Annu. Rev. Astron. Astrophys. 1988. 26:
561-630
Copyright © 1988 by . All rights reserved |

**2.4. The Volume V(r) in Robertson-Walker Spaces**

One supposes that the space that describes any real universe must be homogeneous and isotropic. Otherwise, the notion of extension as applied to material bodies would have a complicated meaning. By this is meant that any material body, transported to any region of the space, must be transformed into itself without tearing or buckling. Such spaces are congruent. They can be rotated into themselves by a coordinate transformation without shear. This is not true for nonhomogeneous, non-isotropic spaces.

Robertson (1929,
1935)
and Walker (1936)
verified that the most
general expression for the geometrical interval *dl*^{2}
between two points in a space of constant curvature with coordinates
*r*,,
, and *r* + *dr*,
+
*d*, and
+
*d* is

(7) |

where *k* is the sign of the space curvature (+1 for *k* >
0, 0 for *k* = 0, -1
for *k* < 0). Various coordinate transformations give a variety of
equivalent forms (e.g.
McVittie 1956,
1965,
Misner et al. 1973).
Equation 7 is particularly convenient in deriving the various
relations between the observable parameters and the geometry in the
standard model.

The *r*, ,
numbers in Equation 7 are
*comoving* coordinates. They are
fixed (constant) for all time for a given galaxy. They are also
dimensions. The factor *R*(*t*) is a scale factor (dimensions
of length) that is a function of time in an expanding or contracting
manifold. *R*(*t*) is independent of *r*,
, and
in a
congruent space (one of constant curvature).

The volume enclosed within the space from the origin (*r* = 0, put at
the observer) and the coordinate value *r* is

(8) |

For *k* = 1 this integrates to

(9) |

for *k* = - 1 to

(10) |

and for *k* = 0 to

(11) |

Note that *r* is not the *interval distance* from the origin
to the point *r*, ,
. The manifold distance in
the space described by Equation 7 is

(12) |

without loss of generality by rotating the coordinate system into the
=
= 0 plane. Hence, the
*coordinate* *r* in Equations 9-11 is given in
terms of the ratio of the measured distance *l* to the scale factor
*R*
(i.e. *l* /*R*), just as in Equations 2 and 4. (There should
be no confusion
about the changed definitions of *l* and *r* between Equations
2, 4, and 8-12). Explicitly,

(13) |

Substituting Equation 13 into Equations 9, 10, and 11, and expanding for clarity to appreciate the dependence on the curvature, gives

(14) |

By analogy with Equations 2 and 4, *kR*^{-2} is called the
curvature of the
space. Note that if *k* = 0, Equation 14 (i.e. Equation 11 also) gives
the Euclidean volume.

The series expansion in Equation 14 illustrates the principle of the
galaxy count-volume test. (In practice, of course, the exact equations
are used.) If the distances *l* to a sample of galaxies were known and
if a *complete* count of galaxies within this distance could be
obtained, the curvature *kR*^{-2} could be measured by the
excess (or
deficiency) of the counts from an *l*^{3} dependence. The
distance *l* and
hence the coordinate value of *r* (from Equation 13) are related to the
redshift via the standard theory, to be developed in the next sections.