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

**2.1. The Necessity for Space Curvature**

Is space curvature real? As an experimental problem, it becomes an
epistemological question because of ambiguities in the definitions
concerning the nature of the measuring rods and the character of the
distances obtained with them
(Section 2.2). As a theoretical problem,
the *reality* of the formalism in the present physics (Einstein's
theory of gravity) must be sought.

The non-Euclidean geometry, foreshadowed by Saccheri
^{(3)}
and invented
by Gauss, Bolyai, and Lobachevski, was largely a curiosity for most
scientists in the mid-nineteenth century, despite its central
importance in this century, lying at the root of our present
understanding of space-time. Unlike Saccheri, Gauss believed in its
reality and proposed methods to measure the spatial
curvature. K. Schwarzschild began such measurements by putting limits
on the value of the curvature using the distribution of stellar
parallaxes.

The intuitive geometry that is fixed on the senses by that outside
spatial frame which gives us our ordinary experience seems Euclidean.
Areas increase strictly as *r*^{2}, volumes as
*r*^{3}, using the apparently
common-Sense definition of *r*. The concept of spatial curvature is
foreign to the intuition and unreal to the nonscientist.

Nevertheless, if we take the structure of general relativity as
*defining* reality, matter really does curve space. Particles move on
straight lines in curved space instead of on curved paths in straight
space. To be sure, we trade one mystery for another. The
*g*_{ij}'s of the
geometrical metric are determined by the distribution of matter,
replacing Newton's force at a distance with geodesics in curved space.
It is in this sense that general relativity has geometrized dynamics.
The question remains, Is the curvature "real?" But what is reality?
Indeed, has the question any verifiable meaning?

As an arguable definition, we could try "*for X to be real requires
that X have effects.*" ^{(4)}
If we observe unmistakable effects we would say
the thing "causing them" is real. It was the absence of predicted
effects that removed the ether from reality. It was the verification
of many predictions of its consequences that made the Lorentz
transformation "real." Yet the Fitzgerald contraction as one
"explanation" of the transformation is not real in this sense, but the
time dilatation debatably is
(Kennedy & Thorndike
1932),
because
*it* is observed, making the relativity of space-time equally real
as long as no other explanation is possible.

On this definition space-time curvature is real. The predictions of
its *effects* via Einstein's equations are well verified [see
Will (1981) and
Backer & Hellings (1986)
for recent reviews]. The curvature
is measured by the non-Euclidean *g*_{ij}'s. Yet areas and
volumes are *not*
measured. What actually is verified is that the formalism of the
equations *works* in certain experimental circumstances (advance of
Mercury's perihelion, time dilatation, a ray bending about the Sun,
gravitational radiation, and perhaps even gravitational lensing).

However, the presence of space curvature would be more convincing if
we had a simple direct proof that volumes fail to increase as
*r*^{3}, or
that the angular sizes of rods fail to decrease as *r*^{-1} in
circumstances where the Riemann-Gauss scalar curvature,
*kc*^{2}/*R*^{2}, is
expected to be nonzero. The full problem of defining relevant
distances in the cosmology of ideal (congruent) spaces then becomes
the central point in deciding the reality of space curvature.

^{3} The grip
that our intuition holds on the mind concerning the
unreality of non-Euclidean geometry prevented Saccheri from believing
what his reason had discovered. E.T. Bell, in his book *Development of
Mathematics*, writes, "[Saccheri's] brilliant failure is one of the
most remarkable instances in the history of mathematical thought of
the mental inertia induced by an education in obedience and orthodoxy,
confirmed in mature life by an excessive reverence for the perishable
works of the immortal dead [Euclid]. With two geometries, each as
valid as Euclid's in his hand, Saccheri threw both away because he was
willfully determined to continue in the obstinate worship of his idol,
despite the insistent promptings of his own sane reason."
Back.

^{4} This is similar to but not identical
with a wider definition often
used that "X is real if it is an essential element of a strongly
confirmed theory." However, with both these definitions, a *reality* of
this kind is ephemeral. If the theory is later found to be inadequate
and must be replaced, the "reality" associated with it must also be
replaced, and hence was not real in the ordinary usage of that word.
Back.