View source for Comoving and proper distances

{{Short description|Measurement of distance}}
{{Redirect|Physical distance|the general concept|Distance (physics)}}
{{Cosmology}}

In [[Big Bang|standard cosmology]], '''comoving distance''' and '''proper distance''' (or physical distance) are two closely related [[distance measures (cosmology)|distance measures]] used by cosmologists to define distances between objects. ''Comoving distance'' factors out the expansion of the universe, giving a distance that does not change in time due to the expansion of space (though this may change due to other, local factors, such as the motion of a galaxy within a cluster).<ref>{{cite book |last1=Huterer |first1=Dragan |title=A Course in Cosmology |date=2023 |publisher=Cambridge University Press |isbn=978-1-316-51359-0}}</ref> ''Proper distance'' roughly corresponds to where a distant object would be at a specific moment of [[cosmological time]], which can change over time due to the [[expansion of the universe]]. Comoving distance and proper distance are defined to be equal at the present time. At other times, the Universe's expansion results in the proper distance changing, while the comoving distance remains constant.

==Comoving coordinates==
[[File:Expansion_of_the_universe,_comoving_coordinates_(Animation).gif|thumb|left|upright=1.6|alt=comoving coordinates|The evolution of the universe and its horizons in comoving distances. The x-axis is distance, in billions of light years; the left-hand y-axis is time, in billions of years since the Big Bang; the right-hand y-axis is the scale factor. This model of the universe includes dark energy which causes an accelerating expansion after a certain point in time, and results in an [[event horizon]] beyond which we can never see.]]

Although general relativity allows the formulation of the laws of physics using arbitrary coordinates, some coordinate choices are more natural or easier to work with. Comoving coordinates are an example of such a natural coordinate choice. They assign constant spatial coordinate values to observers who perceive the universe as [[Isotropy|isotropic]]. Such observers are called "comoving" observers because they move along with the [[Hubble's law|Hubble flow]].
<!-- was: a scientist to [[Formula|formulate]] the laws of physics using an arbitrary system of [[coordinates]], a scientist's job can be simplified by using different coordinate systems that are easy to work with. -->

A comoving observer is the only observer who will perceive the universe, including the [[Cosmic microwave background|cosmic microwave background radiation]], to be isotropic. Non-comoving observers will see regions of the sky systematically [[blue-shift]]ed or [[red-shift]]ed. Thus isotropy, particularly isotropy of the cosmic microwave background radiation, defines a special local [[frame of reference]] called the [[Proper frame|comoving frame]]. The velocity of an observer relative to the local comoving frame is called the [[peculiar velocity]] of the observer.

Most large lumps of matter, such as galaxies, are nearly comoving, so that their peculiar velocities (owing to gravitational attraction) are small compared to their Hubble-flow velocity seen by observers in moderately nearby galaxies, (i.e. as seen from galaxies just outside the [[galaxy group|group]] local to the observed "lump of matter").

[[File:Cosmos-animation Lambda-CDM.gif|thumb|right|upright=1.6|alt=comoving coordinates|Comoving coordinates separate the exactly proportional expansion in a Friedmannian universe in spatial comoving coordinates from the scale factor <math>a(t)~.</math> This example is for the &Lambda;CDM model.]]
The '''comoving time''' coordinate is the elapsed time since the [[Big Bang]] according to a clock of a comoving observer and is a measure of [[Cosmic time|cosmological time]]. The comoving spatial coordinates tell where an event occurs while cosmological time tells when an event occurs. Together, they form a complete [[coordinate system]], giving both the location and time of an event.

Space in comoving coordinates is usually referred to as being "static", as most bodies on the scale of galaxies or larger are approximately comoving, and comoving bodies have static, unchanging comoving coordinates. So for a given pair of comoving galaxies, while the proper distance between them would have been smaller in the past and will become larger in the future due to the expansion of space, the comoving distance between them remains ''constant'' at all times.

The expanding Universe has an increasing [[Scale factor (cosmology)|scale factor]] which explains how constant comoving distances are reconciled with proper distances that increase with time.

==Comoving distance and proper distance==
Comoving distance is the distance between two points measured along a path defined at the present [[cosmological time]]. For objects moving with the Hubble flow, it is deemed to remain constant in time. The comoving distance from an observer to a distant object (e.g. galaxy) can be computed by the following formula (derived using the [[Friedmann–Lemaître–Robertson–Walker metric]]):
<math display="block"> \chi = \int_{t_e}^t c \; \frac{\mathrm{d} t'}{a(t')} </math>
where ''a''(''t''&prime;) is the [[Scale factor (cosmology)|scale factor]], ''t''<sub>e</sub> is the time of emission of the photons detected by the observer, ''t'' is the present time, and ''c'' is the [[speed of light]] in vacuum.

Despite being an [[time integral|integral over time]], this expression gives the correct distance that would be measured by a hypothetical tape measure at fixed time ''t'', i.e. the "proper distance" (as defined below) after accounting for the time-dependent ''comoving speed of light'' via the inverse scale factor term <math>1 / a(t')</math> in the integrand. By "comoving speed of light", we mean the velocity of light ''through'' comoving coordinates [<math>c / a(t')</math>] which is time-dependent even though ''locally'', at any point along the [[null geodesic]] of the light particles, an observer in an inertial frame always measures the speed of light as <math>c</math> in accordance with special relativity. For a derivation see "Appendix A: Standard general relativistic definitions of expansion and horizons" from Davis & Lineweaver 2004.<ref name="D&L_EC">{{cite journal |author=Davis |first1=T. M. |last2=Lineweaver |first2=C. H. |date=2004 |title=Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe |journal=Publications of the Astronomical Society of Australia |volume=21 |issue=1 |pages=97–109 |arxiv=astro-ph/0310808v2 |bibcode=2004PASA...21...97D |doi=10.1071/AS03040 |s2cid=13068122}}</ref> In particular, see ''eqs''. 16–22 in the referenced 2004 paper [note: in that paper the scale factor <math>R(t')</math> is defined as a quantity with the dimension of distance while the radial coordinate <math>\chi </math> is dimensionless.]

===Definitions===
Many textbooks use the symbol <math>\chi</math> for the comoving distance. However, this <math>\chi</math> must be distinguished from the coordinate distance <math>r</math> in the commonly used comoving coordinate system for a [[Friedmann–Lemaître–Robertson–Walker metric|FLRW universe]] where the metric takes the form (in reduced-circumference polar coordinates, which only works half-way around a spherical universe):
<math display="block">ds^2 = -c^2 \, d\tau^2 = -c^2 \, dt^2 + a(t)^2 \left( \frac{dr^2}{1 - \kappa r^2} + r^2 \left(d\theta^2 + \sin^2 \theta \, d\phi^2 \right)\right).</math>

In this case the comoving coordinate distance <math>r</math> is related to <math>\chi</math> by:<ref>{{cite book
 |title=Introduction to Cosmology
 |edition=4th
 |first1=Matts
 |last1=Roos
 |publisher=[[John Wiley & Sons]]
 |year=2015
 |isbn=978-1-118-92329-0
 |page=37
 |url=https://books.google.com/books?id=RkgZBwAAQBAJ}} [https://books.google.com/books?id=RkgZBwAAQBAJ&pg=PA37 Extract of page 37 (see equation 2.39)]</ref><ref>{{cite book
 |title=Measuring the Universe: The Cosmological Distance Ladder
 |edition=illustrated
 |first1=Stephen
 |last1=Webb
 |publisher=[[Springer Science & Business Media]]
 |year=1999
 |isbn=978-1-85233-106-1
 |page=263
 |url=https://books.google.com/books?id=ntZwxttZF-sC}} [https://books.google.com/books?id=ntZwxttZF-sC&pg=PA263 Extract of page 263]</ref><ref>{{cite book
 |title=The Cosmological Background Radiation
 |edition=illustrated
 |first1=Marc
 |last1=Lachièze-Rey
 |first2=Edgard
 |last2=Gunzig
 |publisher=[[Cambridge University Press]]
 |year=1999
 |isbn=978-0-521-57437-2
 |pages=9–12
 |url=https://books.google.com/books?id=3LO75VmI9BMC}} [https://books.google.com/books?id=3LO75VmI9BMC&pg=PA11 Extract of page 11]</ref>
<math display="block">\chi = \begin{cases}
|\kappa|^{-1/2}\sinh^{-1} \sqrt{|\kappa|} r , & \text{if } \kappa<0 \ \text{(a negatively curved ‘hyperbolic’ universe)} \\
r, & \text{if } \kappa=0 \ \text{(a spatially flat universe)}  \\
|\kappa|^{-1/2}\sin^{-1}  \sqrt{|\kappa|} r , & \text{if } \kappa>0 \ \text{(a positively curved ‘spherical’ universe)}
\end{cases}</math>

Most textbooks and research papers define the comoving distance between comoving observers to be a fixed unchanging quantity independent of time, while calling the dynamic, changing distance between them "proper distance". On this usage, comoving and proper distances are numerically equal at the current age of the universe, but will differ in the past and in the future; if the comoving distance to a galaxy is denoted <math>\chi</math>, the proper distance <math>d(t)</math> at an arbitrary time <math>t</math> is simply given by 
<math display="block">d(t) = a(t) \chi</math> 
where <math>a(t)</math> is the scale factor (e.g. Davis & Lineweaver 2004).<ref name=D&L_EC/> The proper distance <math>d(t)</math> between two galaxies at time ''t'' is just the distance that would be measured by rulers between them at that time.<ref>{{Cite arXiv |last=Hogg |first=David W. |date=1999-05-11 |title=Distance measures in cosmology |page=4 |eprint=astro-ph/9905116 |language=en}}</ref>

===Uses of the proper distance===
[[File:Expansion of the universe, proper distances (Animation).gif|thumb|left|alt=proper distances|The evolution of the universe and its horizons in proper distances. The x-axis is distance, in billions of light years; the left-hand y-axis is time, in billions of years since the Big Bang; the right-hand y-axis is the scale factor. This is the same model as in the earlier figure, with dark energy and an event horizon.]]

Cosmological time is identical to locally measured time for an observer at a fixed comoving spatial position, that is, in the local [[comoving frame]]. Proper distance is also equal to the locally measured distance in the comoving frame for nearby objects. To measure the proper distance between two distant objects, one imagines that one has many comoving observers in a straight line between the two objects, so that all of the observers are close to each other, and form a chain between the two distant objects. All of these observers must have the same cosmological time. Each observer measures their distance to the nearest observer in the chain, and the length of the chain, the sum of distances between nearby observers, is the total proper distance.<ref>Steven Weinberg, ''Gravitation and Cosmology'' (1972), p. 415</ref>

It is important to the definition of both comoving distance and proper distance in the cosmological sense (as opposed to [[proper length]] in [[special relativity]]) that all observers have the same cosmological age. For instance, if one measured the distance along a straight line or [[Spacetime#Spacetime in general relativity|spacelike]] [[geodesic]] between the two points, observers situated between the two points would have different cosmological ages when the geodesic path crossed their own [[world line]]s, so in calculating the distance along this geodesic one would not be correctly measuring comoving distance or cosmological proper distance. Comoving and proper distances are not the same concept of distance as the concept of distance in special relativity. This can be seen by considering the hypothetical case of a universe empty of mass, where both sorts of distance can be measured. When the density of mass in the [[Friedmann–Lemaître–Robertson–Walker metric|FLRW metric]] is set to zero (an empty '[[Milne universe]]'), then the cosmological coordinate system used to write this metric becomes a non-inertial coordinate system in the [[Minkowski space]]time of special relativity where surfaces of constant Minkowski proper-time τ appear as [[hyperbola]]s in the [[Minkowski diagram]] from the perspective of an [[inertial frame of reference]].<ref>See the diagram on [https://books.google.com/books?id=1TXO7GmwZFgC&pg=PA28 p. 28] of ''Physical Foundations of Cosmology'' by V. F. Mukhanov, along with the accompanying discussion.</ref> In this case, for two events which are simultaneous according to the cosmological time coordinate, the value of the cosmological proper distance is not equal to the value of the proper length between these same events,<ref>{{cite web |author=Wright |first=E. L. |date=2009 |title=Homogeneity and Isotropy |url=http://www.astro.ucla.edu/~wright/cosmo_02.htm |access-date=28 February 2015}}</ref> which would just be the distance along a straight line between the events in a Minkowski diagram (and a straight line is a [[geodesic]] in flat Minkowski spacetime), or the coordinate distance between the events in the inertial frame where they are [[relativity of simultaneity|simultaneous]].

If one divides a change in proper distance by the interval of cosmological time where the change was measured (or takes the [[derivative]] of proper distance with respect to cosmological time) and calls this a "velocity", then the resulting "velocities" of galaxies or quasars can be above the speed of light, ''c''. Such superluminal expansion is not in conflict with special or general relativity nor the definitions used in [[physical cosmology]]. Even light itself does not have a "velocity" of ''c'' in this sense; the total velocity of any object can be expressed as the sum <math>v_\text{tot} = v_\text{rec} + v_\text{pec}</math> where <math>v_\text{rec}</math> is the recession velocity due to the expansion of the universe (the velocity given by [[Hubble's law]]) and <math>v_\text{pec}</math> is the "peculiar velocity" measured by local observers (with <math>v_\text{rec} = \dot{a}(t) \chi(t)</math> and <math>v_\text{pec} = a(t) \dot{\chi}(t)</math>, the dots indicating a first [[derivative]]), so for light <math>v_\text{pec}</math> is equal to ''c'' (−''c'' if the light is emitted towards our position at the origin and +''c'' if emitted away from us) but the total velocity <math>v_\text{tot}</math> is generally different from&nbsp;''c''.<ref name=D&L_EC/> Even in special relativity the coordinate speed of light is only guaranteed to be ''c'' in an [[inertial frame of reference|inertial frame]]; in a non-inertial frame the coordinate speed may be different from ''c''.<ref>{{cite book |author=Petkov |first=Vesselin |url=https://books.google.com/books?id=AzfFo6A94WEC&pg=PA219 |title=Relativity and the Nature of Spacetime |publisher=Springer Science & Business Media |year=2009 |isbn=978-3-642-01962-3 |page=219}}</ref> In general relativity no coordinate system on a large region of curved spacetime is "inertial", but in the local neighborhood of any point in curved spacetime we can define a "local inertial frame" in which the local speed of light is ''c''<ref>{{cite book |last1=Raine |first1=Derek |url=https://books.google.com/books?id=RK8qDGKSTPwC&pg=PA94 |title=An Introduction to the Science of Cosmology |last2=Thomas |first2=E. G. |publisher=CRC Press |year=2001 |isbn=978-0-7503-0405-4 |page=94}}</ref> and in which massive objects such as stars and galaxies always have a local speed smaller than ''c''. The cosmological definitions used to define the velocities of distant objects are coordinate-dependent – there is no general coordinate-independent definition of velocity between distant objects in general relativity.<ref>{{cite web| author=J. Baez and E. Bunn| title=Preliminaries| url=http://math.ucr.edu/home/baez/einstein/node2.html| publisher=University of California| date=2006| access-date=28 February 2015}}</ref> How best to describe and popularize that expansion of the universe is (or at least was) very likely proceeding &ndash; at the greatest scale &ndash; at above the speed of light, has caused a minor amount of controversy. One viewpoint is presented in Davis and Lineweaver, 2004.<ref name=D&L_EC/>

===Short distances vs. long distances===
Within small distances and short trips, the expansion of the universe during the trip can be ignored. This is because the travel time between any two points for a non-relativistic moving particle will just be the proper distance (that is, the comoving distance measured using the scale factor of the universe at the time of the trip rather than the scale factor "now") between those points divided by the velocity of the particle. If the particle is moving at a relativistic velocity, the usual relativistic corrections for time dilation must be made.

==See also==
* [[Distance measure]] for comparison with other distance measures.
* [[Expansion of the universe]]
* {{slink|Faster-than-light#Universal expansion}}, for the apparent faster-than-light movement of distant galaxies.
* [[Friedmann–Lemaître–Robertson–Walker metric]]
* [[Proper length]]
* [[Redshift]], for the link between comoving distance to redshift.
* [[Shape of the universe]]

==References==
{{Reflist|colwidth=30em}}

==Further reading==
*''Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity''. [[Steven Weinberg]]. Publisher:[[Wiley-VCH]] (July 1972). {{ISBN|0-471-92567-5}}.
*''Principles of Physical Cosmology''. [[Jim Peebles|P. J. E. Peebles]]. Publisher:[[Princeton University Press]] (1993). {{ISBN|978-0-691-01933-8}}.

==External links==
*[https://arxiv.org/abs/astro-ph/9905116 Distance measures in cosmology]
*[http://www.astro.ucla.edu/~wright/cosmo_01.htm Ned Wright's cosmology tutorial]
*[http://icosmos.co.uk/ iCosmos: Cosmology Calculator (With Graph Generation )]
*[https://arxiv.org/abs/astro-ph/9603028 General method, including locally inhomogeneous case] and [[Fortran 77]] software
*[http://www.atlasoftheuniverse.com/redshift.html An explanation from the Atlas of the Universe website of distance].
{{Portal bar|Physics|Stars|Spaceflight|Outer space|Solar System|Science}}

[[Category:Physical cosmology]]
[[Category:Coordinate charts in general relativity]]
[[Category:Physical quantities]]