The comoving distance between two events at the same redshift or distance but separated on the sky by some angle is D_{M} and the transverse comoving distance D_{M} (so-denoted for a reason explained below) is simply related to the line-of-sight comoving distance D_{C}:
where the trigonometric functions sinh and sin account for what is called ``the curvature of space.'' (Space curvature depends on the particular coordinate system chosen, so it is not intrinsic; a change of coordinates makes space flat; the only intrinsic curvature is space-time curvature, which is related to the local mass-energy density or really stress-energy tensor.) For _{} = 0, there is an analytic solution to the equations
(Weinberg 1972, p. 485; Peebles 1993, pp. 320-321). Some (eg., Weedman 1986, pp. 59-60) call this distance measure ``proper distance,'' which, though common usage, is bad style. ^{(1)}
(Although these notes follow the Peebles derivation, there is a qualitatively distinct method using what is known as the development angle , which increases as the Universe evolves. This method is preferred by relativists such as Misner, Thorne & Wheeler 1973, pp. 782-785).
The comoving distance happens to be equivalent to the proper
motion distance (hence the name D_{M}), defined as
the ratio of the
actual transverse velocity (in distance over time) of an object to its
proper motion (in radians per unit time)
(Weinberg 1972,
pp. 423-424). The proper motion distance is plotted in
Figure 1. Proper motion distance is used, for example,
in computing radio jet velocities from knot motion.