Comoving and proper distances

Many textbooks use the symbol ${\displaystyle \chi }$ for the comoving distance. However, this ${\displaystyle \chi }$ must be distinguished from the coordinate distance ${\displaystyle r}$ in the commonly used comoving coordinate system for a FLRW universe where the metric takes the form (in reduced-circumference polar coordinates, which only works half-way around a spherical universe):

${\displaystyle ds^{2}=-c^{2}\,d\tau ^{2}=-c^{2}\,dt^{2}+a(t)^{2}\left({\frac {dr^{2}}{1-\kappa r^{2}}}+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right)\right).}$

In this case the comoving coordinate distance ${\displaystyle r}$ is related to ${\displaystyle \chi }$ by:[3][4][5]

${\displaystyle \chi ={\begin{cases}|\kappa |^{-1/2}\sinh ^{-1}{\sqrt {|\kappa |}}r,&{\text{if }}\kappa <0\ {\text{(a negatively curved ‘hyperbolic’ universe)}}\\r,&{\text{if }}\kappa =0\ {\text{(a spatially flat universe)}}\\|\kappa |^{-1/2}\sin ^{-1}{\sqrt {|\kappa |}}r,&{\text{if }}\kappa >0\ {\text{(a positively curved ‘spherical’ universe)}}\end{cases}}}$

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 ${\displaystyle \chi }$, the proper distance ${\displaystyle d(t)}$ at an arbitrary time ${\displaystyle t}$ is simply given by

${\displaystyle d(t)=a(t)\chi }$

where ${\displaystyle a(t)}$ is the scale factor (e.g. Davis & Lineweaver 2004).[2] The proper distance ${\displaystyle d(t)}$ between two galaxies at time t is just the distance that would be measured by rulers between them at that time.[6]

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.[7]

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 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 lines, 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 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 spacetime of special relativity where surfaces of constant Minkowski proper-time τ appear as hyperbolas in the Minkowski diagram from the perspective of an inertial frame of reference.[8] 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,[9] 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 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 ${\displaystyle v_{\text{tot}}=v_{\text{rec}}+v_{\text{pec}}}$ where ${\displaystyle v_{\text{rec}}}$ is the recession velocity due to the expansion of the universe (the velocity given by Hubble's law) and ${\displaystyle v_{\text{pec}}}$ is the "peculiar velocity" measured by local observers (with ${\displaystyle v_{\text{rec}}={\dot {a}}(t)\chi (t)}$ and ${\displaystyle v_{\text{pec}}=a(t){\dot {\chi }}(t)}$, the dots indicating a first derivative), so for light ${\displaystyle v_{\text{pec}}}$ 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 ${\displaystyle v_{\text{tot}}}$ is generally different from c.[2] Even in special relativity the coordinate speed of light is only guaranteed to be c in an inertial frame; in a non-inertial frame the coordinate speed may be different from c.[10] 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[11] 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.[12] How best to describe and popularize that expansion of the universe is (or at least was) very likely proceeding – at the greatest scale – at above the speed of light, has caused a minor amount of controversy. One viewpoint is presented in Davis and Lineweaver, 2004.[2]

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.