Annu. Rev. Astron. Astrophys. 1988. 26: 561-630
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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 r2, volumes as r3, 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 gij'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 gij'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 r3, or that the angular sizes of rods fail to decrease as r-1 in circumstances where the Riemann-Gauss scalar curvature, kc2/R2, 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.

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