Thinking about the shape of the universe in the context of the standard Big Bang model is simplest using comoving coordinates.
While special relativity states that all inertial reference frames are equivalent, i.e. that there is no favoured set of space-time coordinates, this is only a local theory.
General relativity is also a local theory, but it is used to constrain the local properties of a Riemannian manifold, which itself is global.
In the context of general relativity, the assumption of Weyl's postulate is that a favoured reference frame in space-time can be decided. The most common notion of such coordinates is that of comoving coordinates, where the spatial reference frame is attached to the average positions of galaxies (or any large lumps of matter which are at most moving slowly).
With this set of coordinates, both time and expansion of the Universe can be ignored in order to concentrate on the shape of space (formally speaking, of a spatial hypersurface at constant cosmological time).
Space in comoving coordinates is (on average) static. This is perfectly consistent with the fact that the Universe is expanding. A choice of coordinates is just a choice of labels. There happens to be (according to the standard Big Bang model) a choice of these labels which can be used either for formal calculations or for intuition in which the Universe is static. To get back to thinking about an expanding Universe just requires remembering the scale factor.
This way there is also cosmological time, which for an observer at a fixed spatial point in comoving coordinates is identical to her local measurement of time.
Comoving distance is then the distance in comoving coordinates between two points in space, at a single cosmological time:
where is the scale factor.
equivalent names
Table of contents |
2 Other distances useful in cosmology 3 distances useful on small - galaxy or cluster of galaxies - scales 4 External references |
Comoving distance and cosmological time definitely exist as part of the standard Big Bang model.
However, while cosmological time is identical to locally measured time for an observer at a fixed comoving spatial position, the comoving distance is not, in the general case, identical to a distance as physically experienced by a particle moving slower than or at the speed of light.
If one divides a comoving distance by the present cosmological time (the age of the universe) and calls this a "velocity", then the resulting "velocities" of "galaxies" near the particle horizon or further than the horizon can be above the speed of light.
This is the paradox of the ambiguous phrase space expanding faster than the speed of light. An umambiguous rewording of the phrase can now be made:
for a "galaxy" towards or beyond the horizon, its "velocity", defined as comoving distance from the observer divided by the present cosmological time, can be greater than the speed of light
This is a correct statement. What is debatable is the philosophical interpretation.
Problems according to a strictly empirical point of view (according to which something hidden inside a box does not exist, cf. Bertrand Russell) include:
d_{a} = d_{pm} / (1+z) = d_{L} /(1+z)^{2}
where z is the redshift.
If and only if the curvature is zero, then proper motion distance and comoving distance are identical, i.e. .
For negative curvature,
,
while for positive curvature,
,
where is the (absolute value of the) radius of curvature.
A practical formula for numerically integrating to a redshift for arbitrary values of the matter density parameter , the cosmological constant , and the quintessence parameter is
By using inverse sin and sinh functions, proper distance can be obtained from .
The ordinary distance as experienced by particles travelling slower than or at the speed of light is simply the comoving distance multiplied by the value of the scale factor at the cosmological time studied.
Different names for this include