The Principle of Relativity 6

4. The Comparison between Whitehead's and Einstein's Theories of Relativity
(From the viewpoint of empirical tests)

Eddington first took notice of Whitehead's theory. He proved in his 1924 paper that Whitehead's equation has the same solution (the Schwartzchild solution) as Einstein's in the special case of the stationary gravitational field due to a single mass-point.16 The implication of this equivalence was that Whitehead's theory can pass the standard tests such as periherion precession of Mercury and the bending of light-rays close to the sun. The similar result was obtained by Temple, who also gave a generalized version of Whitehead's theory which holds in the space-time with constant curvature.⁽¹⁷⁾ In the 1920s, the comparison of two theories was mainly on the level of conceptual analysis, for both gave the same results under the limited conditions, and it was difficult to choose between them on experimental grounds then available. The question at issue was the justifiability of Whitehead's theory which, presupposing Minkowski's space globally, rejected the general principle of relativity. For example, Band criticized Whitehead by pointing out that the acceptance of a uniform or "fiat" space was untenable on account of the illegitimate assumption of a standard of absolutely uniform motion.18 But the problem of finding the exact solutions of both theory other than Schwartzchild's was so difficult that the crucial experiment between them was not yet contrived.

In the 1930s and 1940s the main interests of physicists shifted to the realm of quantum mechanics and nuclear physics which developed without relying on any gravitational theory. Here physicists were satisfied only with the special theory of relativity, and kept away from Einstein's later project of relativistic cosmology and the unified theory of fields. Whitehead was regarded as a metaphysician, and his theory of relativity seemed to be virtually ignored during this period. The re-evaluation of Whitehead's theory began in the 1950s, which was due to an Irish physicist, Synge, who esteemed Whitehead's theory for its elegance and originality, and located it between Newton's theory of action-at-a-distance, and Einstein's theory of local action. Setting aside Whitehead's philosophical background, Synge reconstructed mathematical formulae of Whitehead's theory in Einstein's terminology to make them accessible to contemporary physicists. Synge also treated the problem of a continuous static model, and calculated the gravitational field of a finite sphere of uniform density at rest on the basis of Whitehead's theory.⁽¹⁹⁾ Two years later, this result was extended by Rayner to the case of non-static continuous distributions of matter. Calculating the gravitational field of a finite, uniformly rotating, homogeneous sphere, Rayner examined the perturbing effects of the rotation of the central sphere on the orbits of planetary motion, and got similar results to those obtained by Lens and Thirring applying Einstein's theory to the same problem. Rayner also constructed a cosmological model uniformly expanding with homogeneity and isotropy on the basis of Whitehead's theory.⁽²⁰⁾

Whereas Synge and Rayner proved that Whitehead's theory, in spite of the paradigm-difference, had the same conclusions as Einstein's in various applications, Clark for the first time took up the problem of establishing a crucial experiment between the two. Having discussed on the two-body problem, Clark proved that Whitehead's theory of gravitation involves a secular acceleration of the center of mass, and suggested that Whitehead's theory might be refutable by observing the motions of the centers of mass of double stars.⁽²¹⁾ The same problem was also discussed by Schild, who showed that Whitehead's theory can be modified in such a way that linear and angular momentum are rigorously conserved, and the center of mass of any isolated system has no secular acceleration. Schild added an interesting remark that Levi-Civita, using Einstein's theory of general relativity, obtained a similar secular acceleration, but that this was later proved to be in an erroneous calculation.⁽²²⁾

In the 1960s, the confrontation between gravitational theories and experiments again became a matter of concern for physicists. The rapid progress of technology and astronomy made it possible to test various gravitational theories at an unprecedented levels of accuracy. The number of theories in need of testing having increased, the desire to sift them out systematically was intensified. Pioneered by Dicke and Nordtvedt, the various meta-theoretical frameworks of gravitational theories were propounded. Concerning the principle of equivalence on which Einstein founded the general theory of relativity, we must mention the results of redshift experiments in 1965 on the earth by the use of the Moesbauer effect (recoilless emission and absorption of photons). The accuracy of that observation was about twenty times higher than those previously obtained by astronomical observations. This proved to be a strong support for Einstein who had considered the gravitational redshift one of the most important tests of general relativity. Moreover, it is thought by many physicists today that the result of the gravitational redshift proves the so-called Schiff's conjecture that any theory of gravitation must necessarily be a metric theory.

Inspired by Dicke's ideas, Will energetically grappled with the problem of testing in the 1970s, and presented five criteria by which we can eliminate those theories that disagree with experiment. He laid out the "Parametrized Post-Newtonian" framework (PPN) as a meta-theory in which nine metric parameters. varying from theory to theory, made it possible for him to render the various theories of gravitation commensurable. As for Whitehead's theory, Will admitted it was an elegant theory that had been "a thorn in Einstein's side", but claimed that he had now succeeded in refuting it by geophysical effects, the fifth criterion which he had invented.⁽²³⁾

According to Will, Whitehead's theory involves a small anisotropy in the gravitational constant G measured by Cavendish experiments on the earth. As the earth rotates, the anisotropy in G produces "Earth tides", i.e. variations in the acceleration g measured by the gravimeter, which are completely analogous to the tides produced by the moon and the sun. Making use of a simplified model of the galaxy, Will calculated the amplitude of the earth tides on the basis of Whitehead's theory, and got the value, which proved to be 200 times larger than the experimental limit. So he concluded that "Whitehead's theory, after 50 years of life. was "killed" by the geophysical data.⁽²⁴⁾

Will's argument, though accepted by many physicists today as valid, was not without objections concerning the process he used to calculate and estimate the predicted value of earth tides. Fowler claimed that he could reduce the value by a factor of 100 under a different model of the galaxy, and thereby diminish the discrepancy between Whitehead's theory and the geophysical data. Remembering that Whitehead was prepared to adjust his own formulae to take account of new data, Fowler concluded that:

"The real issues between Einstein and Whitehead are not physical but philosophical. No empirical test can decide the issue of the adequacy of Whitehead's basic theory of relativity. This issue must be settled on other grounds."⁽²⁵⁾

The Principle of Relativity 7

5. The Redshift Experiment and Whitehead's Theory

In the previous section we have summarized chronologically the various opinions of physicists concerning the experimental tests of Whitehead's theory as compared with Einstein's. Taking these accounts into considerations, I will try to make clear the meanings of new experimental situations available today so that Whitehead's theory may be reexamined and modified within this context.

First, we should bear in mind the fact that Whitehead's theory contains two levels of arguments. One is propounded as a physical hypothesis, open to the future refutation, and the other is his philosophy of nature which is the guiding principle of his physical theories. For example, his basic equation of gravitation such as



is a refutable hypothesis which Whitehead was able to abandon without altering his background philosophy. In a similar way we can consider it as a hypothesis, and not an inescapable result of his philosophy, to state that "the gravitational forces are propagated along straight lines in Mincowski's space, whereas electromagnetic waves are deflected by the contingencies of the universe," though it is a natural interpretation of Whitehead's formulae. So in one sense it was understandable that Fowler would conclude his remark on Will's alleged refutation by stressing the paradigm-difference.

The present author, however, believes that it is not productive, and is even sterile, to insist too much on the paradigm-priority over observed data. Even today it is logically possible that we believe in the Ptolemaic theory by postulating peripheral hypotheses in order to explain the planetary motion. But physics needs more than logical consistency. We had better consider the refutability of a physical theory on its merit. We always learn something at the time of refutation of our pet-theories.

Take, for example, the thesis that space-time must be "fiat". This thesis was the guiding principle of Whitehead's formulae for gravitation. He admitted openly that he was very willingly to believe that each permanent space is either uniformly elliptic or uniformly hyperbolic, "if any observations are more simply explained by such a hypothesis." But the postulate that the curvature of space-time must be constant was thought by Whitehead to be essential to any satisfactory theory of space-time. It was not a hypothesis, but one of the fundamental principles of Whitehead's theory. I will try to show that the very postulate that the metric structure of space-time must be uniform should be abandoned if we want to learn seriously from experiments which are available today, but were unknown to Whitehead.

We will confine the discussion to the effects of redshift experiments on Whitehead's theory. This does not mean that the problem situations raised by Clark and Will may well be ignored. On the contrary, they should be considered as very important contributions even if there remain some ambiguities concerning their results. Lengthy discussions and mathematical technicalities are involved, if we are to grapple with the problern of earth tides or of conservation laws. Moreover if we abolish the thesis of a uniform metric, we need not insist, as Whitehead did, on the global inertial system, which was responsible for the earth tides in Will's criticism. So it is justifiable first to discuss the problem of the metric structure of space-time.

Einstein, as was pointed out in the first section of this paper, stressed the importance of the gravitational redshifts so much that he dared to say that he would abandon the general theory of relativity if it was not observed. The result, however, of astronomical observations by Freundlich (1930) and others were not satisfactory because of an inaccuracy of measurement. This was one of the reasons why many physicists thought the experimental evidence for the general theory of relativity was not convincing. The situation, however, has changed since 1965, because the aforementioned experiment by the use of the Moesbauer effect gave strong support to Einstein's prediction Since this result is interpreted as verifying the principle of equivalence, some kind of reformulation of Whitehead's theory is necessitated because Whitehead did not accept the principle of equivalence in his original formulations.

Whitehead's theory in its original version (1922), using a simplified model of a radiating atom obtained a gravitational redshift slightly different from that of Einstein's theory by the factor of 7/6. Whitehead also predicted that the values of redshift would depend on the directions of emitted light (Limb Effect), which, at least qualitatively, corresponded to the data of astronomical observation. So it seemed as if the predictive power of Whitehead's theory were equivalent to Einstein's concerning the redshift phenomena, but in fact such an equivalence does not hold. First we must notice the difference of physical interpretations when both theories derive the gravitational redshift. Whereas Whitehead's theory needs additional hypotheses concerning the structure of atomic clock and the nature of interaction between a gravitational field and other force fields, Einstein's theory does not need such auxiliary hypotheses, because the latter postulated that the gravitational fields should directly influence the metric of space-time. One of the most important results of this difference is that whereas Einstein's theory should predict the uniformity of gravitational redshift independently of physical conditions, Whitehead's theory can not expect such a uniformity. This mean that Einstein's theory has a stronger structure than Whitehead's because it runs the risk of being refuted by possible varieties of gravitational redshift. But since Einstein's prediction of redshift has been corroborated by empirical tests since 1965, it gains the advantage of Whitehead's theory because of its completeness.