4.7  The Inertia of Twins

 

A complete clarification of the question raised by you can only be obtained if one forms a picture of the geometric-mechanical constitution of the universe as a whole that is compatible with the theory. This I have tried last year and have found – as it seems to me – a completely satisfying model; but going into it would lead too far.

                                                                                             Einstein, 1918

 

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 have the same value. (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 held by inertial worldlines 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. In the introduction of his 1916 review paper on general relativity he wrote

 

The weakness of the principle of inertia lies in this, that it involves an argument in a circle: a mass moves without acceleration if it is sufficiently far from other bodies; we know that it is sufficiently far from other bodies only by the fact that it moves without acceleration.

 

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...". In his popular book on relativity theory first published in December of 1916 Einstein included a chapter entitled “In what Respects are the Foundations of Classical Mechanics and of the Special Theory of Relativity Unsatisfactory?”, in which he wrote

 

Both in classical mechanics and in the special theory of relativity we differentiate between reference bodies relative to which the recognized “laws of nature” [i.e., the law of inertia] can be said to hold, and reference bodies relative to which these laws do not hold. But no person whose mode of thought is logical can rest satisfied with this condition of thing. He asks “How does it come that certain reference bodies (or their states of motion) are given priority over other reference bodies (or their states of motion)? What is the reason for this preference?

 

In the review paper he illustrates this conundrum 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 essential significance of an experiment is its empirical quality, which a thought experiment obviously doesn't possess. Nevertheless, it's undeniable that scientists have often made good use of this technique.) The puzzling asymmetry of the spinning globes is essentially just another form of the twins paradox, where the twins separate and re-converge (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? In 1916 Einstein thought that

 

The only satisfactory answer must be that the physical system consisting of S1 and S2 reveals within itself no imaginable cause to which the differing behavior of S1 and S2 can be referred. The cause must therefore lie outside the system. We have to take it that the general laws of motion...must be such that the mechanical behavior of S1 and S2 is partly conditioned, in quite essential respects, by distant masses which we have not included in the system under consideration.

 

It should be noted that the strongly Machian attitude conveyed by this passage was subsequently tempered for Einstein when he realized that in the general theory of relativity it may be necessary to attribute the "essential conditioning" to boundary conditions rather than distant masses. Nevertheless, this quotation serves to demonstrate how seriously Einstein took the question, which, of course, is as applicable to the twins paradox as it is to the two-globe paradox.

 

The above “weighty argument from the theory of knowledge” was the first reason cited by Einstein (in 1916) for the need to go beyond special relativity in order to arrive at a suitable conceptual framework. The second reason was the apparent impossibility of doing justice, within the context of special relativity, to the equivalence principle relating gravitation and acceleration. The first of these reasons 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 as soon as we contemplate acceleration - assuming we accept the equivalence principle. From these considerations 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. Einstein wrote a paper in 1918 to explain how the general theory accounts for the asymmetric aging of the twins by means of the “gravitational fields” that appear with respect to accelerated coordinates attached to the traveling twin, and Max Born recounted this analysis in a popular book, concluding 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". Likewise Wolfgang Pauli says of the paradox in his brilliant 1921 treatise on relativity, “Of course, a complete explanation of the problem can only be given within the framework of the general theory of relativity”, again referring to Einstein’s 1918 analysis. These statements are based on the understanding that non-vanishing Christoffel symbols represent a gravitational field, even if the intrinsic curvature in the region is zero. Einstein (and others) always maintained that the equivalence principle was the central insight of general relativity, and this principle in its purest form identifies acceleration of the coordinates with a homogeneous gravitational field. In opposition to this view, some have argued that only inhomogeneous metrical fields, i.e., fields with non-vanishing curvature, should be regarded as exhibiting “gravity”, and that there is no need to invoke general relativity in the absence of curvature.

 

No one disputes that it is permissible to analyze the twins in the context of general relativity. The question is whether it is necessary. 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 (ala 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. 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. As Einstein repeatedly emphasized in his writings, in general relativity the Minkowski metric of flat spacetime is not regarded as an a priori structure (as it is in special relativity), but as a solution of the gravitational field equations for a particularly simple set of boundary conditions. In this context, the question as to what determines the local metrical field (whether it is flat or curved) is unavoidable, which is why, as soon as the general theory was completed, Einstein was immediately led (in 1917) to consider the cosmological boundary conditions.

 

The typical statement of the twins paradox does not stipulate how the galaxies in the universe along with the cosmological boundary conditions that determine the metrical field are dynamically configured relative to the twins. If every galaxy in the universe were “moving” in tandem with the "traveling twin", which (if either) of the twins' reference frames would be considered inertial?  Obviously special relativity is silent on this point, and even general relativity 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 generally require some additional information (such as boundary conditions) in order to yield definite results. The existence of cosmological models in which the average matter of the universe rotates (a fact proven by Kurt Gödel) 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, which is not surprising in view of the possibility of gravitational waves. 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. (The same point can be made by referring to a flat cylindrical spacetime.) 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.) Likewise the identification of gravity with local spacetime curvature is untenable, as shown by the fact that a suitable arrangement of gravitating masses can produce an extended region of flat spacetime in which the metrical field is nevertheless accelerating in the global sense, and we surely would not regard such a region as free of gravitation.

 

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 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 epistemological foundation of those formulas must be sought in the context of the general theory. In this regard, Einstein remarked that

 

No fairer destiny could be allotted to any physical theory, than that it should of itself point out the way to the introduction of a more comprehensive theory, in which it lives on as a limiting case.

 

Notice that the general theory is operative even in flat spacetime, because, as noted above, all of spacetime (whether flat or curved) is to be regarded as a solution of the field equations, rather than as some a priori structure. 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.

 

We should also mention another common objection to the relativistic treatment of the twins, based not on any empirical disagreement, but on linguistic and metaphysical preferences. It is pointed out that we can, without logical contradiction, posit the existence of a unique, absolute, and true metaphysical time at every location, and we can account for the differences between the elapsed times on clocks that have followed different paths simply by stipulating that the rate of a clock depends on its absolute state of motion (defined relative to, for instance, the local frame in which the presumably global cosmic background radiation is maximally isotropic). Indeed this was essentially the view advocated by Lorentz. However, as discussed at the end of Section 1.5, postulating a metaphysical “truth” along with whatever physical laws are necessary to account for why the observed facts differ from the postulated “truth” is not generally useful, except as a way of artificially reconciling our experience with any particular metaphysical truth that we might select. The relativistic point of view is based on purely local concepts, such as that of an “ideal clock” corrected for all locally sensible conditions, recommended to us by the empirical fact that all observable aspects of local physical phenomena – including the rates of temporal progression – exhibit the same dependence on their state of inertial motion (which is not a locally sensible condition). This is the physical symmetry presented to us, and we are certainly justified in exploiting this symmetry to simplify and clarify the enunciation of physical laws.

 

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