5. COMOVING DISTANCE (TRANSVERSE)

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:

(15)

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

(16)

(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).

Figure 1. The dimensionless proper motion distance DM / DH. The three curves are for the three world models, Einstein-de Sitter (M, ) = (1, 0), solid; low-density, (0.05, 0), dotted; and high lambda, (0.2, 0.8), dashed.

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.