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

**2.2. The Idea of Geometrical Experiments**

How then are we to measure deviations from Euclidean predictions? What
rules concerning the properties of measuring rods do we adopt, and by
what rules do we assess whether an experiment has given a
non-Euclidean result? Early on, Poincaré denied the reality of actual
curved space by stating that in any measurement that appeared to give
a non-Euclidean result, one is at liberty to redefine the properties
of the measuring rods in such a way as to recover a Euclidean
prediction. A particularly interesting example of this, involving
nonuniformly heated metal measuring rods, is given by
Robertson (1949).
Poincaré's point has been variously debated (cf.
Whittaker 1958,
Reichenbach 1958)
with the consensus opinion being that
contrived (unreasonable) explanations of changes in the measuring
rods, if they are required to save the Euclidean case, are less
desirable than a real Riemann-Lobachevski geometry. The debate then
changes to the meaning of *contrived and unreasonable*. Consider again
the Fitzgerald contraction of fast-moving measuring rods, and
ultimately the reality of the Lorentz transformation. The issue is now
resolved in
Einstein's (1905)
favor in that his deeper interpretation
of space-time is viewed as more *reasonable* than the Fitzgerald
explanation, which is now viewed as *contrived*.

In cosmology we are faced with similar problems. We cannot measure
distances by placing rigid rods end to end. Rather, operational
definitions of distance "by angular size," "by apparent luminosity,"
"by light travel time," or "by redshift" are perforce employed. Their
use then requires a theory that connects the observables (luminosity,
redshift, angular size) with the various notions of distances
(McVittie 1974).
One of the great initial surprises is that these
distances differ from one another at large redshift, yet all have
clear operational definitions. Which distance is "correct?" *All* are
correct, of course, each consistent with their definition. Clearly,
then, distance is a construct in the sense of
Margenau (1950),
operationally defined entirely by its method of measurement.

The best that astronomers can do is to connect the observables by a
theory and test predictions of that theory when the equations are
written in terms of the observables alone. To this end, the concept of
distance becomes of heuristic value only. It is simply an auxiliary
*parameter* that must drop from the final predictive equations.

But spatial curvature appears on a different footing. Although it
too cannot be directly measured without a covering theory of
"luminosity distance" or "redshift distance" to relate "volumes" to
"distance," the curvature does enter as a *primary* parameter (not
to be dropped from the equations) in the predictive relations between the
observables (luminosity, angular diameter, and redshift). The curvature is

(1) |

if = 0. The
parameter *q*_{0} (or
_{0}) enters
into all the equations
connecting redshift, luminosity, angular size, and number counts. In
this sense, the curvature is measurable and therefore is "real,"
because it has observable effects on the *m*(*z*),
(*z*), and
*N*(*z*) relations.

Direct experimental geometry is then a possibility, provided that we
are willing to accept the equations that connect the
*q*_{0} measure of
curvature with angles, areas, volumes, and redshifts - equations derived
from some adopted cosmology.