4.7 The Inertia of Twins

The most commonly discussed "paradox" associated with the theory of relativity concerns the differing lapses of proper time along two different paths between two fixed events. This is often expressed in terms of a pair of twins, one moving inertially from event A to event B, and the other moving inertially from event A to an intermediate event M, where he changes his state of motion, and then moves inertially from M to B, where it is found that the total elapsed time of the first twin exceeds that of the second. Much of the popular confusion over this sequence of events is simply due to specious reasoning. For example, if x,t and x',t' denote inertial rest frame coordinates respectively of the first and second twin (on either the outbound or inbound leg of his journey), some people are confused by the elementary fact that if those two coordinate systems are related according to the Lorentz transformation, then the partials (�t'/�t)_x and (�t/�t')_x' both equal 1/(1-v²)^1/2, where v is the relative velocity between the two inertial frames. (For example, the unfortunate Herbert Dingle spent his retirement years on a pitiful crusade to convince the scientific community that those two partial derivatives must be the reciprocals of each other, and that therefore special relativity is logically inconsistent.) Other people struggle with the equally elementary algebraic fact that the proper time along any given path between two events is invariant under arbitrary Lorentz transformations. (The inability to grasp this has actually led some eccentrics to waste years in a futile effort to prove special relativity inconsistent by finding a Lorentz transformation that does not leave the proper time along some path invariant.)

Despite the obvious fallacies underlying these popular confusions, and despite the manifest logical consistency of special relativity, it is nevertheless true that the so-called twins paradox, interpreted in a more profound sense, does highlight a fundamental epistemological shortcoming of the principle of inertia, on which both Newtonian mechanics and special relativity are based. Naturally if we simply stipulate that one of the twins is in inertial motion the entire time and the other is not, then the resolution of the "paradox" is trivial, but the stipulation of "inertial motion" for one of the twins begs the very question that motivates the paradox (in its more profound form), namely, how are inertial worldlines distinguished from the set of all possible worldlines? In a sense, the only answer special relativity can give is that the inertial worldline between two events is the one with the greatest lapse of proper time, which is clearly of no help in resolving which of the twins' worldlines is "inertial", because we don't know a priori which twin has the greater lapse of proper time - that's what we're trying to determine!

This circularity in the definition of inertia and the inability to justify the privileged position which inertial worldlines hold in special relativity were among the problems that led Einstein in the years following 1905 to seek a broader and more coherent context for the laws of physics. The same kind of circular reasoning arises whenever we critically examine the concept of inertia. For example, when trying to decide if our region of spacetime is really flat, so that "straight lines" exist, we face the same difficulty. As Einstein said:

We could equally well substitute [has the greatest lapse of proper time] for [is sufficiently far from other bodies]. In either case the point is the same: special relativity postulates the existence of inertial frames and assigns to them a preferred role, but it gives no a priori way of establishing the correct mapping between this concept and anything in reality. This is what Einstein was referring to when he said "In classical mechanics, and no less in the special theory of relativity, there is an inherent epistemological defect...". He illustrates this with a famous thought experiment involving two relatively spinning globes, discussed in Chapter 4.1. (The term "thought experiment" might be regarded as an oxymoron, since the epistemological significance of an experiment is its empirical quality, which a thought experiment obviously doesn't possess. Nevertheless, it's undeniable that scientists have made good use of this technique - along with occasionally making bad use of it.) The puzzling asymmetry of the spinning globes is essentially just another form of the twins paradox, where the twins separate and reconverge (one accelerates away and back while the other remains stationary), and they end up with asymmetric lapses of proper time. How can the asymmetry be explained? According to Einstein

This is as true of the twins paradox as it is of the two-globe paradox. Zahar notes that "Einstein had ... two reasons for giving up special relativity as a suitable framework for the whole of physics: first, philosophical dissatisfaction with having given a privileged status to the set of inertial frames; second the...impossibility of reconciling E = mc² with gravitation." It is the first of these reasons that bears most directly on the twins paradox, although the problem of reconciling acceleration with gravity inevitably enters the picture as well, since we can't avoid the issue of gravitation if acceleration is allowed - assuming we accept the equivalence principle. (It's ironic that although special relativity strongly suggests the equivalence of mass and energy, it turns out to be incompatible with it.) In retrospect it's clear that special relativity could never have been more than a transitional theory, since it was not comprehensive enough to justify its own conclusions.

The question of whether general relativity is required to resolve the twins paradox has long been a subject of spirited debate. On one hand, Max Born concluded his detailed examination of this topic with the statement that "the clock paradox is due to a false application of the special theory of relativity, namely, to a case in which the methods of the general theory should be applied". On the other hand, many people object vigorously to any suggestion that special relativity is inadequate to satisfactorily resolve the twins paradox. Ultimately the answer depends on what sort of satisfaction is being sought, viz., on whether the paradox is being presented as a challenge to the consistency of special relativity (as is Dingle's fallacy) or to the completeness of special relativity. If we're willing to accept uncritically the existence and identifiability of inertial frames, and their preferred status, and if we are willing to exclude any consideration of gravity or the equivalence principle, then we can reduce the twins paradox to a trivial exercise in special relativity. However, if it is the completeness (rather than the consistency) of special relativity that is at issue, then the naive acceptance of inertial frames is precisely what is being challenged. Also, in this context, we can hardly justify the exclusion of gravitation, considering that the very same metrical field which determines the inertial worldlines also represents the gravitational field.

Notice that the typical statement of the twins paradox does not stipulate how the galaxies in the universe are moving relative to the twins. If every galaxy in the universe (other than our own) were moving in tandem with the "travelling twin", which (if either) of the twins' reference frames would be considered inertial? Obviously special relativity is silent on this point, and even GR does not give an unequivocal answer. Weinberg asserts that "inertial frames are determined by the mean cosmic gravitational field, which is in turn determined by the mean mass density of the stars", but the second clause is not necessarily true, because the field equations require some additional information (such as boundary conditions) in order to yield any definite results. The existence of cosmological models in which the average matter of the universe rotates (a fact proven by Kurt Goedel) shows that even general relativity is incomplete, in the sense that it is subject to global conditions with considerable freedom. General relativity may not even give a unique field for a given (non-spherically symmetric) set of boundary conditions and mass distribution (perhaps not surprising in view of the possibility of geons - massive objects constructed entirely of gravitation). Thus even if we sharpen the statement of the twins paradox to specify how the twins are moving relative to the rest of the matter in the universe, the theory of relativity still doesn't enable us to say for sure which twin is inertial.

Furthermore, once we recognize that the inertial and gravitational field are one and the same, the twins paradox becomes even more acute, because we must then acknowledge that within the theory of relativity it's possible to contrive a situation in which two identical clocks in identical local circumstances (i.e., without comparing their positions to any external reference) can nevertheless exhibit different lapses in proper time between two given events. The simplest example is to place the twins in intersecting orbits, one circular and the other highly elliptical. Each twin is in freefall continuously between their periodic meetings, and yet they experience different lapses of proper time. Thus the difference between the twins is not a consequence of local effects; it is a global effect. At any point along those two geodesic paths the local physics is identical, but the paths are embedded differently within the global manifold, and it is the different embedding within the manifold that accounts for the difference in proper length. This more general form of the twins paradox compels us to abandon the view that physical phenomena are governed solely by locally sensible influences. (Notice, however, that we are forced to this conclusion not by logical contradiction, but only by our philosophical devotion to the principle of sufficient cause, which requires us to assign like physical causes to like physical effects.)

It is fundamentally misguided to exercise such epistemological concerns within the framework of special relativity, because special relativity was always a provisional theory with recognized epistemological short-comings. As mentioned above, one of Einstein's two main two reasons for abandoning special relativity as a suitable framework for physics was the fact that, no less than Newtonian mechanics, special relativity is based on the unjustified and epistemologically problematical assumption of a preferred class of reference frames, precisely the issue raised by the twins paradox. Today the "special theory" exists only (aside from its historical importance) as a convenient set of widely applicable formulas for important limiting cases of the general theory, but the phenomenological justification for those formulas can only be found in the general theory.

This is true even if we posit the absence of gravitational effects, because the question at issue is essentially the origin of inertia, i.e., why one worldline is inertial while another is not, and the answer unavoidably involves the origin and significance of the background metric, even in the absence of curvature. The special theory never claimed, and was never intended, to address such questions. The general theory attempts to provide a coherent framework within which to answer such questions, but it's not clear whether the attempt is successful. The only context in which general relativity can give (at least arguably) a complete explanation of inertia is a closed, finite, unbounded cosmology, but the observational evidence doesn't (at present) clearly support this hypothesis, and any alternative cosmology requires some principle(s) outside of general relativity to determine the metrical configuration of the universe.

Thus the twins paradox is ultimately about the origin and significance of inertia, and the existence of a definite metrical structure with a preferred class of worldlines (geodesics). In the general theory of relativity, spacetime is not simply the totality of all the relations between material objects. The spacetime metric field is endowed with its own ontological existence, as is clear from the fact that gravity itself is a source of gravity. In a sense, the non-linearity of general relativity is an expression of the ontological existence of spacetime itself. In this context it's not possible to draw the classical distinction between relational and absolute entities, because spatio-temporal relations themselves are active elements of the theory.

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