Ray on the Twin Paradox
                         _______________________

                             V. Alan White

                       Department of Philosophy 
                 University of Wisconsin-Manitowoc 
                           awhite@uwc.edu

                    (received: August 2, 1997)


 
	Recently in a stimulating and perceptive book Christopher Ray 
discusses at some length the nature and resolution of the now infamous twin 
paradox ([3], p. 36-44). Although Ray does much good work to set aright many 
misconceptions about this subtle thought experiment, I do part with him on 
what exactly constitutes sufficiency of explanation in this and a related 
case.

	First,  a brief reprise of the paradox. Beginning on Earth, two 
biologically identical twins are separated by one twin's departure on a space 
journey during which the travelling twin achieves near-lightspeed velocity 
(relative to the stay-at-home twin). Upon returning, the wayfaring twin finds 
that she is younger than her earthbound sibling. The extent of the age 
differential depends on the combined factors of relative velocity achieved and 
distance traveled. Morbid versions have it that the astronaut twin returns 
after a high-speed journey of many light-years to find that her sister has 
died of old age.
 
 	Although many have contested the result of this thought experiment, 
there appears to be no reason to believe that such aging differences would not 
occur under these circumstances (see [1], pp. 180-192 and [2] for a rather 
comprehensive history). However, some discussants have argued about whether 
special relativity theory (SR) is sufficient to account for the twin effect, 
or whether general relativity (GR) is required to do so ([3], p. 24). Without 
invoking distracting detail, one main point of contention here has been over 
the role of the accelerations or inertial forces involved in the traveling 
twin's trip:  are they essential for the asymmetrical time-dilation to occur?
 
	A very ingenious revision of the standard twin scenario was concocted 
by defenders of the sufficiency of SR to make their case ([3], p. 41-44; [2], 
p. 113). In this triplets version, a brother X remains on earth while brothers 
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 brother 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 experienced by Y and Z.
 
	Ray concurs with this strategy and its apparent implications for the 
irrelevance of accelerations or inertial forces in accounting not only for the 
time dilation effect, but for the aging _asymmetry_ as well. Referring to the 
fact that GR is based on the local equivalence of inertial and gravitational 
forces, he chides those who would attempt to explain the twin paradox by 
invoking a necessary reference to those forces in terms of GR:
 
	[E]ven if we are tempted to move to GR to explain the 
	asymmetry in the story of the twins because inertial forces 
	are involved in that thought experiment, the tale of the 
	triplets indicates that the asymmetry has nothing to do with 
	forces-for no forces are involved in any way.  ([3], p. 43)
 
	Ray then goes on to criticize those who would turn to "positivist" 
accounts of inertial forces and spacetime-Machians and neo-Machians (see [3], 
chapter seven)-in order to explain the asymmetry of the twins' aging:
 
	[T]he [Machian] positivist claims that inertial forces 
	arise when objects accelerate relative to some average of 
	the material contents of the universe.  But, since we can 
	imagine this 'triplet' thought experiment taking place in 
	an otherwise empty spacetime, what might we cite as being 
	responsible for the asymmetry?  Each of the three paths 
	involved is inertial. And we have no reason to single out 
	any one path for preferential treatment in any way. Even if 
	we do give some preferred status to one path, we would clearly 
	have some non-material reason for doing so. The positivist 
	might still try to argue that only in the context of a complete 
	dynamical theory which sets all motions within a realistic, 
	gravitational context may we talk sensibly about such effects. 
	However, even if we set the thought experiment within the 
	observed universe, since none of the participants experiences 
	any forces and all three may be moving relatively to the 
	material contents of the universe, we may only point to the 
	fact of relative motion to explain the resulting asymmetry. 
	And why, from a materialist point of view, should objects in 
	relative motion not behave in the same way? What is true 
	materially for one object may also be true for the other objects 
	involved. The only significant differences seem to be the 
	_directions_ of the three motions. ([3], p. 44; Ray's emphasis)
 
	Ray is asserting here that a SR spacetime volume which contains the 
three separate inertial motions of the triplets is sufficiently determined in 
structure to account for both the phenomenon of relative time dilation among 
the triplets as well as the aging disparity between one and the other two, and 
that this asymmetry must be accounted for specifically by the geometry of the 
spacetime through which the triplets move ('We appeal to geometry to resolve 
the "paradox"-to the geometrical ideas of spacetime paths and world lines. . 
.'  [3], p. 44). Most simply put, Ray appears to observe that one triplet does 
not change direction in SR spacetime at all; the other two, in combination, 
do.
 
	Though Ray clearly attempts to account wholly for the dilation and 
asymmetry by appeal to SR, what is unclear is _how_ this reliance on the 
triplets' differences in direction establishes an asymmetrical relationship 
among them. If we recall that SR (and GR as well) holds that the longest 
proper time interval between timelike events is a geodesic, and that SR 
spacetime is flat, then these facts of the case, in combination with a 
postulate that time is unidirectional for all inertial frames, do suffice to 
distinguish one triplet from the others in the required manner, which 
presumably is what Ray alludes to in his remark about the significantly 
distinct directions of the triplets' spacetime travels. This is because only 
one triplet can occupy the unique geodesic connecting the earliest event of 
encounter (by all three inertial frames) with the latest;  the worldlines of 
the other two siblings are, in combination, necessarily longer in connecting 
these events ([6], p. 46). Therefore, the corresponding combined proper time 
interval of these latter two triplets is necessarily shorter than that of the 
other sibling occupying the requisite geodesic ([4], p. 156-158 nicely 
develops this path-integral inequality).

	To this point we can see that Ray is quite correct:  SR is logically 
sufficient to account for the triplets case as described. But the 
trouble with such _gedankenexperiments_ is that they are of necessity 
highly abstract, stylized, and contextual. Sometimes these 
constraints guarantee that one derives from them pretty much what one 
puts into them. Beyond the isolated context of the triplets example, 
however, note that all of these scenarios require relative observers 
whose dynamic histories exhibit or imply acceleration-presumably even 
in the case of the triplets antecedent to where the 
_gedankenexperiment_ narrative begins (how else could related 
siblings be involved?).  To make this clearer, consider Ray's 
appealingly simple reasoning.  Recast into a brief argument it is: 

	(i) The triplet's thought experiment removes any narrative 
		reference to accelerations or inertial forces between
 		the relevant events.
 	(ii) All of the events of the triplets' case can be mapped
 		onto a SR (flat) spacetime.
 	(iii) One triplet's inertial geodesic can be distinguished
 		from his siblings' combined worldlines between the
 		meeting events by considering only SR spacetime geometry.
 	(iv) SRT's spacetime metric requires an inequality of the proper 
		times of the triplets' worldlines between the meeting
 		events such that the sibling inhabiting the geodesic
 		has the longest proper time.
 	(v) Therefore, SRT provides a sufficient explanatory account of 
		the triplets' case and, by showing the irrelevance
 		of acceleration to that case, provides a sufficient
		explanatory account of the twins' case as well
 		(if actual accelerations are ignored).

	My point is that (i) does the crucial work for Ray, and that it simply 
screens off any inquiry into how the triplets got into their relative inertial 
motions.  Ray could try to counter this response by stating that the example 
need not involve historically-related participants.  One could just stipulate 
that, _ex hypothesi_, the three inertial observers appeared out of nowhere or 
were primordially moving as they are, and that they contacted one another 
pre-encounter and arranged the required clock synchronizations.  While this is 
correct and establishes (i) in another way, it just constitutes a _petitio 
principii_ as far as demonstrating the superiority of SR over GR explanation 
in this case is concerned.  I particularly find it troubling that many of the 
pioneers and brilliant expositors of relativity--including Einstein 
himself--provided explanations of twin paradox scenarios that explicitly 
favored a GR-based account. Clearly many of these were not merely motivated to 
demonstrate the mere compatibility of GR with SR explanations either--the 
_superiority or priority_ of the GR account over the SR account is intended.  
Surely it is arrogant and dismissive to treat these instances as irrelevant or 
mistaken without further examination.

	If such accelerations are relevant in any way to the resolution of 
the triplets case (and Ray admits this as a possibility, though only 
in the dubious sense-which he acknowledges as dubious-that 
acceleration _causes_ the aging differential [3] p. 232, note 15; 
compare Dennis Sciama in [5], p. 14-16), then SR offers nothing 
whatever to satisfy our curiosity about, for instance, whether 
accelerations are accountable in terms of absolutist or relationist 
spacetime-only GR (or some variant or refutation of it) can.  And 
there is good reason to believe that it is only if we satisfy our 
curiosity about such a question that we can finally go along with Ray 
or a GR advocate or the positivists in fully assessing the 
significance of the triplets case.

	Why this is so should now be fairly evident. SR's spacetime geometry 
alone provides a sufficient account of time dilation between two of 
the triplets and their other sibling, but it is _not_ clear that a 
GR-type account of acceleration and spacetime geometry isn't required 
to provide the requisite asymmetry among them in the first place. 
After all, it's instructive to note that the triplets case is 
classically used to demonstrate _only_ the sufficiency of SR to 
account for the time dilation in terms of an SR spacetime metric. In 
all these _gedankenexperiments_ the asymmetry issue is usually pried 
apart from the issue of time dilation in one of two ways. The first 
way just brutely attributes noninertial motion to one twin or 
triplet-pair (e.g., [4], p. 159; compare Ray's own remarks [3], p. 
38), which of course entails the inequality of proper time intervals 
between the first and final meeting-events mentioned above. The 
second way keeps the asymmetry issue separate by empirically 
stipulating that it arises from a relative acceleration of one twin 
or triplet pair which is detectable by an accelerometer (and thus 
involves GR to account for this, see e.g., [6], p. 47; [5] p. 14). 
Certainly the explanatory import of at least the latter of these two 
means of dividing the dilation and asymmetry issues is not adequately 
addressed by Ray's commentary on the triplets case. Perhaps the chief 
reason for this is that Ray casts the 'positivist' in a rather 
caricatured role as a critic of the SR account of the paradoxes.

	Typically positivists argue that only those things which we 
	can observe in some straightforward way should be included in 
	our physical descriptions of the universe. We can "see" neither 
	space nor time nor spacetime geometry. So positivist inclinations 
	would naturally lead us to discount explanations of physical 
	effects in terms of geometry and to look for something more 
	concrete such as inertial forces with their origin in some 
	kind of material interaction.    ([3], p. 40)

 	Though some positivists have doubtlessly-and wrongly-attributed the 
resolution of the paradoxes to the causal influence of inertial 
forces, this cannot be said of neo-Machians like Dennis Sciama who 
divide the issues of time-dilation and asymmetry of aging in their 
accounts. For while they concur that SR spacetime structure (or some 
such structure with the apposite metric) suffices to account for 
symmetrical time-dilation between inertial movers, they also insist 
that the final asymmetry in the paradox cases is ultimately tied to 
the source of spacetime itself:  the relational properties of 
material in the universe. Sciama and other neo-Machians maintain that 
the twins and triplets age asymmetrically because of asymmetrical 
acceleration which occurs in their histories, thus placing the 
various participants onto disparate inertial and noninertial 
spacetime paths. The crucial 'positivist' claim they make is that 
acceleration itself is only possible due to the presence of distant 
matter forming the relational basis of all spacetime (again, [6], p. 
47; [5] p. 14).  Ray's criticism fails to address such claims.

 	So I have no doubt that SR is logically sufficient to explain both 
the twins and triplets paradoxes in terms of the geometry of SR 
spacetime _given_ that the participants of the scenarios are already 
in their respective relative motions. However, it is not clear that 
further explanations having to do with forces, matter distribution, 
and GR spacetime structure are devoid of any role in accounting for 
the asymmetries of aging, particularly since distinct motions arise 
in the real world through asymmetrical acceleration. Perhaps 
Machians, or at least GR advocates of twin paradox explanation, 
should take heart.(1)
 _______________________________

 (1) I thank Peter Smith and an anonymous reader--who strongly 
disagreed with my view--for very useful criticism.  To assuage that 
reader's concerns a bit, I should stress that I am keenly aware that 
some Machian variants of GR (I have Brans-Dicke in mind here in 
particular) have not fared well against classical Einsteinian GR in 
experimental tests of various kinds (see [7]).  But there may well be 
other Machian variants, or Machian interpretations of classical GR 
foundations (e.g., for the "dragging" effect of rotational frames 
discussed in [7], p. 227-241), that theorists should not reject out 
of hand. 
                        ---------------------------  

References
 
[1] Henri Arzelies, _Relativistic Kinematics_ (Oxford: Pergamon Press, 1966).
 
[2] L. Marder, _Time and the Space-Traveller_  (Philadelphia: University of 
Pennsylvania Press, 1971).
 
[3] Christopher Ray, _Time, Space, and Philosophy_ (London: Routledge, 1991).
 
[4] Robert Resnick and David Halliday, _Basic Concepts in Relativity_ (New 
York:  Macmillan, 1992).
 
[5] Dennis Sicama, 'Time "Paradoxes" in Relativity', in Raymond Flood and 
Michael Lockwood, eds., _The Nature of Time_ (New York:  Basil Blackwell, 
1988).
 
[6] Albert Shadowitz, _Special Relativity_ (New York:  Dover, 1988).
 
[7] Clifford Will, _Was Einstein Right?_ second edition (New York: Basic 
Books, 1993).