sig=1Y0ETpNemutmPYNF8KbWSbLrF7E&q=%22Ballot%22 Doppler Sonography: A Brief History]" in ''[http://www.springer.com/west/home/medicine/gynecology?SGWID=4-10066-22-46625046-0 Doppler Ultrasound in Obstetrics And Gynecology]'' (2005) by Dev (EDT) Maulik, Ivica Zalud</ref> Doppler correctly predicted that the phenomenon should apply to all [[wave]]s, and in particular suggested that the varying [[color]]s of [[star]]s could be attributed to their motion with respect to the Earth.<ref>{{MacTutor Biography|id=Doppler}}</ref> While this attribution turned out to be incorrect (stellar colors are indicators of a star's [[color temperature|temperature]], not motion), Doppler would later be vindicated by verified redshift observations.

The first Doppler redshift was described in 1848 by French physicist [[Hippolyte Fizeau|Armand-Hippolyte-Louis Fizeau]], who pointed to the shift in [[spectral line]]s seen in stars as being due to the Doppler effect. The effect is sometimes called the "Doppler-Fizeau effect". In 1868, British astronomer [[William Huggins]] was the first to determine the velocity of a star moving away from the Earth by this method.<ref name=Huggins>William Huggins, "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1868RSPT..158..529H&amp;db_key=AST&amp;data_type=HTML&amp;format=&amp;high=42ca922c9c03088 Further Observations on the Spectra of Some of the Stars and Nebulae, with an Attempt to Determine Therefrom Whether These Bodies are Moving towards or from the Earth], Also Observations on the Spectra of the Sun and of Comet II." (1868) ''Philosophical Transactions of the Royal Society of London'', Volume 158, pp. 529–564</ref>

In 1871, optical redshift was confirmed when the phenomenon was observed in [[Fraunhofer lines]] using solar rotation, about 0.1 Å in the red.<ref>Reber, G., "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1995Ap%26SS.227...93R&amp;db_key=AST&amp;data_type=HTML&amp;format=&amp;high=4521318e0222293 Intergalactic Plasma]"(1995) Astrophysics and Space Science, v. 227, p. 93–96.</ref> In 1901 [[Aristarkh Apollonovich Belopolsky|Aristarkh Belopolsky]] verified optical redshift in the laboratory using a system of rotating mirrors.<ref>Bélopolsky, A., "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1901ApJ....13...15B&amp;db_key=AST&amp;data_type=HTML&amp;format=&amp;high=4521318e0220955 On an Apparatus for the Laboratory Demonstration of the Doppler-Fizeau Principle]" (1901) Astrophysical Journal, vol. 13, p.15 </ref>

The earliest occurrence of the term "red-shift" in print (in this hyphenated form), appears to be by American astronomer [[Walter S. Adams]] in 1908, where he mentions "Two methods of investigating that nature of the nebular red-shift".<ref>Adams, Walter S., "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1908CMWCI..22....1A No. 22. Preliminary catalogue of lines affected in sun-spots]" (1908) ''Contributions from the Mount Wilson Observatory'' / Carnegie Institution of Washington, vol. 22, pp.1–21</ref> The word doesn't appear unhyphenated, perhaps indicating a more common usage of its German equivalent, ''Rotverschiebung'', until about 1934 by [[Willem de Sitter]].<ref>W. de Sitter, "[http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?journal=BAN&year=1934&volume=7&page_ind=210&letter=.&type=SCREEN_GIF On distance, magnitude, and related quantities in an expanding universe], (1934) ''Bulletin of the Astronomical Institutes of the Netherlands'', Vol. 7, p.205. He writes: "It thus becomes urgent to investigate the effect of the redshift and of the metric of the universe on the apparent magnitude and observed numbers of nebulae of given magnitude"</ref>

Beginning with observations in 1912, [[Vesto Slipher]] discovered that most [[spiral nebula]]e had considerable redshifts.<ref>Slipher first reports on his measurement in the inaugural volume of the Lowell Observatory Bulletin, pp.2.56-2.57[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1913LowOB...1b..56S&amp;db_key=AST&amp;data_type=HTML&amp;format=&amp;high=448f04e38822894]. His article entitled ''The radial velocity of the Andromeda Nebula'' reports making the first Doppler measurement on September 17, 1912. In his report, Slipher writes: "The magnitude of this velocity, which is the greatest hitherto observed, raises the question whether the velocity-like displacement might not be due to some other cause, but I believe we have at present no other interpretation for it." Three years later, in the journal ''Popular Astronomy'', Vol. 23, p. 21–24 [http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1915PA.....23...21S&amp;db_key=AST&amp;data_type=HTML&amp;format=&amp;high=448f04e38822894], Slipher wrote a review entitled ''Spectrographic Observations of Nebulae''. In it he states, "The early discovery that the great Andromeda spiral had the quite exceptional velocity of - 300 km(/s) showed the means then available, capable of investigating not only the spectra of the spirals but their velocities as well." Slipher reported the velocities for 15 spiral nebulae spread across the entire [[celestial sphere]], all but three having observable "positive" (that is recessional) velocities.</ref> Subsequently, [[Edwin Hubble]] discovered an approximate relationship between the redshift of such "nebulae" (now known to be [[galaxy|galaxies]] in their own right) and the [[distance]] to them with the formulation of his eponymous [[Hubble's law]].<ref>Hubble, Edwin, "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1929PNAS...15..168H&amp;db_key=AST&amp;data_type=HTML&amp;format=&amp;high=42ca922c9c30954 A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae]" (1929) ''Proceedings of the National Academy of Sciences of the United States of America'', Volume 15, Issue 3, pp. 168–173 ([http://www.pnas.org/cgi/reprint/15/3/168 Full article], PDF)</ref> These observations corroborated [[Alexander Friedman]]'s 1922 work, in which he derived the famous [[Friedmann equations]].<ref>Friedman, A: Über die Krümmung des Raumes, Z. Phys. 10 (1922), 377–386. (English translation in: Gen. Rel. Grav. 31 (1999), 1991–2000.)</ref> They are today considered strong evidence for an [[expanding universe]] and the [[Big Bang]] theory.<ref name=Eddington>This was recognized early on by physicists and astronomers working in cosmology in the 1930s. The earliest layman publication describing the details of this correspondence was [[Sir Arthur Eddington]]'s book ''The Expanding Universe: Astronomy's 'Great Debate', 1900–1931'', published by Press Syndicate of the University of Cambridge in 1933.</ref>

==Measurement, characterization, and interpretation==
The [[visible spectrum|spectrum]] of light that comes from a single source (see idealized spectrum illustration top-right) can be measured. To determine the redshift, features in the spectrum such as [[spectral line|absorption lines]], [[spectral line|emission lines]], or other variations in [[light intensity]], are searched for. If found, these features can be compared with known features in the spectrum of various chemical compounds found in experiments where that compound is located on earth. A very common [[chemical element|atomic element]] in space is [[hydrogen]]. The spectrum of originally featureless light shined through hydrogen will show a [[hydrogen spectrum|signature spectrum]] specific to hydrogen that has features at regular intervals. If restricted to absorption lines it would look similar to the illustration (top right). If the same pattern of intervals is seen in an observed spectrum from a distant source but occurring at shifted wavelengths, it can be identified as hydrogen too. If the same spectral line is identified in both spectra but at different wavelengths then the redshift can be calculated using the table below. Determining the redshift of an object in this way requires a frequency- or wavelength-range. In order to calculate the redshift one has to know the wavelength of the emitted light in the rest frame of the source, in other words, the wavelength that would be measured by an observer located adjacent to and comoving with the source. Since in astronomical applications this measurement cannot be done directly, because that would require travelling to the distant star of interest, the method using spectral lines described here is used instead. Redshifts cannot be calculated by looking at unidentified features whose rest-frame frequency is unknown, or with a spectrum that is featureless or [[white noise]] (random fluctuations in a spectrum).<ref>See, for example, this 25 May 2004 [http://heasarc.gsfc.nasa.gov/docs/swift/about_swift/redshift.html press release] from [[NASA]]'s [[Swift Gamma-Ray Burst Mission|Swift]] [[space telescope]] that is researching [[gamma-ray burst]]s: "Measurements of the gamma-ray spectra obtained during the main outburst of the GRB have found little value as redshift indicators, due to the lack of well-defined features. However, optical observations of GRB afterglows have produced spectra with identifiable lines, leading to precise redshift measurements."</ref>

Redshift (and blue shift) may be characterized by the relative difference between the observed and emitted wavelengths (or frequency) of an object. In astronomy, it is customary to refer to this change using a [[dimensionless]] quantity called ''z''. If ''λ'' represents wavelength and ''f'' represents frequency (note, ''λf'' = ''c'' where ''c'' is the [[speed of light]]), then ''z'' is defined by the equations:

{| class="wikitable" align=center
|+ '''Calculation of redshift, <math>z</math>'''
! '''Based on wavelength''' !! '''Based on frequency'''
|- align=center
| <math>z = \frac{\lambda_{\mathrm{obsv}} - \lambda_{\mathrm{emit}}}{\lambda_{\mathrm{emit}}}</math>
| <math>z = \frac{f_{\mathrm{emit}} - f_{\mathrm{obsv}}}{f_{\mathrm{obsv}}}</math>
|- align=center
| <math>1+z = \frac{\lambda_{\mathrm{obsv}}}{\lambda_{\mathrm{emit}}}</math>
| <math>1+z = \frac{f_{\mathrm{emit}}}{f_{\mathrm{obsv}}}</math>
|}

After ''z'' is measured, the distinction between redshift and blue shift is simply a matter of whether ''z'' is positive or negative. See the [[#Redshift forumlas|formulas section]] below for some basic interpretations that follow when either a redshift or blue shift is observed. For example, [[Doppler effect]] blue shifts (''z'' < 0) are associated with objects approaching (moving closer to) the observer with the light shifting to greater [[energy|energies]]. Conversely, Doppler effect redshifts (''z'' > 0) are associated with objects receding (moving away) from the observer with the light shifting to lower energies. Likewise, gravitational blue shifts are associated with light emitted from a source residing within a weaker [[gravitational field]] observed within a stronger [[gravitational field]], while gravitational redshifting implies the opposite conditions.

==Redshift formulas==
In general relativity one can derive several important special-case formulas for redshift in certain special spacetime geometries, as summarized in the following table. In all cases the magnitude of the shift (the value of ''z'') is independent of the wavelength.<ref name=basicastronomy />
[[Image:Suzredshift.gif|thumb|Doppler effect, [[yellow]] (~575 [[nanometer|nm]] wavelength) ball appears [[green]]ish (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 5200 [[kilometer per second|km/s]], or about 75 times faster than the speed record for the [[Helios II|fastest manmade space probe]].]]
{| class="wikitable" align=center
|+ '''Redshift Summary'''
! Redshift type !! Geometry !! Formula<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>
|- align=center
| Relativistic Doppler || [[Minkowski space]] (flat spacetime) || <math>1 + z = \gamma \left(1 + \frac{v_{\parallel}}{c}\right)</math><br>(<math>z \approx \frac{v_{\parallel}}{c}</math> for small <math>v</math>)
|- align=center
| Cosmological redshift || [[Friedmann-Lemaître-Robertson-Walker|FLRW spacetime]] (expanding Big Bang universe) || <math>1 + z = \frac{a_{\mathrm{now}}}{a_{\mathrm{then}}}</math>
|- align=center
| Gravitational redshift || any [[stationary spacetime]] (e.g. the [[Schwarzschild geometry]]) || <math>1 + z = \sqrt{\frac{g_{tt}(\text{receiver})}{g_{tt}(\text{source})}}</math><br>(for the Schwarzschild geometry, <math>1 + z = \sqrt{\frac{1 - \frac{2GM}{r_{\text{receiver}}}}{1 - \frac{2GM}{r_{\text{source}}}}}</math>)
|}

===Doppler effect===
{{main|Doppler effect}}
If a source of the light is moving away from an observer, then redshift (''z'' > 0) occurs; if the source moves towards the observer, then [[blue shift]] (''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]] ''v'', then, ignoring relativistic effects, the redshift is given by

:<math>z \approx \frac{v}{c}</math> &nbsp; &nbsp; (Since <math>\gamma \approx 1</math>, [[#Relativistic Doppler effect|see below]])

where ''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.

===Relativistic Doppler effect===
{{main|Relativistic Doppler effect}}
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]] ''γ'' into the classical Doppler formula as follows:

:<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>H. Ives and G. Stilwell, An Experimental study of the rate of a moving atomic clock, J. Opt. Soc. Am. 28, 215–226 (1938) [http://www.opticsinfobase.org/abstract.cfm?URI=josa-28-7-215] </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. Consequently, for an object moving at an angle ''θ'' to the observer (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 (θ = 0°), this equation reduces to:

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

For the special case that the source is moving at [[right angle]]s (θ = 90°) to the detector, the relativistic redshift is known as the [[Transverse 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 if 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 blue shift 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]" at the University of Queensland</ref>

[[File:Velocity-redshift.JPG|thumb |250px |A variety of possible recession velocity vs. redshift functions including the simple linear relation ''v = cz''; a variety of possible shapes from theories of [[Metric expansion of space|cosmological expansion]] related to general relativity; and a curve that does not permit speeds faster than light in accordance with special relativity. All curves are linear at low redshifts. See Davis and Lineweaver.<ref name=D&L>{{cite journal |author=Tamara M. Davis, Charles H. Lineweaver |title=Superluminal Recessional Velocities |url=http://arxiv.org/abs/astro-ph/0011070v2 |journal=ArXiv preprint |year=2000}}</ref> ]]

===Expansion of space===
{{main|Metric expansion of space}}
In the early part of the twentieth century, Slipher, Hubble and others made the first measurements of the redshifts and blue shifts of galaxies beyond the [[Milky Way]]. They initially interpreted these redshifts and blue shifts as due solely to the Doppler effect, but later Hubble discovered a rough correlation between the increasing redshifts and the increasing distance of galaxies. Theorists almost immediately realized that these observations could be explained by a different mechanism for producing redshifts. [[Hubble's law]] of the correlation between redshifts and distances is required by models of cosmology derived from general relativity that have a [[metric expansion of space]].<ref name=Eddington /> As a result, photons propagating through the expanding space are stretched, creating the [[cosmological redshift]]. This differs from the Doppler effect redshifts described above because the velocity boost (i.e. the [[Lorentz transformation]]) between the source and observer is not due to classical [[momentum]] and [[energy]] transfer, but instead the photons increase in wavelength and redshift as the space through which they are traveling expands.<ref name=Harrison1>The distinction is made clear in {{cite book |author=Edward Robert Harrison |year= 2000 |title=Cosmology: The Science of the Universe  |publisher=Cambridge University Press |edition=2 |url=http://books.google.com/books?id=-8PJbcA2lLoC&pg=PA315#PPA306,M1 |pages=306''ff'' |isbn=052166148X}}</ref> The observational consequences of this effect can be derived using [[Friedmann-Robertson-Walker metric|the equations]] from [[general relativity]] that describe a [[cosmological principle|homogeneous and isotropic universe]].

To derive the redshift effect, use the [[geodesic equation]] for a light wave, which is

:<math>ds^2=0=-c^2dt^2+\frac{a^2 dr^2}{1-kr^2}</math>

where

*<math>ds</math> is the Lorentzian [[line element]]
*<math>dt</math> is the time interval
*<math>dr</math> is the spatial interval
*<math>c</math> is the speed of light
*<math>a</math> is the time-dependent cosmic [[scale factor (Universe)|scale factor]] or [[Friedmann-Lemaître-Robertson-Walker metric|Robertson-Walker scale factor]]
*<math>k</math> is the [[curvature]] per unit area.

For an observer observing the crest of a light wave at a position <math>r=0</math> and time <math>t=t_\mathrm{now}</math>, the crest of the light wave was emitted at a time <math>t=t_\mathrm{then}</math> in the past and a distant position <math>r=R</math>. Integrating over the path in both space and time that the light wave travels yields:

:<math>
c \int_{t_\mathrm{then}}^{t_\mathrm{now}} \frac{dt}{a}\; = 
    \int_{R}^{0} \frac{dr}{\sqrt{1-kr^2}}\,.
</math>

In general, the wavelength of light is not the same for the two positions and times considered due to the changing properties of the metric. When the wave was emitted, it had a wavelength <math>\lambda_\mathrm{then}</math>. The next crest of the light wave was emitted at a time

:<math>t=t_\mathrm{then}+\lambda_\mathrm{then}/c\,.</math>

The observer sees the next crest of the observed light wave with a wavelength <math>\lambda_\mathrm{now}</math> to arrive at a time

:<math>t=t_\mathrm{now}+\lambda_\mathrm{now}/c\,.</math>

Since the subsequent crest is again emitted from <math>r=R</math> and is observed at <math>r=0</math>, the following equation can be written:

:<math>
c \int_{t_\mathrm{then}+\lambda_\mathrm{then}/c}^{t_\mathrm{now}+\lambda_\mathrm{now}/c} \frac{dt}{a}\; = 
    \int_{R}^{0} \frac{dr}{\sqrt{1-kr^2}}\,.
</math>

The right-hand side of the two integral equations above are identical which means

:<math>
c \int_{t_\mathrm{then}+\lambda_\mathrm{then}/c}^{t_\mathrm{now}+\lambda_\mathrm{now}/c} \frac{dt}{a}\; =
c \int_{t_\mathrm{then}}^{t_\mathrm{now}} \frac{dt}{a}\,
</math>

or, alternatively,

:<math>
\int_{t_\mathrm{now}}^{t_\mathrm{now}+\lambda_\mathrm{now}/c} \frac{dt}{a}\; =
\int_{t_\mathrm{then}}^{t_\mathrm{then}+\lambda_\mathrm{then}/c} \frac{dt}{a}\,.
</math>

For very small variations in time (over the period of one cycle of a light wave) the scale factor is essentially a constant (<math>a=a_\mathrm{now}</math> today and <math>a=a_\mathrm{then}</math> previously). This yields

:<math>\frac{t_\mathrm{now}+\lambda_\mathrm{now}/c}{a_\mathrm{now}}-\frac{t_\mathrm{now}}{a_\mathrm{now}}\; = \frac{t_\mathrm{then}+\lambda_\mathrm{then}/c}{a_\mathrm{then}}-\frac{t_\mathrm{then}}{a_\mathrm{then}}
</math>

which can be rewritten as

:<math>\frac{\lambda_\mathrm{now}}{\lambda_\mathrm{then}}=\frac{a_\mathrm{now}}{a_\mathrm{then}}\,.</math>

Using the definition of redshift provided [[#Measurement, characterization, and interpretation|above]], the equation

:<math>1+z = \frac{a_\mathrm{now}}{a_\mathrm{then}}</math>

is obtained. In an expanding universe such as the one we inhabit, the scale factor is [[monotonic function|monotonically increasing]] as time passes, thus, z is positive and distant galaxies appear redshifted. This type of redshift is called the ''[[cosmological redshift]]'' or ''Hubble redshift''. If the universe were contracting instead of expanding, we would see distant galaxies blue shifted 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 blue shifted and other objects that "approach" are redshifted. For more on this bizarre result see Davis, T. M., Lineweaver, C. H., and Webb, J. K. "[http://arxiv.org/abs/astro-ph/0104349/ Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blue shifted objects]", ''[[American Journal of Physics]]'' (2003), '''71''' 358–364.</ref>

These galaxies are not receding simply by means of a physical velocity in the direction away from the observer; instead, the intervening space is stretching, which accounts for the large-scale isotropy of the effect demanded by the [[cosmological principle]].<ref name="Peebles-1993">Peebles (1993).</ref> For cosmological redshifts of z < 0.01 the effects of [[spacetime]] expansion are minimal and cosmological redshifts can be dominated by additional Doppler redshifts and blue shifts caused by the peculiar motions of the galaxies relative to one another.<ref>Measurements of the peculiar velocities out to 5 [[parsec|Mpc]] using the [[Hubble Space Telescope]] were reported in 2003 by Karachentsev et al. ''Local galaxy flows within 5 Mpc''. 02/2003 ''[[Astronomy and Astrophysics]]'', '''398''', 479-491.[http://arxiv.org/abs/astro-ph/0211011]</ref> The difference between physical velocity and space expansion can be illustrated by the [[Metric expansion of space#Expanding rubber sheet model|Expanding Rubber Sheet Universe]], a common cosmological analogy used to describe the expansion of space. If two objects are represented by ball bearings and spacetime by a stretching rubber sheet, the Doppler effect is caused by rolling the balls across the sheet to create peculiar motion. The cosmological redshift occurs when the ball bearings are stuck to the sheet and the sheet is stretched. (Obviously, there are dimensional problems with the model, as the ball bearings should be ''in'' the sheet, and cosmological redshift produces higher velocities than Doppler does if the distance between two objects is large enough.)

Using a 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 Ω<sub>0</sub>:

[[File:Two redshifts.JPG|thumb |350px |Two different sources of redshift: ''Top, Doppler shift'': the star moving to the left emits light that is blue shifted in the direction of the receiving antenna that the star approaches, and red shifted in the direction of the receiving antenna that the star is leaving. ''Center and bottom panels: cosmological expansion'': The distance between the emitting star and both antennas increases while the light is propagating, increasing the wavelength of the light seen by both antennas. Each panel shows the propagation of light over two periods: in the center panel at time of emission the wavelength is short, and in the bottom panel at time of reception the wavelength has stretched with the expansion of space. Patterned after Koupelis & Kuhn.<ref name=Kuhn>{{cite book |title=In Quest of the Universe |author=Theo Koupelis, Karl F. Kuhn |edition=5 |url=http://books.google.com/books?id=6rTttN4ZdyoC&pg=PA556#PPA557,M1 |page=557 |publisher=Jones & Bartlett Publishers |year=2007 |isbn=0763743879}}</ref> The reader is cautioned that too literal an interpretation of this figure can be misleading.<ref name=Lewis>"It is perfectly valid to interpret the equations of relativity in terms of an expanding space. The mistake is to push analogies too far and imbue space with physical properties that are not consistent with the equations of relativity." {{cite journal |title=Cosmological Radar Ranging in an Expanding Universe |url=http://arxiv.org/abs/0805.2197v1 |journal=Monthly Notices of the Royal Astronomical Society |volume 388 |author=Geraint F. Lewis ''et al.'' |year=2008 |pages=960-964 }}</ref> In particular, the Doppler shift panel is seen from a single inertial frame, that of the two antennas. The cosmological expansion panels are a concatenation of local inertial frames spanning the space between the various locally stationary objects.<ref name=Chodorowski>{{cite journal |author=Michal Chodorowski  |title=Is space really expanding? A counterexample |year=2007 |url=http://arxiv.org/abs/astro-ph/0601171v2 |journal=Concepts Phys |volume=4 |pages=17-34}}</ref>]]

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

with 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 thousand liters of space.<ref Name=Weinberg>{{cite book |author=Steven Weinberg |edition=2 |title=The First Three Minutes: A Modern View of the Origin of the Universe |url=http://books.google.com/books?id=oxfoF_gasvsC&pg=PA34 |page=34 |isbn=0465024378 |year=1993 |publisher=Basic Books}}</ref> At large redshifts one finds:

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

where ''H<sub>0</sub>'' = present-day [[Hubble constant]], and ''z'' = redshift.<ref name=Bergström>{{cite book |author=Lars Bergström, Ariel Goobar |title=Cosmology and Particle Astrophysics |url=http://books.google.com/books?id=CQYu_sutWAoC&pg=PA77 |page=77, Eq.4.79 |isbn=3540329242 |publisher=Springer |edition=2|year=2006}}</ref><ref name = Longair>{{cite book |title=Galaxy Formation |author=M.S. Longair |url=http://books.google.com/books?id=2ARuLT-tk5EC&pg=PA161 |page=161 |isbn=3540637850 |publisher=Springer |year=1998}}</ref><ref name=Sanchez>{{cite book |editor=Norma Sanchez |page=223 |title=Current Topics in Astrofundamental Physics |url=http://books.google.com/books?id=GOJoas-Dg7QC&pg=PA223 |isbn=0792368568 |year=2001 |publisher=Springer 
|chapter=The High Redshift Radio Universe |author=Yu N Parijskij}}</ref>

The redshifts of galaxies include both a component related to recessional velocity from expansion of the universe, and a component related to [[peculiar motion]] (Doppler shift). 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. Harrison provides a summary of this and related distinctions.<ref name=Harrison2>{{cite journal |title=The redshift-distance and velocity-distance laws |author=Edward Harrison |year=1992 |url=http://adsabs.harvard.edu/abs/1993ApJ...403...28H |journal=Astrophysical Journal, Part 1 |pages=28-31 |volume=403}}. A pdf file can be found [http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1993ApJ...403...28H&amp;data_type=PDF_HIGH&amp;whole_paper=YES&amp;type=PRINTER&amp;filetype=.pdf  here].</ref> Describing the cosmological expansion origin of redshift, Harrison says: “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.” See Harrison, p. 315.<ref name=Harrison1/> “The increase of wavelength from emission to absorption of light does not depend on the rate of change of ''a(t)'' [here ''a(t)'' is the [[Friedmann-Lemaître-Robertson-Walker metric|Robertson-Walker scale factor]]] at the times of emission or absorption, but on the increase of ''a(t)'' in the whole period from emission to absorption.”<ref name=Weinberg_Cosmology>{{cite book |url=http://books.google.com/books?id=48C-ym2EmZkC&pg=PA11 |author=Steven Weinberg |title=Cosmology |publisher=Oxford University Press |page=11 |year=2008 |isbn=9780198526827}}</ref>

Nonetheless, popular literature often uses the expression "Doppler redshift" instead of "cosmological redshift" to describe the redshift of galaxies dominated by the expansion of spacetime, despite the fact that the redshift is not found using the relativistic Doppler equation.<ref>Odenwald & Fienberg 1993</ref> In particular, Doppler redshift is bound by [[special relativity]]; thus ''v > c'' is impossible while, in contrast, ''v > c'' is possible for cosmological redshift because the space which separates the objects (for example, a quasar from the Earth) can expand faster than the speed of light.<ref>Speed faster than light is allowed because the [[metric expansion of space|expansion]] of the [[spacetime]] [[Metric (mathematics)|metric]] is described by [[general relativity]] in terms of sequences of only locally valid inertial frames as opposed to a global [[Minkowski metric]]. Expansion faster than light is an integrated effect over many local inertial frames and is allowed because no single inertial frame is involved. The speed-of-light limitation applies only locally. See {{cite journal |author=Michal Chodorowski  |title=Is space really expanding? A counterexample |year=2007 |url=http://arxiv.org/abs/astro-ph/0601171v2 |journal=Concepts Phys |volume=4 |pages=17-34}} </ref> More mathematically, the viewpoint that "distant galaxies are receding" and the viewpoint that "the space between galaxies is expanding" are related by changing [[coordinate system]]s. Expressing this precisely requires working with the mathematics of the [[Friedmann-Robertson-Walker metric]].<ref>M. Weiss, What Causes the Hubble Redshift?, entry in the Physics [[FAQ]] (1994), available via [[John Baez]]'s [http://math.ucr.edu/home/baez/physics/Relativity/GR/hubble.html website]</ref>

===Gravitational redshift===
{{main|Gravitational redshift}}
In the theory of [[general relativity]], there is time dilation within a gravitational well. This is known as the [[gravitational redshift]] or ''Einstein Shift''.<ref>See for example, Chant, C. A., "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1930JRASC..24..390C&amp;db_key=AST&amp;data_type=HTML&amp;format=&amp;high=42ca922c9c27309 Notes and Queries (Telescopes and Observatory Equipment-The Einstein Shift of Solar Lines)]" (1930) ''Journal of the Royal Astronomical Society of Canada'', Vol. 24, p.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 [[electrical charge|uncharged]], [[rotation|nonrotating]], [[spherical symmetry|spherically symmetric]] mass:

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

where
* <math>G\,</math> is the [[gravitational constant]],
* <math>M\,</math> is the [[mass]] of the object creating the gravitational field,
* <math>r\,</math> is the radial coordinate of the observer (which is analogous to the classical distance from the center of the object, but is actually a [[Schwarzschild coordinates|Schwarzschild coordinate]]), and
* <math>c\,</math> 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 | authorlink = Albert Einstein | year = 1907 | title = Unknown title | journal = Jahrbuch der Radioaktivität und Elektronik | volume = 4 | pages = 411&ndash;?}}</ref>

The effect is very small but measurable on Earth using the [[Mössbauer effect]] and was first observed in the [[Pound-Rebka experiment]].<ref>R. V. Pound and G. A. Rebka Jr., Apparent weight of photons, ''Phys. Rev. Lett.'' '''4''', 337 (1960). [http://prola.aps.org/abstract/PRL/v4/i7/p337_1] 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 | last = Sachs | first = R. K. | authorlink = Rainer Kurt Sachs|coauthors = [[Arthur Michael Wolfe|Wolfe, A. M.]] | year = 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 | pages = 73 }}</ref>

==Observations in astronomy==
The redshift observed in astronomy can be measured because the [[emission spectrum|emission]] and [[absorption spectrum|absorption]] spectra for [[atom]]s are distinctive and well known, calibrated from [[spectroscopy|spectroscopic experiments]] in [[laboratory|laboratories]] on Earth. When the redshift of various absorption and emission lines from a single astronomical object is measured, ''z'' is found to be remarkably constant. Although distant objects may be slightly blurred and lines broadened, it is by no more than can be explained by [[thermal motion|thermal]] or [[motion (physics)|mechanical motion]] of the source. For these reasons and others, the consensus among astronomers is that the redshifts they observe are due to some combination of the three established forms of Doppler-like redshifts. Alternative hypotheses and explanations for redshift such as [[tired light]] are not generally considered plausible.<ref name=reboul>When cosmological redshifts were first discovered, [[Fritz Zwicky]] proposed an effect known as tired light. While usually considered for historical interests, it is sometimes, along with [[intrinsic redshift]] suggestions, utilized by [[nonstandard cosmologies]]. In 1981, H. J. Reboul summarised many [http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1981A%26AS...45..129R&db_key=AST&data_type=HTML&format=&high=42ca922c9c23806 alternative redshift mechanisms] that had been discussed in the literature since the 1930s. In 2001, [[Geoffrey Burbidge]] remarked in a [http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2001PASP..113..899B&db_key=AST&data_type=HTML review] that the wider astronomical community has marginalized such discussions since the 1960s. Burbidge and [[Halton Arp]], while investigating the mystery of [[Quasar#History of quasar observation|the nature of quasars]], tried to develop alternative redshift mechanisms, and very few of their fellow scientists acknowledged let alone accepted their work. Moreover, Goldhaber ''et al.'' 2001; "Timescale Stretch Parameterization of Type Ia Supernova B-Band Lightcurves", ApJ, 558:359-386, 2001 September 1 pointed out that alternative theories are unable to account for timescale stretch observed in [[type Ia supernovae]]</ref>

Spectroscopy, as a measurement, is considerably more difficult than simple [[photometry (astronomy)|photometry]], which measures the [[brightness]] of astronomical objects through certain [[filter (optics)|filter]]s.<ref>For a review of the subject of photometry, consider Budding, E., ''Introduction to Astronomical Photometry'', Cambridge University Press (September 24, 1993), ISBN 0-521-41867-4</ref> When photometric data is all that is available (for example, the [[Hubble Deep Field]] and the [[Hubble Ultra Deep Field]]), astronomers rely on a technique for measuring [[photometric redshift]]s.<ref>The technique was first described by Baum, W. A.: 1962, in G. C. McVittie (ed.), ''Problems of extra-galactic research'', p. 390, IAU Symposium No. 15</ref> Due to the filter being sensitive to a range of wavelengths and the technique relying on making many assumptions about the nature of the spectrum at the light-source, [[observational error|error]]s for these sorts of measurements can range up to δ''z'' = 0.5, and are much less reliable than spectroscopic determinations.<ref>Bolzonella, M.; Miralles, J.-M.; Pelló, R., [http://arxiv.org/abs/astro-ph/0003380 Photometric redshifts based on standard SED fitting procedures], ''[[Astronomy and Astrophysics]]'', '''363''', p.476–492 (2000).</ref> However, photometry does allow at least for a qualitative characterization of a redshift. For example, if a sun-like spectrum had a redshift of ''z'' = 1, it would be brightest in the [[infrared]] rather than at the yellow-green color associated with the peak of its [[blackbody spectrum]], and the light intensity will be reduced in the filter by a factor of two (1+''z'') (see [[K correction]] for more details on the photometric consequences of redshift).<ref>A pedagogical overview of the K-correction by David Hogg and other members of the [[Sloan Digital Sky Survey|SDSS]] collaboration can be found at [http://arxiv.org/abs/astro-ph/0210394 astro-ph].</ref>

===Local observations===

[[Image:LASCO C1a.png|thumb|200px|A picture of the solar corona taken with the [[Solar and Heliospheric Observatory|LASCO C1]] coronagraph. The picture is a color coded image of the doppler shift of the FeXIV 5308 Å line, caused by the coronal plasma velocity towards or away from the satellite.]]

In nearby objects (within our [[Milky Way]] galaxy) observed redshifts are almost always related to the [[Line-of-sight propagation|line-of-sight]] velocities associated with the objects being observed. Observations of such redshifts and blue shifts have enabled astronomers to measure [[velocity|velocities]] and parametrize the [[mass]]es of the [[orbit (celestial mechanics)|orbiting]] [[star]]s in [[Binary star#Spectroscopic binaries|spectroscopic binaries]], a method first employed in 1868 by British astronomer [[William Huggins]].<ref name=Huggins /> Similarly, small redshifts and blue shifts detected in the spectroscopic measurements of individual stars are one way astronomers have been able to [[Methods of detecting extrasolar planets#Radial velocity|diagnose and measure]] the presence and characteristics of [[extrasolar planet|planetary systems]] around other stars.<ref>The [[Exoplanet Tracker]] is the newest observing project to use this technique, able to track the redshift variations in multiple objects at once, as reported in Ge, Jian et al. [http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2006ApJ...648..683G&link_type=ARTICLE&db_key=AST The First Extrasolar Planet Discovered with a New-Generation High-Throughput Doppler Instrument], ''[[The Astrophysical Journal]]'', 2006 '''648''', Issue 1, pp. 683-695.[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2006ApJ...648..683G&amp;db_key=AST&amp;data_type=HTML&amp;format=&amp;high=445165d66216614]</ref> Measurements of redshifts to fine detail are used in [[helioseismology]] to determine the precise movements of the [[photosphere]] of the [[Sun]].<ref>Libbrecht, Ken G., [http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1988SSRv...47..275L&amp;data_type=PDF_HIGH&amp;type=PRINTER&amp;filetype=.pdf Solar and stellar seismology], ''Space Science Reviews'', 1988 '''37''' n. 3–4, 275–301.</ref> Redshifts have also been used to make the first measurements of the [[Rotation#Astronomy|rotation rates]] of [[planet]]s,<ref>In 1871 [[Hermann Carl Vogel]] measured the rotation rate of [[Venus]]. [[Vesto Slipher]] was working on such measurements when he turned his attention to spiral nebulae.</ref> velocities of [[interstellar cloud]]s,<ref>An early review by [[Jan Hendrick Oort|Oort, J. H.]] on the subject: [http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1970A%26A.....7..381O&amp;data_type=PDF_HIGH&amp;type=PRINTER&amp;filetype=.pdf The formation of galaxies and the origin of the high-velocity hydrogen], [[Astronomy and Astrophysics]], '''7''', 381 (1970) [http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1970A%26A.....7..381O&amp;db_key=AST&amp;data_type=HTML&amp;format=&amp;high=4504216b9b04182].</ref> the [[galaxy rotation problem|rotation of galaxies]],<ref name=basicastronomy /> and the [[dynamics (mechanics)|dynamics]] of [[accretion theory|accretion]] onto [[neutron star]]s and [[black hole]]s which exhibit both Doppler and gravitational redshifts.<ref>Asaoka, Ikuko, [http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1989PASJ...41..763A&amp;data_type=PDF_HIGH&amp;type=PRINTER&amp;filetype=.pdf X-ray spectra at infinity from a relativistic accretion disk around a Kerr black hole], ''Astronomical Society of Japan, Publications'' (ISSN 0004-6264), ''41'' no. 4, 1989, p. 763–778 [http://adsabs.harvard.edu/cgi-bin/bib_query?1989PASJ...41..763A]</ref> Additionally, the [[temperature]]s of various emitting and absorbing objects can be obtained by measuring [[Doppler broadening]] — effectively redshifts and blue shifts over a single emission or absorption line.<ref>Rybicki, G. B. and A. R. Lightman, ''Radiative Processes in Astrophysics'', John Wiley & Sons, 1979, p. 288 ISBN 0-471-82759-2</ref> By measuring the broadening and shifts of the 21-centimeter [[hydrogen line]] in different directions, astronomers have been able to measure the [[recessional velocity|recessional velocities]] of [[interstellar gas]], which in turn reveals the [[rotation curve]] of our Milky Way.<ref name=basicastronomy /> Similar measurements have been performed on other galaxies, such as [[Andromeda (galaxy)|Andromeda]].<ref name=basicastronomy /> As a diagnostic tool, redshift measurements are one of the most important [[astronomical spectroscopy|spectroscopic measurements]] made in astronomy.

===Extragalactic observations===
{{Cosmology}}
The most distant objects exhibit larger redshifts corresponding to the [[Hubble's law|Hubble flow]] of the universe. The largest observed redshift, corresponding to the greatest distance and furthest back in time, is that of the [[cosmic microwave background radiation]]; the numerical value of its redshift is about ''z'' = 1089 (''z'' = 0 corresponds to present time), and it shows the state of the [[Universe]] about 13.7 billion years ago, and 379,000 years after the initial moments of the [[Big Bang]].<ref>An accurate measurement of the cosmic microwave background was achieved by the [[COBE]] experiment. The final published temperature of 2.73 K was reported in this paper: Fixsen, D. J.; Cheng, E. S.; Cottingham, D. A.; Eplee, R. E., Jr.; Isaacman, R. B.; Mather, J. C.; Meyer, S. S.; Noerdlinger, P. D.; Shafer, R. A.; Weiss, R.; Wright, E. L.; Bennett, C. L.; Boggess, N. W.; Kelsall, T.; Moseley, S. H.; Silverberg, R. F.; Smoot, G. F.; Wilkinson, D. T.. (1994). "Cosmic microwave background dipole spectrum measured by the COBE FIRAS instrument", ''Astrophysical Journal'', 420, 445. The most accurate measurement as of 2006 was achieved by the [[Wilkinson Microwave Anisotropy Probe|WMAP]] experiment.</ref>

The luminous point-like cores of [[quasar]]s were the first "high-redshift" (<math>z > 0.1</math>) objects discovered before the improvement of telescopes allowed for the discovery of other high-redshift galaxies.

For galaxies more distant than the [[Local Group]] and the nearby [[Virgo Cluster]], but within a thousand [[parsec|megaparsecs]] or so, the redshift is approximately proportional to the galaxy's distance. This correlation was first observed by [[Edwin Hubble]] and has come to be known as [[Hubble's law]]. [[Vesto Slipher]] was the first to discover galactic redshifts, in about the year 1912, while Hubble correlated Slipher's measurements with distances he [[cosmic distance ladder|measured by other means]] to formulate his Law. In the widely accepted cosmological model based on [[general relativity]], redshift is mainly a result of the expansion of space: this means that the farther away a galaxy is from us, the more the space has expanded in the time since the light left that galaxy, so the more the light has been stretched, the more redshifted the light is, and so the faster it appears to be moving away from us. [[Hubble's law]] follows in part from the [[Copernican principle]].<ref name="Peebles-1993"/> Because it is usually not known how [[luminosity|luminous]] objects are, measuring the redshift is easier than more direct distance measurements, so redshift is sometimes in practice converted to a crude distance measurement using Hubble's law.

[[Gravitation|Gravitational interactions]] of galaxies with each other and clusters cause a significant [[variance|scatter]] in the normal plot of the Hubble diagram. The [[peculiar velocity|peculiar velocities]] associated with galaxies superimpose a rough trace of the [[mass]] of [[virial theorem|virialized objects]] in the universe. This effect leads to such phenomena as nearby galaxies (such as the [[Andromeda Galaxy]]) exhibiting blue shifts as we fall towards a common [[barycenter]], and redshift maps of clusters showing a [[Fingers of God|Finger of God]] effect due to the scatter of peculiar velocities in a roughly spherical distribution.<ref name="Peebles-1993"/> This added component gives cosmologists a chance to measure the masses of objects independent of the ''[[mass to light ratio]]'' (the ratio of a galaxy's mass in solar masses to its brightness in solar luminosities), an important tool for measuring [[dark matter]].<ref>{{cite book|first=James|last=Binney|coauthors=and Scott Treimane|title=Galactic dynamics|publisher=Princeton University Press|isbn=0-691-08445-9|year=1994}}</ref>

The Hubble law's linear relationship between distance and redshift assumes that the rate of expansion of the universe is constant. However, when the universe was much younger, the expansion rate, and thus the Hubble "constant", was larger than it is today. For more distant galaxies, then, whose light has been travelling to us for much longer times, the approximation of constant expansion rate fails, and the Hubble law becomes a non-linear integral relationship and dependent on the history of the expansion rate since the emission of the light from the galaxy in question. Observations of the redshift-distance relationship can be used, then, to determine the expansion history of the universe and thus the matter and energy content.

While it was long believed that the expansion rate has been continuously decreasing since the Big Bang, recent observations of the redshift-distance relationship using [[Type Ia supernova]]e have suggested that in comparatively recent times the expansion rate of the universe has [[Accelerating universe|begun to accelerate]].

===Highest redshifts===
Currently, the objects with the highest known redshifts are galaxies and the objects producing gamma ray bursts. The most reliable redshifts are from [[spectroscopic]] data, and the highest confirmed [[spectroscopic]] redshift of a galaxy is that of [[IOK-1]],<ref> {{cite journal 
| author=Masanori Iye, ''et al.''
| title=A galaxy at a redshift z = 6.96
| url=http://www.nature.com/nature/journal/v443/n7108/abs/nature05104.html
| journal=Nature
| volume=443
| issue=7108
| pages=186–188
| doi=10.1038/nature05104
| year=2006
}}
</ref> at a redshift z = 6.96. Slightly less reliable are [[Lyman-alpha forest|Lyman-break]] redshifts, the highest of which is the lensed galaxy A1689-zD1 at a redshift z = 7.6<ref>Bradley, L.., et al., Discovery of a Very Bright Strongly Lensed Galaxy Candidate at z ~ 7.6, ''[[Astrophysical Journal|The Astrophysical Journal]]'' (2008), Volume 678, Issue 2, pp. 647-654. [http://adsabs.harvard.edu/abs/2008ApJ...678..647B</ref> and the next highest being <math>z=7.0</math><ref>Egami, E., et al., Spitzer and Hubble Space Telescope Constraints on the Physical Properties of the z~7 Galaxy Strongly Lensed by A2218, ''[[Astrophysical Journal|The Astrophysical Journal]]'' (2005), v. 618, Issue 1, pp. L5-L8 [http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2005ApJ...618L...5E&amp;db_key=AST&amp;data_type=HTML&amp;format=&amp;high=43a73989ff16910].</ref> while as-yet unconfirmed reports by Ellis R. et al. from a [[gravitational lens]] observed in a distant [[galaxy groups and clusters|galaxy cluster]] may indicate a galaxy with a redshift of <math>z=10.1</math>.

The most distant observed gamma ray burst was [[GRB 090423]], which had a redshift of 8.2.<ref>{{cite journal
|title=GRB 090423 reveals an exploding star at the epoch of re-ionization
|author= Salvaterra, R. ''et al.''
|journal=Submitted to Nature
|url=http://adsabs.harvard.edu/abs/2009arXiv0906.1578S
|year=2009
}}</ref>

The most distant known quasar, CFHQS J2329-0301, is at <math>z = 6.43 </math>.<ref>Willott, Chris J. et al., Four Quasars Above Redshift 6 Discovered by the Canada-France High-z Quasar Survey, ''[[Astronomical Journal|The Astronomical Journal]]'' (2007), v. 134, pp. 2435–2450 [http://adsabs.harvard.edu/abs/2007AJ....134.2435W&amp;db_key=AST&amp;data_type=HTML&amp;format=].</ref> The highest known redshift radio galaxy (TN J0924-2201) is at a redshift z = 5.2<ref>Klamer et al., 2005, ApJ 621, L1</ref> and the highest known redshift molecular material is the detection of emission from the CO molecule from the quasar SDSS J1148+5251 at z = 6.42<ref>Walter, Fabian et al., Molecular gas in the host galaxy of a quasar at redshift z = 6.42, ''[[Nature]]'' (2003), v. 424, pp. 406–408 [http://adsabs.harvard.edu/cgi-bin/bib_query?2003Natur.424..406W&amp;db_key=AST&amp;data_type=HTML&amp;format=].</ref>

''Extremely red objects'' (EROs) are astronomical sources of radiation that radiate energy in the red and near infrared part of the electromagnetic spectrum. These may be starburst galaxies that have  a high redshift accompanied by reddening from intervening dust, or they could be highly redshifted elliptical galaxies with an older (and therefore redder) stellar population.<ref>{{cite journal
 | author=Smail, Ian; Owen, F. N.; Morrison, G. E.; Keel, W. C.; Ivison, R. J.; Ledlow, M. J.
 | total=The Diversity of Extremely Red Objects
 | journal=The Astrophysical Journal | volume=581 | issue=2
 | pages=844&ndash;864 | doi=10.1086/344440 | bibcode=2002ApJ...581..844S }}</ref> Objects that are even redder than EROs are termed ''hyper extremely red objects'' (HEROs).<ref>{{cite journal
 | author=Totani, Tomonori; Yoshii, Yuzuru; Iwamuro, Fumihide; Maihara, Toshinori; Motohara, Kentaro
 | title=Hyper Extremely Red Objects in the Subaru Deep Field: Evidence for Primordial Elliptical Galaxies in the Dusty Starburst Phase
 | journal=The Astrophysical Journal | volume=558 | issue=2
 | year=2001 | pages=L87&ndash;L91 | doi=10.1086/323619
 | bibcode=2001ApJ...558L..87T }}</ref>

===Redshift surveys===
{{main|Redshift survey}}
[[Image:2dfgrs.png|left|thumb|300px|Rendering of the 2dFGRS data]]

With the advent of automated [[telescope]]s and improvements in [[astronomical spectroscopy|spectroscopes]], a number of collaborations have been made to map the universe in redshift space. By combining redshift with angular position data, a redshift survey maps the 3D distribution of matter within a field of the sky. These observations are used to measure properties of the [[large-scale structure of the cosmos|large-scale structure]] of the universe. The [[Great Wall (astronomy)|Great Wall]], a vast [[supercluster]] of galaxies over 500 million [[light-year]]s wide, provides a dramatic example of a large-scale structure that redshift surveys can detect.<ref>M. J. Geller & J. P. Huchra, ''Science'' '''246''', 897 (1989). [http://www.sciencemag.org/cgi/content/abstract/246/4932/897 online]</ref>

The first redshift survey was the [[CfA Redshift Survey]], started in 1977 with the initial data collection completed in 1982.<ref>See the official CfA [http://cfa-www.harvard.edu/~huchra/zcat/ website] for more details.</ref> More recently, the [[2dF Galaxy Redshift Survey]] determined the large-scale structure of one section of the Universe, measuring ''z''-values for over 220,000 galaxies; data collection was completed in 2002, and the final [[data set]] was released 30 June 2003.<ref>{{cite paper|title=The 2dF galaxy redshift survey: Power-spectrum analysis of the final dataset and cosmological implications|author=Shaun Cole ''et al.'' (The 2dFGRS Collaboration)|journal=Mon. Not. Roy. Astron. Soc.|volume=362|pages=505&ndash;34|year=2005|url=http://www.arxiv.org/abs/}} [http://msowww.anu.edu.au/2dFGRS/ 2dF Galaxy Redshift Survey homepage]</ref> (In addition to mapping large-scale patterns of galaxies, 2dF established an upper limit on [[neutrino]] mass.) Another notable investigation, the [[Sloan Digital Sky Survey]] (SDSS), is ongoing as of 2005 and aims to obtain measurements on around 100 million objects.<ref>[http://www.sdss.org/ SDSS Homepage]</ref> SDSS has recorded redshifts for galaxies as high as 0.4, and has been involved in the detection of [[quasar]]s beyond ''z'' = 6. The [[DEEP2 Redshift Survey]] uses the [[Keck telescopes]] with the new "DEIMOS" [[spectrograph]]; a follow-up to the pilot program DEEP1, DEEP2 is designed to measure faint galaxies with redshifts 0.7 and above, and it is therefore planned to provide a complement to SDSS and 2dF.<ref>{{cite conference|title=Science objectives and early results of the DEEP2 redshift survey|author=Marc Davis ''et al.'' (DEEP2 collaboration)|year=2002|booktitle=Conference on Astronomical Telescopes and Instrumentation, Waikoloa, Hawaii, 22–28 Aug 2002| url=http://arxiv.org/abs/astro-ph/0209419 }}</ref>

==Effects due to physical optics or radiative transfer==
The interactions and phenomena summarized in the subjects of [[radiative transfer]] and [[physical optics]] can result in shifts in the wavelength and frequency of electromagnetic radiation. In such cases the shifts correspond to a physical energy transfer to matter or other photons rather than being due to a transformation between reference frames. These shifts can be due to such physical phenomena as [[Wolf effect|coherence effects]] or the [[scattering]] of [[electromagnetic radiation]] whether from [[electric charge|charged]] [[elementary particle]]s, from particulates, or from fluctuations of the [[index of refraction]] in a [[dielectric medium]] as occurs in the radio phenomenon of [[Whistler (radio)|radio whistlers]].<ref name=basicastronomy /> While such phenomena are sometimes referred to as "redshifts" and "blue shifts", in astrophysics light-matter interactions that result in energy shifts in the radiation field are generally referred to as "reddening" rather than "redshifting" which, as a term, is normally reserved for the [[#Redshift formulas|effects discussed above]].<ref name=basicastronomy />

In many circumstances scattering causes radiation to redden because [[entropy]] results in the predominance of many low-[[energy]] photons over few high-energy ones (while [[conservation of energy|conserving total energy]]).<ref name=basicastronomy /> Except possibly under carefully controlled conditions, scattering does not produce the same relative change in wavelength across the whole spectrum; that is, any calculated ''z'' is generally a [[function (mathematics)|function]] of wavelength. Furthermore, scattering from [[randomness|random]] [[matter|media]] generally occurs at many [[angle]]s, and ''z'' is a function of the scattering angle. If multiple scattering occurs, or the scattering particles have relative motion, then there is generally distortion of [[spectral line]]s as well.<ref name=basicastronomy />

In [[interstellar medium|interstellar astronomy]], [[visible spectrum|visible spectra]] can appear [[red]]der due to scattering processes in a phenomenon referred to as [[interstellar reddening]]<ref name=basicastronomy /> — similarly [[Rayleigh scattering]] causes the [[Earth's atmosphere|atmospheric]] reddening of the [[Sun]] seen in the [[sunrise]] or [[sunset]] and causes the rest of the [[sky]] to have a [[blue]] color. This phenomenon is distinct from red''shift''ing because the [[atomic spectral line|spectroscopic lines]] are not shifted to other wavelengths in reddened objects and there is an additional [[extinction (astronomy)|dimming]] and distortion associated with the phenomenon due to photons being scattered in and out of the [[Line-of-sight propagation|line-of-sight]].

''For a list of scattering processes, see [[Scattering]].''

==References==
===Notes===
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===Articles===
*Odenwald, S. & Fienberg, RT. 1993; "Galaxy Redshifts Reconsidered" in ''Sky & Telescope'' Feb. 2003; pp31–35 (This article is useful further reading in distinguishing between the 3 types of redshift and their causes.)
*Lineweaver, Charles H. and Tamara M. Davis, "[http://www.sciam.com/article.cfm?chanID=sa006&colID=1&articleID=0009F0CA-C523-1213-852383414B7F0147 Misconceptions about the Big Bang]", ''[[Scientific American]]'', March 2005. (This article is useful for explaining the cosmological redshift mechanism as well as clearing up misconceptions regarding the physics of the expansion of space.)

===Book references===
* {{cite book | last=Binney|first=James|coauthors=and Michael Merrifeld|title=Galactic Astronomy|publisher=Princeton University Press|year=1998|isbn=0-691-02565-7}}
* {{cite book | author=Carroll, Bradley W. and Dale A. Ostlie| title=An Introduction to Modern Astrophysics| publisher=Addison-Wesley Publishing Company, Inc.| year=1996| isbn=0-201-54730-9}}
* {{cite book | author=Feynman, Richard; Leighton, Robert; Sands, Matthew | title=[[The Feynman Lectures on Physics|Feynman Lectures on Physics]]. Vol. 1 | publisher=Addison-Wesley | year=1989 | isbn=0-201-51003-0}}
* {{cite book | last = Grøn | first = Øyvind |authorlink=Øyvind Grøn| coauthors = Hervik, Sigbjørn | title = Einstein's General Theory of Relativity | location = New York | publisher = Springer | year = 2007 | isbn = 978-0-387-69199-2}}
* {{cite book | author=Kutner, Marc | title=Astronomy: A Physical Perspective | publisher=Cambridge University Press | year=2003 | isbn=0-521-52927-1}}
* {{cite book | last = Misner | first = Charles | coauthors = Thorne, Kip S. and Wheeler, John Archibald | title = Gravitation | location = San Francisco | publisher = W. H. Freeman | year = 1973 | isbn = 0-7167-0344-0}}
* {{cite book | first = P. J. E. | last = Peebles | title = Principles of Physical Cosmology | publisher = Princeton University Press | year = 1993 | isbn = 0-691-01933-9 }}
* {{cite book | author=Taylor, Edwin F.; [[John Archibald Wheeler|Wheeler, John Archibald]] | title=Spacetime Physics: Introduction to Special Relativity (2nd ed.) | publisher=W.H. Freeman | year=1992 | isbn=0-7167-2327-1}}
* {{cite book | first = Steven | last = Weinberg | title = Gravitation and Cosmology | publisher = John Wiley | year = 1971 | isbn = 0-471-92567-5}}
* See also [[physical cosmology#Textbooks|physical cosmology textbooks]] for applications of the cosmological and gravitational redshifts.

==External links==
{{Commons|Redshift}}
*[http://www.astro.ucla.edu/~wright/doppler.htm Ned Wright's Cosmology tutorial]
*[http://www.space.com/scienceastronomy/redshift.html Article on redshift from SPACE.com]
*[http://coolcosmos.ipac.caltech.edu/cosmic_classroom/cosmic_reference/redshift.html Cosmic reference guide entry on redshift]
*[http://www.asterism.org/tutorials/tut29-1.htm Mike Luciuk's Astronomical Redshift tutorial]
*[http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/Images/hu_animexp.gif Animated GIF of Cosmological Redshift] by Wayne Hu

{{featured article}}

[[Category:Astronomical spectroscopy]]
[[Category:Doppler effects]]
[[Category:Physical cosmology]]
[[Category:Physical quantities]]

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