Editing Comoving and proper distances (section)

===Uses of the proper distance===
[[File:Spacetime-diagram-flat-universe-proper-coordinates.png|thumb|left|upright=1.2|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 y-axis is time, in billions of years since the Big Bang. 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/>