Conclusion

I have made no more progress in the general theory of relativity. The electric field still remains unconnected. Overdeterminism does not work. Nor have I produced anything for the electron problem. Does the reason have to do with my hardening brain mass, or is the redeeming idea really so far away?

                                                                                                Einstein to Ehrenfest, 1920

Despite the spectacular success of the theory of relativity, it is sometimes said that tests of Bell's inequalities and similar quantum phenomena have demonstrated that nature is, in some sense, incompatible with the local realism on which relativity is based. However, as we’ve seen, Bell's inequalities apply only to a strictly non-deterministic theory containing "free choice", and only under the assumption of strong temporal asymmetry. Bell himself noted that the observed quantum phenomena are not incompatible with "local realism" for a fully deterministic theory, nor for a temporally symmetrical theory. The entire framework of classical relativity, with its unified spacetime and partial ordering of events, is founded on a strictly deterministic and temporally symmetrical basis, so the relevance of Bell's inequalities is dubious at best. Furthermore, since no energy or information is propagated faster than light, even when Bell’s inequalities are violated, quantum mechanics actually provides a striking confirmation of the principles on which Einstein chose to base special relativity. Admittedly the phenomena of quantum mechanics are incompatible with at least some aspect of the intuitive metrical idea of locality, but this should not be surprising, because (as discussed in the preceding sections) the metrical idea of locality is already inconsistent with the pseudo-metrical structure of spacetime, which forms the basis of modern relativity.

While modern relativity initiated a revolution in our thinking about the (pseudo-Riemannian) metrical structure of spacetime, with its null rays and non-transitive locality, the concomitant revolution in our thinking about the local topology of spacetime has lagged behind. We've learned that the measurably invariant intervals between the events of spacetime cannot be accurately represented as the distances between the points of a Euclidean metric space, but we continue to assume that the local topology of the set of spacetime events is nevertheless the topology of a Euclidean metric space. This incongruous state of affairs may be due in part to the historical circumstance that Einstein's special relativity was originally viewed as simply an elegant interpretation of Lorentz's ether theory. According to Lorentz, spacetime really is a Euclidean manifold with the metric and topology of E⁴, on top of which is superimposed a set of functions representing apparent temporal and spatial measures arising from distortions in our measuring devices. From this point of view, the fact that all the effects of relativity can be represented as purely kinematic features of a Minkowski metric is just a mathematical oddity of no fundamental significance. In this context the continued use of the E⁴ topology is understandable, especially if we suspect (as Lorentz did) that Lorentz invariance breaks down at some point, so that the "singularities" represented by null intervals of Minkowski spacetime are not realized. However, if we accept perfect Lorentz invariance, as is suggested both by all the empirical evidence and by the success of quantum field theory (of which Lorentz invariance is a fundamental ingredient), then the E⁴ topology is arguably no longer the most useful representation of spacetime. Einstein said in 1920 that the "ether" had finally been deprived of its last mechanical property when we recognized that a state of motion could not be attributed to it, i.e., that its parts could not be tracked through time. However, one could argue that the last vestige of a mechanical property is actually the presumed Euclidean topology, which continues to be accepted to the present day, despite its incongruity with the pseudo-Riemannian metric of spacetime.

Some authors have claimed that general relativity is not actually a generalization of special relativity. Admittedly general relativity teaches us that the principles of special relativity are strictly applicable only over infinitesimal regions (in the presence of gravitation), so in this sense the general theory restricts rather than generalizes the special theory. However, it’s more meaningful to regard special relativity as the theory of flat four-dimensional spacetime, characterized by the Minkowski metric (in suitable coordinates), and the general theory then generalizes this by allowing the spacetime manifold to be curved, as represented by a wider class of metric tensors. It’s remarkable that this generalization, which is so simple and natural from the geometrical standpoint, leads almost uniquely to a viable theory of gravitation. Echoing Minkowski’s comment about ‘staircase wit’, it’s tempting to claim that if we hadn’t already known about the existence of gravity from our direct experience, some fancy-free mathematician might have predicted it, based purely on abstract considerations of the class of intelligible spatio-temporal metrical relations. (Interestingly, a similar claim has been made about modern “string theory”, i.e., that it “predicts” the existence of gravitation.)

Just as the special theory is sometimes interpreted in the Lorentzian sense, an analogous situation exists with regard to the general theory, which is sometimes interpreted as a field theory rather than as a geometrical theory. Those who hold this view usually discount or downplay the equivalence principle, and regard the fact that gravitation can be represented by curvature of spacetime as merely a mathematical oddity of no fundamental significance. One of the motivations for this approach is the desire to make the theory of gravitation fit into the same “linear” mold as the theories of the other fundamental forces (albeit with non-linear corrections to match the field equations), usually with the hope that this might help us to see how the theory could be quantized like the other theories. From considerations of the spectrum of gravitational waves in equilibrium inside a container, analogous to Planck’s analysis of blackbody electromagnetic radiation, one would expect that the gravitational field must be quantized, and yet no convincing quantization of the non-linear field equations of general relativity has emerged. This may not be so surprising when we consider that a quantum of gravitational radiation would also have to be a source of gravitation – analogous to a photon with electric charge. Only a few physicists (including Einstein himself) have tried to approach the problem from the opposite direction, taking the non-linear geometrical basis of general relativity as the starting point, and seeking some further generalization or restriction that would yield quantum phenomena.

Of course, it's entirely possible that the theory of relativity is simply wrong on some fundamental level where quantum mechanics "takes over". In fact, this is probably the majority view among physicists today, who hope that eventually a theory uniting gravity and quantum mechanics will be found which will explain precisely how and in what circumstances the classical theory of relativity fails to accurately represent the operations of nature, while at the same time explaining why it seems to work as well as it does. However, it may be worthwhile to remember previous periods in the history of physics when the principle of relativity was judged to be fundamentally inadequate to account for observed phenomena. Recall Ptolemy's arguments against a moving Earth, or the 19th century belief that electromagnetism necessitated a luminiferous ether, or the early-20th century view that special relativity could never be reconciled with gravity. In each case a truly satisfactory resolution of the difficulties was eventually achieved, not by discarding relativity, but by re-interpreting and extending it, thereby gaining a fuller understanding of its logical content and consequences.

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