When Alfred North Whitehead published The Principle of Relativity in 1922 as an alternative to Einstein’s 1905 and 1916 expositions, it immediately suffered the same ignoble stillbirth of Hume’s first Treatise. Unlike Hume’s work, however, no subsequent resurrection of Whitehead’s book or theory occurred. The reasons for this appear to be quite understandable. Whitehead’s mathematical treatment of special relativity doesn’t differ from Einstein’s, and where it does with respect to general relativity, the available evidence hasn’t been kind to Whitehead. Perhaps most significantly, however, is that the conceptual base for this theoretical stew—Whitehead’s concept of the relativity of simultaneity—has a distinctly Newtonian flavor. The enthusiasm among philosophers and physicists for this recipe might best be compared to that engorged Thanksgivers exhibit for warmed-over turkey casserole.

It is surprising, however, that there has been little explicit analysis of why Whiteheadian relativistic simultaneity should be found inferior to Einstein’s. Further, of what analysis and criticism on this topic that has appeared, much is seriously defective. To help clear the air, I propose to examine an egregiously overlooked little paper by Whitehead in which he resolves the famous twin paradox from the perspective of his own theory of relativity. I believe that much can be discovered about what is wrong with Whiteheadian relativity by examining precisely why in this paper his treatment of the paradox is flawed. Additionally, a necessary by-product of this examination will be some critical commentary on some of the more persistent disputes about the nature and resolution of the paradox itself.

The twin paradox is notorious to those with even a passing acquaintance with relativity. It springs directly from Einstein’s 1905 Annalen der Physik paper, and though it was originally known as "the clock paradox", in 1911 Paul Langevin personified it by replacing the wayfaring clock with a human being making a long round trip at fantastic speed. The substitution of biologically identical twins for incipiently synchronous clocks by later commentators finally lent it its most popular name and most controversial form.

The mature version goes something like this. Starting with identical twins on earth, one twin is shipped off on a space journey at a velocity only fractionally less that of light. Upon returning, the astronaut twin finds that she is younger than her earthbound sibling. The final age differential may be more or less, depending on the total distance travelled by the wayfaring twin. In morbid versions she returns after an extended journey to find that her sister has died of old age.

It is this counterintuitive difference in aging that originally was termed "paradoxical". However, given the rather spare and reasonable assumptions required by special relativity—the constancy for all inertial observers of light velocity in a vacuum and the preservation of physical laws for all inertial frames—and given that the paradox is resolvable by the application of special relativity alone, the tale is logically self-consistent and thus its final result is only surprising, not truly paradoxical. Whether other, deeper paradox resides in the tale will be discussed in due course.

The occasion for Whitehead’s essay on the paradox was an Aristotelian Society symposium convened in 1923 and expansively entitled "The Problem of Simultaneity: Is There a Paradox in the Principle of Relativity in Regard to the Relation of Time Measured to Time Lived?" (Hereafter this will be cited by page number only) The other symposiasts were H. Wildon Carr and R. A. Sampson. Since the text of Whitehead’s paper reveals that he was installed in the role of commentator, it will be necessary to establish exactly what the context of his response was.

It is painfully apparent that Whitehead regarded Sampson’s contribution to be nearly worthless. Throughout his paper Whitehead’s only references are to remarks made by Carr, except for his deft and cordial opening apology that " in concentrating on one or two points I must not be presumed to undervalue the importance of or interest in the remainder" (34). However, even a cursory reading of Sampson’s paper reveals that Whitehead was justified in his snub. Sampson, a mathematician, conflated issues (particularly the use of a scale factor in comparing measurements between distinct inertial observers with time dilation as it physically affects the reunited twins) so that he ended up dispelling all paradox by concluding that both twins age at the same universal rate.

Both Carr and Whitehead concur, on the other hand, that the narrative of the tale does entail that differential aging occurs (17; 36). Ultimately, however, Whitehead does not accept Carr’s theoretical explanation of why this should be so. As in the case with Sampson, one may tend toward some sympathy with Whitehead’s reaction. Though the precise details cannot concern us here, suffice it to say that Carr was not the "orthodox relativist" that Whitehead at one point claims him to be (38). However, it is clear that Whitehead directs his criticism not so much at Carr’s idiosyncratic analysis of the paradox, but at what he takes to be the orthodox, Einsteinian account of it.

The paradox is presented by Carr as follows [author’s warning: though neither Whitehead nor Carr actually discuss the paradox in terms of twin brothers, I will construe their remarks as if their masculine references were to male twins]:

Carr cites Langevin, Bergson, and Berquerel as his sources of this paradox, though curiously he does not mention its origin in section I.4 of Einstein’s 1905 paper or the physicist’s 1918 discussion of it from the viewpoint of general relativity (17). Also, Carr does not actually make it clear whether his analysis of the twin narrative is based purely on tenets of special relativity or includes those of general relativity, though one receives the impression that only a treatment by the more elementary account is intended. Further, neither Carr nor Whitehead present any actual calculations. However, providing some latitude for estimations, slight miscalculations and/or misprints, the figures they use do jibe with the mathematics of special relativity.

IV. Whitehead’s Resolution of the Paradox

Whitehead’s resolution of the paradox, which mirrors many contemporary accounts, is as follows. (35-36)

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Whitehead’s explanation of the asymmetry is simple, elegant, and, at least overall if not in specific detail, quite correct. The change of motion of the one twin at the star does effectively render the spacetime length H1H2 irrelevant to any objective comparison of the twins’ proper times. As will be apparent later, however, it’s not at all clear that this pictorial resolution of the paradox is compatible with the conceptual interpretation Whitehead puts upon it.

V. Whitehead’s Criticisms of the Paradox

After the diplomatic preliminaries averred to earlier in the opening of his paper, Whitehead immediately attacks Carr’s resolution of the paradox as incomplete or arbitrary:

This point recurs throughout discussion of the paradox over the decades, and has led most recent commentators, like Whitehead, to declare that it is in virtue of this seeming absurdity—that both twins have a right to say that the other experiences time-dilation—that the narrative is paradoxical. Obviously such an interpretation holds either that special relativity contains some internal inconsistency, or that special relativity, while perhaps consistent and correct in application to restricted situations in nature, must be augmented to resolve the story of the twins. As it will become clear, Whitehead wishes to level a bit of both charges at an Einsteinian resolution of the puzzle.

Clearly Whitehead is disenchanted with what he perceives to be the pure relativity of motion which the standard theory attributes to the twins:

Is Whitehead correct in his assertion that the standard Einsteinian account of the paradox—which we may assume encompasses only special relativity theory (SR)--cannot explain the asymmetrical aging of the twins? The basis for his complaint is that the standard view appears to admit that only purely relative motion exists in the adventures of the twins, thus providing no grounds for concluding that an asymmetrical age difference must obtain for the reunited twins.

One component of Whitehead’s criticism here is fair; another, unfair. It is fair to inquire whether SR provides the conceptual wherewithal to ground any claim of asymmetry for the twins which explains the aging diference. It is unfair to insinuate that the relativistic equivalence of inertial frames under SR itself undermines asymmetry, which the above passage seems to do. Clearly that latter insinuation is, if genuine on Whitehead’s part, too strong, for it would clearly conflict with Whitehead’s own need for such equivalence in his own system of relativity. Therefore, we need only address the fair complaint in Whitehead’s remarks.

The standard reply generally given to this complaint is that whereas the stay-at-home twin A occupies only one inertial frame during the relevant time interval, the traveller B does not. Hence, between the events of departure and return B’s worldline has a shorter proper time because it is not a geodesic—that is, it is not the shortest spacetime path between those events. Due to the affine and metrical structure of the SR manifold, such a geodesic necessarily has the longest proper time between its endpoint events over any other possible worldline path between them. Note that this explanation suffices to account for the asymmetry given that (i) A and B’s motions occur as specified in Carr’s narrative, (ii) the causal-temporal sequence of events—the departure, the turnaround, the reuniting—can be specified unambiguously on SR spacetime so that the unique geodesic of A alone obtains, and (iii) the properties of SR spacetime are sufficient to produce the required symmetrical dilation of time.

Certainly W yields (i). Though he doesn’t use more contemporary terminology, he probably would yield (iii) as well, though in conceptual terms of his own version of relativity, which includes a quasi-absolute simultaneity of events relativized to inertial frames (more on this later). His disagreement cited above seems to be over the standard SR account of (ii).

Usually standard SR accounts of (ii) often depend upon empirical grounds for distinguishing A and B in their tracks through spacetime. That is, A and B can be equipped with accelerometers, and it can be established that B experiences accelerations, whereas A doesn’t. Since the noninertial motion can thus be unambiguously assigned to B (along with other reasonable assumptions about spacetime, e.g., temporal anistrophy across inertial frames), the direct consequence of this is that the events of departure, reunification and those between obtain for A in a single inertial frame. Therefore A’s worldline between those events is a geodesic.

Though this account satisfies (ii), it does not resolve all questions about the adequacy of standard SR to solve the twin paradox in terms of spacetime structure alone, for a version of the puzzle can be constructed which dispenses with empirical reference to accelerations altogether.

In this triplets version, a brother X remains on earth while sisters Y and Z zip away on near-lightspeed journeys. However, after Y has achieved his cruising speed (relative to X), he passes by earth so that he can synchronize clocks with X. Then, much later and farther along, Y crosses paths with sister Z, without either decelerating, and they note one another’s times. Z then continues home to meet X. Of course, the narrative is constructed so that at the points of clock-synchronization and during the travel between encounters, no relative accelerations are involved (no pun intended!). However, the path-integral equations of SR still sum the proper times of Y’s and Z’s travels so that the combined time registered by their clocks is less than the time recorded by X from when he last saw Y’s clock to when he sees Z’s (and by the same discrepancy predicted for the twins, neglecting spacetime-path differences due to accelerations in the twin example). Therefore, relative accelerations between X, Y, and Z appear to play no essential part in accounting for the time dilation of Y and Z’s proper-time clocks as compared to X’s. This in fact is a very useful thought-experiment for showing that GR—which is needed for understanding the physical equivalence of gravitational and accelerated systems—is not necessary for explaining the paradox either with the triplets or the twins.

The essential steps in this argument are:

Obviously, given the validity of the argument, only the truth of the premises is at issue. Which might be questionable? (iii) is a prime candidate, for it might seem to contain ineliminable reference to some sort of empirical claim, implicitly or explicitly. And, it appears that (ii) depends upon the means of determining (iii). The truth of (i) is in part stipulative, and bears much of the weight of any claim against the relevance of forces to the triplets’ case. (iv) is the least controversial, from the viewpoint of the coherence of the premises. Given some reasonable ground for the applicability of SR to the triplets, its claim is intratheoretically analytic. Aside from (i) then, (iii) is the crucial premise, for it supplies the crucial point about asymmetry.

Granted that (1) we may not distinguish between the adventures of X, Y, and Z in an empirical way by reference to accelerations and inertial forces and (2) that the conceptual apparatus of SR includes well-known topological and metrical properties of a flat Minkowski manifold, there are nonetheless purely logical distinctions that preserve the assertability of (iii). Consider the three events of the meetings of X-Y, Y-Z, and X-Z. By hypothesis triplet X’s worldline contains the first and third of these, but not the second. The combined worldlines of Y and Z, however, contain all three. Since the occurrence of event Y-Z falls within (or nearly on) the future light-cone of X-Y, and the occurrence of X-Z falls within (or nearly on) the future light-cones of Y-Z and X-Y, then Y-Z occurs temporally between events X-Y and X-Z for both worldline X and the combined worldline of Y and Z. Now recalling that by SR (and the example’s stipulation) events such as X-Y, Y-Z, and X-Z represent relative rectilinear motions involving distinct inertial frames, then between X-Y and X-Z the Yand Z combined worldline contains a frame difference not involved in the worldline of X between X-Y and X-Z. Parsimony thus requires that of these two worldlines between X-Y and X-Z, only X’s may qualify for being a single inertial frame.

This logical analysis reveals purely geometrical reasons for discriminating the worldline of X as opposed to Y and Z in combination with only minimal assumptions about temporal anisotropy and time sequence. Therefore SR need not rely on empirical references to accelerations to produce the required asymmetry, and the tenability of (iii) above stands.

While the previous section indicates that Whitehead’s broadest and most general charge against the consistency of application of SR is not cogent, there are at least two other criticisms that Whitehead levies at SR which are more revisionist and reconstructionist in their outlook. It is thus in these latter remarks that Whitehead begins at once to dismantle certain conceptual parts of SR and replace them with ones of his own, and in the process sketching out his own theory of relativity.

In the process of working toward his revisions, however, Whitehead surprisingly appears to dismiss any attempt to analyze the paradox by means of transmitted (and Doppler-shifted) time signals:

Since Whitehead does not dispute the final asymmetry between the twins ("But we are all agreed that the traveller will have counted 730 days (two years)." (37)), one may be left quite puzzled by these remarks. Surely Whitehead is not concurring with Sampson, who in his contribution says something similar in concluding that the traveller "receives. . .on his outward journey, only one year’s news from earth, so that he will only have enjoyed one additional birthday. . .but he will make up for it on returning by enjoying one hundred and ninety-nine. . ." (32). Sampson’s reasoning, of course confuses the question of the ordinal reception of such signals with the question of whether the comparative proper time intervals of transmission and reception are equivalent (of course, they are not). Since Whitehead clearly agrees that the comparative proper time intervals of the twins are not equivalent, we must charitably assume that Whitehead believes that "there is no disagreement" on the mere ordinality of any such signals received in a one-way transmission from earth to the traveller.

Thus disposing of what can only be described as a quirky distraction to Whitehead’s train of thought, he turns to his central objection to SR’s resolution of the paradox. That is, the proper time intervals of both twins must be comparable throughout their separation in some fundamental and meaningful way for the asymmetry found at the reunification to exist, and it is Whitehead’s deep conviction that SR cannot supply a ground for such comparison. In brief, he realizes that, although the stay-at-home twin must have experienced more days (as a composite time-interval) than the star-faring sibling, any pair of days taken from both world-lines themselves must be of equal length--congruent--in some sense.

Further, Whitehead argues that the standard SR account of the congruence of time intervals is arbitrary or meaningless:

His own remedy for this ill is by way of an objective, non-operationalist, but thoroughly relativized conception of simultaneity:

Although Whitehead does not elaborate here on the manner in which this concept of simultaneity secures the congruence of comparable nonlocal physical events, he does so elsewhere. For clarification, we should turn briefly to those sources--primarily The Principle of Relativity (hereafter R).

First, it is quite clear that for Whitehead simultaneity is an inertial frame-dependent but quasi-Newtonian spatiotemporal relation.

Note that Whitehead distinguishes between physical relations, which participate in our scheme of knowledge as real but contingent adjectival relatedness, and spatiotemporal relations, which are real, uniform, but noncontingent (necessary?), nonadjectival relations of significance.

Still, one gets the impression that such spatiotemporal relations, distinct and (apparently) more ontologically fundamental than contingent adjectives, are something reminiscent of Locke’s substantival je ne ce qua: they are the superstructure of all reality, but the contingent adjectives (objects) they uphold do not affect these relations (note the similarity with Newtonian absolute space). Though Whitehead insists that his spacetime geometry isn’t a priori, the hints that it is abound.

How does Whitehead secure his concept of congruence within this scheme? Though he holds that the geometrical signature of congruence is found by means of the opposite side of parallelograms in spacetime geometry, the ultimate foundation is in the matching of events:

So simultaneity is a relation that ties together all events which occupy the same moment within a time-system, and it is the relations of all the various time-systems that produces the relation of space for any given time-system at a given moment within that system (R 333). Hence the relation of simultaneity is extensionally equivalent (as concerns the events related and only in a sense of set-membership) to the Newtonian-like spatial spread of a moment within a given time-system. Since Whitehead assumes on Occamist grounds that a flat spacetime exists in the universe, then his spacetime diagrams, such as used above to explicate his resolution of the twin paradox, employ simultaneity lines in much the same way as they usually function metrically on Minkowski diagrams: they are spacelike extensions of a given moment which are orthogonal to the time-axis of an event’s inertial frame. This implies that for Whitehead’s diagrams a simultaneity-line from one inertial frame intersects another inertial frame’s time-axis at a metrical point which is related to the former time-axis by the usual Lorentz contraction factor.

It seems that it is this relatedness of distinct time-systems by simultaneity is Whitehead’s basis for claiming that two events of these systems are congruent: "My own belief is that the congruence of time-lapses expresses an important relation of time-lapses founded upon the intrinsic character of timc exhibited in the fundamental fact of simultaneity." ( 39-40 ) On reflection, however, such a claim for the foundation of congruence is murky at best; incoherent at worst. Consider a pair of events (not event-points) in the usual twin scenario, say, the time-interval of one minute of proper time on the two twins’ worldlines which are (in some way) relatable by simultaneity lines. Surely Whitehead isn’t claiming that two such minutes are durationally equal as related by simultaneity lines, for (i) there is no point-to-point equality of spacetime points within the durations (measured as proper times) which can be connected by simultaneity lines because (ii) the Lorentz contraction assures that the twins cannot measure each other’s time-intervals to be of equal length! (Recall that this situation is symmetrically possible because the Lorentz factor involves observations of nonlocal events.) Therefore the relation of two such events by simultaneity lines does not yield a metric congruence between them. Yet this is certainly what is suggested by Whitehead’s references to simultaneity.

Therefore it is not apparent that frame-dependent Newtonian simultaneity can secure the sort of nonlocal congruence that Whitehead desires. Is SR better off here? Apparently so--because foundational to SR formalism is the postulate that light always if measured as a constant c in any inertial frame. Such a postulate is made consistent with Galilean relativity (the other postulate of SR) only by assuming that physical processes are metrically uniform--i.e., congruent as events, which is what Whitehead himself requires--locally and universally. While this assumption is the very one which makes Whitehead uncomfortable enough to charge SR with circularity, clearly it is a more consistent position conceptually than an a prioristic spacetime structure whose complex simultaneity relations don’t yield the congruence promised!

Finally, Whitehead is ready to close in for the kill.

Here Whitehead appears to make one or perhaps two complaints about "orthodox" relativity. The general claim is that relativity--apparently whether SR or GR--cannot make sense of the acceleration of the traveling twin. The second may be specifically directed against GR’s principle of equivalence, which maintains the empirical indistinguishability of gravitation and acceleration. Whitehead’s idea is simple enough--if acceleration is an absolute way to distinguish the twins’ experiences and produce an asymmetry between them, and if relativity cannot do that, then those explanations are inadequate.

If Whitehead means to say that SR is conceptually incapable of handling accounts of accelerated entities at all, then he is certainly mistaken. Any such account need only translate statements about accelerations into statements about successive infinitesimal inertial frames, and go the SR route from there. What SR cannot do, however, is to account for accelerations in a way that yields insight into how such phenomena (second derivative velocity displacements) are related to gravity and spacetime, which is the very reason GR was developed.

GR, using the principle of equivalence, asserts that gravitation and acceleration are locally indistinguishable via transformations. It does not erode the logical or empirical ground distinguishing inertial and noninertial worldlines, however. Therefore, contra Whitehead, we may distinguish between the twins' worldlines in GR. The question of whether accelerations are only accountable with a substantival spacetime, such as Whitehead seems to require, remains open however.

As one might have expected, Whitehead’s treatment of the twin paradox is a partial success, but, as an occasion to fault SR and GR, a resounding failure. Though his spacetime-diagram resolution is adequate in the sense that it is entirely compatible with any usual account of the paradox in SR-Minkowski four-space, Whitehead’s conceptual interpretation of simultaneity not only fails to supplant Einstein’s, but in its failure reinforces the subtle brilliance of his adversary’s account.

Why does Whitehead finally fail? His reliance on the fundamental nature of congruence as dependent upon a neo-Newtonian simultaneity lends an important clue: Whitehead cannot free himself of treating spatiotemporal existence in Euclidean conceptual terms (despite the fact that Minkowski four-space has a non-Euclidean metric!). That underlying prejudice is why the incongruence of nonlocal events related by simultaneity lines escapes his notice, at least in The Principle of Relativity. In later works this and other inconsistencies drove him to revise some of his central concepts--including simultaneity--so that they largely became more compatible with standard SR accounts. His unshakable commitment to the doctrine of uniform significance, however, could never allow him to abandon his dogmatic insistence on the smooth--and uncurved--nature of spacetime itself.