# Redshift | redshift formulae

## Redshift formulae

In general relativity one can derive 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 z) is independent of the wavelength.[2]

Redshift summary
Redshift type Geometry Formula[22]
Relativistic Doppler Minkowski space (flat spacetime)

For motion completely in the radial or line-of-sight direction:
${\displaystyle 1+z=\gamma \left(1+{\frac {v_{\parallel }}{c}}\right)={\sqrt {\frac {1+{\frac {v_{\parallel }}{c}}}{1-{\frac {v_{\parallel }}{c}}}}}}$
${\displaystyle z\approx {\frac {v_{\parallel }}{c}}}$ for small ${\displaystyle v_{\parallel }}$
For motion completely in the transverse direction:
${\displaystyle 1+z={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$

Cosmological redshift FLRW spacetime (expanding Big Bang universe) ${\displaystyle 1+z={\frac {a_{\mathrm {now} }}{a_{\mathrm {then} }}}}$
Gravitational redshift Any stationary spacetime (e.g. the Schwarzschild geometry) ${\displaystyle 1+z={\sqrt {\frac {g_{tt}({\text{receiver}})}{g_{tt}({\text{source}})}}}}$
For the Schwarzschild geometry, ${\displaystyle 1+z={\sqrt {\frac {1-{\frac {2GM}{c^{2}r_{\text{receiver}}}}}{1-{\frac {2GM}{c^{2}r_{\text{source}}}}}}}}$

### Doppler effect

Doppler effect, yellow (~575 nm wavelength) ball appears greenish (blueshift to ~565 nm wavelength) approaching observer, turns 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 km/s, or about 75 times faster than the speed record for the fastest manmade space probe.

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 blueshift (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, which is much less than the speed of light (vc), the redshift is given by

${\displaystyle z\approx {\frac {v}{c}}}$     (since ${\displaystyle \gamma \approx 1}$)

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.

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 (for motion solely in the line of sight):

${\displaystyle 1+z=\left(1+{\frac {v}{c}}\right)\gamma .}$

This phenomenon was first observed in a 1938 experiment performed by Herbert E. Ives and G.R. Stilwell, called the Ives–Stilwell experiment.[23]

Since the Lorentz factor is dependent only on the 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 projection of the movement of the source into the line-of-sight which yields different results for different orientations. If θ is the angle between the direction of relative motion and the direction of emission in the observer's frame[24] (zero angle is directly away from the observer), the full form for the relativistic Doppler effect becomes:

${\displaystyle 1+z={\frac {1+v\cos(\theta )/c}{\sqrt {1-v^{2}/c^{2}}}}}$

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

${\displaystyle 1+z={\sqrt {\frac {1+v/c}{1-v/c}}}}$

For the special case that the light is approaching at right angles (θ = 90°) to the direction of relative motion in the observer's frame,[25] the relativistic redshift is known as the transverse redshift, and a redshift:

${\displaystyle 1+z={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}}$

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.[26]

### Expansion of space

In the early part of the twentieth century, Slipher, Hubble and others made the first measurements of the redshifts and blueshifts of galaxies beyond the Milky Way. They initially interpreted these redshifts and blueshifts as due to random motions, 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 mechanism for producing redshifts seen in certain cosmological solutions to Einstein's equations of general relativity. Hubble's law of the correlation between redshifts and distances is required by all such models that have a metric expansion of space.[18] As a result, the wavelength of photons propagating through the expanding space is stretched, creating the cosmological redshift.

There is a distinction between a redshift in cosmological context as compared to that witnessed when nearby objects exhibit a local Doppler-effect redshift. Rather than cosmological redshifts being a consequence of the relative velocities that are subject to the laws of special relativity (and thus subject to the rule that no two locally separated objects can have relative velocities with respect to each other faster than the speed of light), the photons instead increase in wavelength and redshift because of a global feature of the spacetime metric through which they are traveling. One interpretation of this effect is the idea that space itself is expanding.[27] Due to the expansion increasing as distances increase, the distance between two remote galaxies can increase at more than 3×108 m/s, but this does not imply that the galaxies move faster than the speed of light at their present location (which is forbidden by Lorentz covariance).

#### Mathematical derivation

The observational consequences of this effect can be derived using the equations from general relativity that describe a homogeneous and isotropic universe.

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

${\displaystyle ds^{2}=0=-c^{2}dt^{2}+{\frac {a^{2}dr^{2}}{1-kr^{2}}}}$

where

For an observer observing the crest of a light wave at a position r = 0 and time t = tnow, the crest of the light wave was emitted at a time t = tthen in the past and a distant position r = R. Integrating over the path in both space and time that the light wave travels yields:

${\displaystyle c\int _{t_{\mathrm {then} }}^{t_{\mathrm {now} }}{\frac {dt}{a}}\;=\int _{R}^{0}{\frac {dr}{\sqrt {1-kr^{2}}}}\,.}$

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 λthen. The next crest of the light wave was emitted at a time

${\displaystyle t=t_{\mathrm {then} }+\lambda _{\mathrm {then} }/c\,.}$

The observer sees the next crest of the observed light wave with a wavelength λnow to arrive at a time

${\displaystyle t=t_{\mathrm {now} }+\lambda _{\mathrm {now} }/c\,.}$

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

${\displaystyle c\int _{t_{\mathrm {then} }+\lambda _{\mathrm {then} }/c}^{t_{\mathrm {now} }+\lambda _{\mathrm {now} }/c}{\frac {dt}{a}}\;=\int _{R}^{0}{\frac {dr}{\sqrt {1-kr^{2}}}}\,.}$

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

${\displaystyle c\int _{t_{\mathrm {then} }+\lambda _{\mathrm {then} }/c}^{t_{\mathrm {now} }+\lambda _{\mathrm {now} }/c}{\frac {dt}{a}}\;=c\int _{t_{\mathrm {then} }}^{t_{\mathrm {now} }}{\frac {dt}{a}}\,}$

Using the following manipulation:

{\displaystyle {\begin{aligned}0&=\int _{t_{\mathrm {then} }}^{t_{\mathrm {now} }}{\frac {dt}{a}}-\int _{t_{\mathrm {then} }+\lambda _{\mathrm {then} }/c}^{t_{\mathrm {now} }+\lambda _{\mathrm {now} }/c}{\frac {dt}{a}}\\&=\int _{t_{\mathrm {then} }}^{t_{\mathrm {then} }+\lambda _{\mathrm {then} }/c}{\frac {dt}{a}}+\int _{t_{\mathrm {then} }+\lambda _{\mathrm {then} }/c}^{t_{\mathrm {now} }}{\frac {dt}{a}}-\int _{t_{\mathrm {then} }+\lambda _{\mathrm {then} }/c}^{t_{\mathrm {now} }+\lambda _{\mathrm {now} }/c}{\frac {dt}{a}}\\&=\int _{t_{\mathrm {then} }}^{t_{\mathrm {then} }+\lambda _{\mathrm {then} }/c}{\frac {dt}{a}}-\left(\int _{t_{\mathrm {now} }}^{t_{\mathrm {then} }+\lambda _{\mathrm {then} }/c}{\frac {dt}{a}}+\int _{t_{\mathrm {then} }+\lambda _{\mathrm {then} }/c}^{t_{\mathrm {now} }+\lambda _{\mathrm {now} }/c}{\frac {dt}{a}}\right)\\&=\int _{t_{\mathrm {then} }}^{t_{\mathrm {then} }+\lambda _{\mathrm {then} }/c}{\frac {dt}{a}}-\int _{t_{\mathrm {now} }}^{t_{\mathrm {now} }+\lambda _{\mathrm {now} }/c}{\frac {dt}{a}}\end{aligned}}}

we find that:

${\displaystyle \int _{t_{\mathrm {now} }}^{t_{\mathrm {now} }+\lambda _{\mathrm {now} }/c}{\frac {dt}{a}}\;=\int _{t_{\mathrm {then} }}^{t_{\mathrm {then} }+\lambda _{\mathrm {then} }/c}{\frac {dt}{a}}\,.}$

For very small variations in time (over the period of one cycle of a light wave) the scale factor is essentially a constant (a = anow today and a = athen previously). This yields

${\displaystyle {\frac {t_{\mathrm {now} }+\lambda _{\mathrm {now} }/c}{a_{\mathrm {now} }}}-{\frac {t_{\mathrm {now} }}{a_{\mathrm {now} }}}\;={\frac {t_{\mathrm {then} }+\lambda _{\mathrm {then} }/c}{a_{\mathrm {then} }}}-{\frac {t_{\mathrm {then} }}{a_{\mathrm {then} }}}}$

which can be rewritten as

${\displaystyle {\frac {\lambda _{\mathrm {now} }}{\lambda _{\mathrm {then} }}}={\frac {a_{\mathrm {now} }}{a_{\mathrm {then} }}}\,.}$

Using the definition of redshift provided above, the equation

${\displaystyle 1+z={\frac {a_{\mathrm {now} }}{a_{\mathrm {then} }}}}$

is obtained. In an expanding universe such as the one we inhabit, the scale factor is monotonically increasing as time passes, thus, z is positive and distant galaxies appear redshifted.

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 Ω0:

${\displaystyle \Omega _{0}={\frac {\rho }{\rho _{\text{crit}}}}\ ,}$

with ρcrit 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.[28] At large redshifts one finds:

${\displaystyle t(z)={\frac {2}{3H_{0}{\Omega _{0}}^{1/2}(1+z)^{3/2}}}\ ,}$

where H0 is the present-day Hubble constant, and z is the redshift.[29][30][31]

#### Distinguishing between cosmological and local effects

For cosmological redshifts of z < 0.01 additional Doppler redshifts and blueshifts due to the peculiar motions of the galaxies relative to one another cause a wide scatter from the standard Hubble Law.[32] The resulting situation can be illustrated by the 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.[33][34][35]

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).[36] 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.[37] 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..."[38] Steven Weinberg clarified, "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 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."[39]

Popular literature often uses the expression "Doppler redshift" instead of "cosmological redshift" to describe the redshift of galaxies dominated by the expansion of spacetime, but the cosmological redshift is not found using the relativistic Doppler equation[40] which is instead characterized by special relativity; thus v > c is impossible while, in contrast, v > c is possible for cosmological redshifts because the space which separates the objects (for example, a quasar from the Earth) can expand faster than the speed of light.[41] More mathematically, the viewpoint that "distant galaxies are receding" and the viewpoint that "the space between galaxies is expanding" are related by changing coordinate systems. Expressing this precisely requires working with the mathematics of the Friedmann-Robertson-Walker metric.[42]

If the Universe were contracting instead of expanding, we would see distant galaxies blueshifted by an amount proportional to their distance instead of redshifted.[43]

### 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.[44] The theoretical derivation of this effect follows from the Schwarzschild solution of the Einstein equations which yields the following formula for redshift associated with a photon traveling in the gravitational field of an uncharged, nonrotating, spherically symmetric mass:

${\displaystyle 1+z={\frac {1}{\sqrt {1-{\frac {2GM}{rc^{2}}}}}},}$

where

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.[45]

The effect is very small but measurable on Earth using the Mössbauer effect and was first observed in the Pound–Rebka experiment.[46] 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).[47][48]

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