In his first paper on the special theory of relativity, Einsteinindicated that the question of whether or not two spatially separatedevents were simultaneous did not necessarily have a definite answer,but instead depended on the adoption of a convention for itsresolution. Some later writers have argued that Einstein's choice of aconvention is, in fact, the only possible choice within the frameworkof special relativistic physics, while others have maintained thatalternative choices, although perhaps less convenient, are indeedpossible.

• 1. The Conventionality Thesis
• 2. Phenomenological Counterarguments
• 3. Transport of Clocks
• 4. Malament's Theorem
• 5. Other Considerations
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. The Conventionality Thesis

The debate about the conventionality of simultaneity is usuallycarried on within the framework of the special theory of relativity.Even prior to the advent of that theory, however, questions had beenraised (see, e.g., Poincaré 1898) as to whether simultaneitywas absolute; i.e., whether there was a unique event at location Athat was simultaneous with a given event at location B. In his firstpaper on relativity, Einstein (1905) asserted that it was necessary tomake an assumption in order to be able to compare the times ofoccurrence of events at spatially separated locations (Einstein 1905,38–40 of the Dover translation or 125–127 of the Princetontranslation; but note Scribner 1963, for correction of an error in theDover translation). His assumption, which defined what is usuallycalled standard synchrony, can be described in terms of the followingidealized thought experiment, where the spatial locationsAandB are fixed locations in some particular, but arbitrary,inertial (i.e., unaccelerated) frame of reference: Let a light ray,traveling in vacuum, leaveA at timet₁(as measured by a clock at rest there), and arrive atBcoincident with the eventE atB. Let the ray beinstantaneously reflected back toA, arriving at timet₂. Then standard synchrony is defined by sayingthatE is simultaneous with the event atA thatoccurred at time (t₁ +t₂)/2. This definition is equivalent to therequirement that the one-way speeds of the ray be the same on the twosegments of its round-trip journey betweenA andB.

It is interesting to note (as pointed out by Jammer (2006, 49), in hiscomprehensive survey of virtually all aspects of simultaneity) thatsomething closely analogous to Einstein's definition of standardsimultaneity was used more than 1500 years earlier by St. Augustine inhisConfessions (written in 397 CE). He was arguing againstastrology by telling a story of two women, one rich and one poor, whogave birth simultaneously but whose children had quite different livesin spite of having identical horoscopes. His method of determiningthat the births, at different locations, were simultaneous was to havea messenger leave each birth site at the moment of birth and travel tothe other, presumably with equal speeds. Since the messengers met atthe midpoint, the births must have been simultaneous. Jammer commentsthat this “may well be regarded as probably the earliestrecorded example of anoperational definition of distantsimultaneity.”

The thesis that the choice of standard synchrony is a convention,rather than one necessitated by facts about the physical universe(within the framework of the special theory of relativity), has beenargued particularly by Reichenbach (see, for example, Reichenbach1958, 123–135) and Grünbaum (see, for example,Grünbaum 1973, 342–368). They argue that the onlynonconventional basis for claiming that two distinct events are notsimultaneous would be the possibility of a causal influence connectingthe events. In the pre-Einsteinian view of the universe, there was noreason to rule out the possibility of arbitrarily fast causalinfluences, which would then be able to single out a unique eventatA that would be simultaneous withE. In anEinsteinian universe, however, no causal influence can travel fasterthan the speed of light in vacuum, so from the point of view ofReichenbach and Grünbaum, any event atA whose time ofoccurrence is in the open interval betweent₁ andt₂ could be defined to be simultaneouswithE. In terms of the ε-notation introduced byReichenbach, any event atA occurring at atimet₁ + ε(t₂ −t₁), where 0 < ε < 1, could besimultaneous withE. That is, the conventionality thesisasserts that any particular choice of ε within its statedrange is a matter of convention, including the choice ε=1/2(which corresponds to standard synchrony). If ε differs from1/2, the one-way speeds of a light ray would differ (in anε-dependent fashion) on the two segments of its round-tripjourney betweenA andB. If, more generally, weconsider light traveling on an arbitrary closed path inthree-dimensional space, then (as shown by Minguzzi 2002,155–156) the freedom of choice in the one-way speeds of lightamounts to the choice of an arbitrary scalar field (although twoscalar fields that differ only by an additive constant would give thesame assignment of one-way speeds).

It might be argued that the definition ofstandard synchrony makes use only of the relation of equality (of theone-way speeds of light in different directions), so that simplicitydictates its choice rather than a choice that requires thespecification of a particular value for a parameter. Grünbaum(1973, 356) rejects this argument on the grounds that, since theequality of the one-way speeds of light is a convention, this choicedoes not simplify the postulational basis of the theory but only givesa symbolically simpler representation.

2. Phenomenological Counterarguments

Many of the arguments against the conventionality thesis make use ofparticular physical phenomena, together with the laws of physics, toestablish simultaneity (or, equivalently, to measure the one-way speedof light). Salmon (1977), for example, discusses a number of suchschemes and argues that each makes use of a nontrivial convention. Forinstance, one such scheme uses the law of conservation of momentum toconclude that two particles of equal mass, initially located halfwaybetweenA andB and then separated by an explosion,must arrive atA andBsimultaneously.Salmon (1977, 273) argues,however, that the standard formulation of the law of conservation ofmomentum makes use of the concept of one-way velocities, which cannotbe measured without the use of (something equivalent to) synchronizedclocks at the two ends of the spatial interval that is traversed;thus, it is a circular argument to use conservation of momentum todefine simultaneity.

It has been argued (see, for example, Janis 1983, 103–105, andNorton 1986, 119) that all such schemes for establishingconvention-free synchrony must fail. The argument can be summarized asfollows: Suppose that clocks are set in standard synchrony, andconsider the detailed space-time description of the proposedsynchronization procedure that would be obtained with the use of suchclocks. Next suppose that the clocks are reset in some nonstandardfashion (consistent with the causal order of events), and consider thedescription of the same sequence of events that would be obtained withthe use of the reset clocks. In such a description, familiar laws maytake unfamiliar forms, as in the case of the law of conservation ofmomentum in the example mentioned above. Indeed, all of specialrelativity has been reformulated (in an unfamiliar form) in terms ofnonstandard synchronies (Winnie 1970a and 1970b). Since the proposedsynchronization procedure can itself be described in terms of anonstandard synchrony, the scheme cannot describe a sequence of eventsthat is incompatible with nonstandard synchrony. A comparison of thetwo descriptions makes clear what hidden assumptions in the scheme areequivalent to standard synchrony. Nevertheless, editors of respectedjournals continue to accept, from time to time, papers purporting tomeasure one-way light speeds; see, for example, Greavesetal. (2009). Application of the procedure just described showswhere their errors lie.

3. Transport of Clocks

A phenomenological scheme that deserves special mention, because ofthe amount of attention it has received over the course of many years,is to define synchrony by the use of clocks transported betweenlocationsA andB in the limit of zerovelocity. Eddington (1924, 15) discusses this method of synchrony, andnotes that it leads to the same results as those obtained by the useof electromagnetic signals (the method that has been referred to hereas standard synchrony). He comments on both of these methods asfollows (1924, 15–16): “We can scarcely consider thateither of these methods of comparing time at different places is anessential part of our primitive notion of time in the same way thatmeasurement at one place by a cyclic mechanism is; therefore they arebest regarded as conventional.”

One objection to the use of the slow-transport scheme to synchronizeclocks is that, until the clocks are synchronized, there is no way ofmeasuring the one-way velocity of the transported clock. Bridgman(1962, 26) uses the “self-measured” velocity, determinedby using the transported clock to measure the time interval, to avoidthis problem. Using this meaning of velocity, he suggests (1962,64–67) a modified procedure that is equivalent to Eddington's,but does not require having started in the infinite past. Bridgmanwould transport a number of clocks fromA toB atvarious velocities; the readings of these clocks atB woulddiffer. He would then pick one clock, say the one whose velocity wasthe smallest, and find the differences between its reading and thereadings of the other clocks. Finally, he would plot these differencesagainst the velocities of the corresponding clocks, and extrapolate tozero velocity. Like Eddington, Bridgman does not see this scheme ascontradicting the conventionality thesis. He says (1962, 66),“What becomes of Einstein's insistence that his method forsetting distant clocks — that is, choosing the value 1/2 forε — constituted a ‘definition’ of distantsimultaneity? It seems to me that Einstein's remark is by no meansinvalidated.”

Ellis and Bowman (1967) take a different point of view. Their means ofsynchronizing clocks by slow transport (1967, 129–130) is againsomewhat different from, but equivalent to, those already mentioned.They would place clocks atA andB with arbitrarysettings. They would then place a third clock atA andsynchronize it with the one already there. Next they would move thisthird clock toB with a velocity they refer to as the“intervening ‘velocity’”, determined by usingthe clocks in place atA andB to measure the timeinterval. They would repeat this procedure with decreasing velocitiesand extrapolate to find the zero-velocity limit of the differencebetween the readings of the clock atB and the transportedclock. Finally, they would set the clock atB back by thislimiting amount. On the basis of their analysis of this procedure,they argue that, although consistent nonstandard synchronizationappears to be possible, there are good physical reasons (assuming thecorrectness of empirical predictions of the special theory ofrelativity) for preferring standard synchrony. Their conclusion (assummarized in the abstract of their 1967, 116) is, “The thesisof the conventionality of distant simultaneity espoused particularlyby Reichenbach and Grünbaum is thus either trivialized orrefuted.”

A number of responses to these views of Ellis and Bowman (see, forexample, Grünbaumet al. 1969; Winnie 1970b,223–228; and Redhead 1993, 111–113) argue that nontrivialconventions are implicit in the choice to synchronize clocks by theslow-transport method. For example, Grünbaum(Grünbaumet al. 1969, 5–43) argues that it is anontrivial convention to equate the time interval measured by theinfinitely slowly moving clock traveling fromA toBwith the interval measured by the clock remaining atA and instandard synchrony with that atB, and the conclusion of vanFraassen (Grünbaumet al. 1969, 73) is, “Ellisand Bowman have not proved that the standard simultaneity relation isnonconventional, which it is not, but have succeeded in exhibitingsomealternative conventions which also yield thatsimultaneity relation.” Winnie (1970b), using his reformulationof special relativity in terms of arbitrary synchrony, showsexplicitly that synchrony by slow-clock transport agrees withsynchrony by the standard light-signal method when both are describedin terms of an arbitrary value of ε within the range 0 <ε < 1, and argues that Ellis and Bowman err in havingassumed the ε=1/2 form of the time-dilation formula in theirarguments. He concludes (Winnie 1970b, 228) that “it is notpossible that the method of slow-transport, or any other synchronymethod, could, within the framework of thenonconventional ingredients of the Special Theory, result infixingany particular value of ε to the exclusion ofany other particular values.” Redhead (1993) also argues thatslow transport of clocks fails to give a convention-free definition ofsimultaneity. He says (1993, 112), “There is no absolute factualsense in the term ‘slow.’ If we estimate‘slow’ relative to a moving frameK′, thenslow-clock-transport will pick out standard synchronyinK′, but this …corresponds to nonstandardsynchrony inK.”

An alternative clock-transport scheme, which avoids the issue ofslowness, is to have the clock move fromA toB andback again (along straight paths in each direction) with the sameself-measured speed throughout the round trip (Mamone Capria 2001,812–813; as Mamone Capria notes, his scheme is similar to thoseproposed by Brehme 1985, 57–58, and 1988, 811–812). If themoving clock leavesA at timet₁ (asmeasured by a clock at rest there), arrives atB coincidentwith the eventE atB, and arrives back atA at thetimet₂, then standard synchrony is obtained bysaying thatE is simultaneous with the event atAthat occurred at the time (t₁ +t₂)/2. It would seem that this transport scheme issufficiently similar to the slow-transport scheme that it couldengender much the same debate, apart from those aspects of the debatethat focussed specifically on the issue of slowness.

4. Malament's Theorem

An entirely different sort of argument against the conventionalitythesis has been given by Malament (1977), who argues that standardsynchrony is the only simultaneity relation that can be defined,relative to a given inertial frame, from the relation of (symmetric)causal connectibility. Let this relation be represented by κ,let the statement that events p and q are simultaneous be representedbyS(p,q), and let the given inertial framebe specified by the world line,O, of some inertialobserver. Then Malament's uniqueness theorem shows that ifSis definable from κ andO, if it is an equivalencerelation, if pointsp onO andq notonO exist such thatS(p,q) holds,and ifS is not the universal relation (which holds for allpoints), thenS is the relation of standard synchrony.

Some commentators have taken Malament's theorem to have settled thedebate on the side of nonconventionality. For example, Torretti (1983,229) says, “Malament proved that simultaneity by standardsynchronism in an inertial frameF is theonlynon-universal equivalence between events at different pointsofF that is definable (‘in any sense of“definable” no matter how weak’) in terms of causalconnectibility alone, for a givenF”; and Norton(Salmonet al. 1992, 222) says, “Contrary to mostexpectations, [Malament] was able to prove that the central claimabout simultaneity of the causal theorists of time was false. Heshowed that the standard simultaneity relation was the only nontrivialsimultaneity relation definable in terms of the causal structure of aMinkowski spacetime of special relativity.”

Other commentators disagree with such arguments, however.Grünbaum (2010) has written a detailed critique of Malament'spaper. He first cites Malament's need to postulate that S is anequivalence relation as a weakness in the argument, a view alsoendorsed by Redhead (1993, 114). Grünbaum's main argument,however, is based on an earlier argument by Janis (1983,107–109) that Malament's theorem leads to a unique (butdifferent) synchrony relative to any inertial observer, that thislatitude is the same as that in introducing Reichenbach's ε,and thus Malament's theorem should carry neither more nor less weightagainst the conventionality thesis than the argument(mentionedabove in the last paragraph of thefirst section of this article) that standard synchrony is the simplestchoice. Grünbaum concludes “that Malament's remarkableproof has not undermined my thesis that, in the STR, relativesimultaneity is conventional, as contrasted with itsnon-conventionality in the Newtonian world, which I have articulated!Thus, I do not need to retract the actual claim I made in1963…” Somewhat similar arguments are given by Redhead(1993, 114) and by Debs and Redhead (2007, 87–92).

Havas (1987, 444) says, “What Malament has shown, in fact, isthat in Minkowski space-time … one can always introducetime-orthogonal coordinates … , an obvious and well-knownresult which implies ε=1/2.” In a comprehensive reviewof the problem of the conventionality of simultaneity, Anderson,Vetharaniam, and Stedman (1998, 124–125) claim thatMalament's proof is erroneous. Although they appear to be wrong inthis claim, the nature of their error highlights the fact thatMalament's proof, which uses the time-symmetric relation κ,would not be valid if a temporal orientation were introduced intospace-time (see, for example, Spirtes 1981, Ch. VI, Sec. F; andStein 1991, 153n).

Sarkar and Stachel (1999) argue that there is no physical warrant forthe requirement that a simultaneity relation be invariant undertemporal reflections. Dropping that requirement, they show thatMalament's other criteria for a simultaneity relation are then alsosatisfied if we fix some arbitrary event in space-time and say eitherthat any pair of events on its backward null cone are simultaneous or,alternatively, that any pair of events on its forward null cone aresimultaneous. They show further that, among the relations satisfyingthese requirements, standard synchrony is the unique such relationthat is independent of the position of an observer and thehalf-null-cone relations are the unique such relations that areindependent of the motion of an observer. If the backward-conerelation were chosen, then simultaneous events would be those seensimultaneously by an observer at the cone's vertex. As Sarkar andStachel (1999, 209) note, Einstein (1905, 39 of the Dovertranslation or 126 of the Princeton translation) considered thispossibility and rejected it because of its dependence on the positionof the observer. Since the half-null-cone relations define causallyconnectible events to be simultaneous, it would seem that they wouldalso be rejected by adherents of the views of Reichenbach andGrünbaum.

Ben-Yami (2006) also argues against Malament's requirement ofinvariance under temporal reflections, but for different reasons thanthose of Sarkar and Stachel. Ben-Yami (2006, 461) takes hisfundamental causal relation to be the following: “Ifevente₁ is a cause ofevente₂, thene₂ does notprecedee₁.” He thus allows events to besimultaneous with their causes, and consequently the range ofReichenbach's ε is extended to include both 0 and1. Ben-Yami's causal relation is not time-symmetric, which is hisreason for rejecting the requirement of invariance under temporalreflections. He concludes (Ben-Yami 2006, 469-470) that, with hismodified causal relation, there are “infinitely many possiblesimultaneity relations: any space-like or light-like conichypersurface of an event onO defines a simultaneity relationfor that event relative toO, and then, by translations, forany event onO.” However he then goes on to argueagainst the assumption that an observer, represented byO,would remain inertial forever, and ultimately concludes not only thatstandard simultaneity cannot be defined but that the only twosimultaneity relations that can be defined relative to an event arethose determined by its future and past light cones.

Giulini (2001, 653) argues that it is too strong a requirement toask that a simultaneity relation be invariant under causaltransformations (such as scale transformations) that are not physicalsymmetries, which Malament as well as Sarkar and Stachel do. Using“Aut” to refer to the appropriate invariance group and“nontrivial” to refer to an equivalence relation onspacetime that is neither one in which all points are in the sameequivalence class nor one in which each point is in a differentequivalence class, Giulini (2001, 657–658) defines two types ofsimultaneity: Absolute simultaneity is a nontrivial Aut-invariantequivalence relation on spacetime such that each equivalence classintersects any physically realizable timelike trajectory in at mostone point, and simultaneity relative to some structureX inspacetime (for Malament,X is the world line of an inertialobserver) is a nontrivialAut_X-invariantequivalence relation on spacetime such that each equivalence classintersects any physically realizable timelike trajectory in at mostone point, whereAut_X is the subgroupofAut that preservesX. First takingAutto be the inhomogeneous (i.e., including translations) Galileantransformations, Giulini (2001, 660–662) shows that standardGalilean (i.e., pre-relativistic) simultaneity is the unique absolutesimultaneity relation. Then takingAut to be theinhomogeneous Lorentz transformations (also known as thePoincaré transformations), Giulini (2001, 664–666) showsthat there is no absolute simultaneity relation and that standardEinsteinian synchrony is the unique relative simultaneitywhenX is taken to be a foliation of spacetime by straightlines (thus, like Malament, singling out a specific inertial frame,but in a way that is different from Malament's choiceofX).

5. Other Considerations

Since the conventionality thesis rests upon the existence of a fastestcausal signal, the existence of arbitrarily fast causal signals wouldundermine the thesis. If we leave aside the question of causality, forthe moment, the possibility of particles (called tachyons) moving witharbitrarily high velocities is consistent with the mathematicalformalism of special relativity (see, for example, Feinberg 1967).Just as the speed of light in vacuum is an upper limit to the possiblespeeds of ordinary particles (sometimes called bradyons), it would bea lower limit to the speeds of tachyons. When a transformation is madeto a different inertial frame of reference, the speeds of bothbradyons and tachyons change (the speed of light in vacuum being theonly invariant speed). At any instant, the speed of a bradyon can betransformed to zero and the speed of a tachyon can be transformed toan infinite value. The statement that a bradyon is moving forward intime remains true in every inertial frame (if it is true in one), butthis is not so for tachyons. Feinberg (1967) argues that this does notlead to violations of causality through the exchange of tachyonsbetween two uniformly moving observers because of ambiguities in theinterpretation of the behavior of tachyon emitters and absorbers,whose roles can change from one to the other under the transformationbetween inertial frames. He claims to resolve putative causalanomalies by adopting the convention that each observer describes themotion of each tachyon interacting with that observer's apparatus insuch a way as to make the tachyon move forward in time. However, allof Feinberg's examples involve motion in only one spatialdimension. Pirani (1970) has given an explicit two-dimensional examplein which Feinberg's convention is satisfied but a tachyon signal isemitted by an observer and returned to that observer at an earliertime, thus leading to possible causal anomalies.

A claim that no value of ε other than 1/2 is mathematicallypossible has been put forward by Zangari (1994). He argues thatspin-1/2 particles (e.g., electrons) must be representedmathematically by what are known as complex spinors, and that thetransformation properties of these spinors are not consistent with theintroduction of nonstandard coordinates (corresponding to values ofε other than 1/2). Gunn and Vetharaniam (1995), however,present a derivation of the Dirac equation (the fundamental equationdescribing spin-1/2 particles) using coordinates that are consistentwith arbitrary synchrony. They argue that Zangari mistakenly requireda particular representation of space-time points as the only oneconsistent with the spinorial description of spin-1/2 particles.

Another argument for standard synchrony has been given by Ohanian (2004),who bases his considerations on the laws of dynamics. He argues that anonstandard choice of synchrony introduces pseudoforces into Newton'ssecond law, which must hold in the low-velocity limit of specialrelativity; that is, it is only with standard synchrony that net forceand acceleration will be proportional. Macdonald (2005) defends theconventionality thesis against this argument in a fashion analagous tothe argument used by Salmon (mentionedabove inthe first paragraph of the second section of this article) against theuse of the law of conservation of momentum to define simultaneity:Macdonald says, in effect, that it is a convention to require Newton'slaws to take their standard form.

Many of the arguments against conventionality involve viewing thepreferred simultaneity relation as an equivalence relation that isinvariant under an appropriate transformation group. Mamone Capria(2012) has examined the interpretation of simultaneity as an invariantequivalence relation in great detail, and argues that it does not haveany bearing on the question of whether or not simultaneity isconventional in special relativity.

A vigorous defense of conventionality has been offered by Rynasiewicz(2012). He argues that his approach “has the merit of nailingthe exact sense in which simultaneity is conventional. It isconventional in precisely the same sense in which the gauge freedomthat arises in the general theory of relativity makes the choicebetween diffeomorphically related models conventional.” Hebegins by showing that any choice of a simultaneity relation isequivalent to a choice of a velocity in the equation for local time inH.A. Lorentz'sVersuch theory (Lorentz 1895). Then, beginningwith Minkowski space with the standard Minkowski metric, he introducesa diffeomorphism in which each point is mapped to a point with thesame spatial coordinates, but the temporal coordinate is that of aLorentzian local time expressed in terms of the velocity as aparameter. This mapping is not an isometry, for the light cones aretilted, which corresponds to anisotropic light propagation. Heproceeds to argue, using the hole argument (see, for example, Earmanand Norton 1987) as an analogy, that this parametric freedom is justlike the gauge freedom of general relativity. As the tilting of thelight cones, if projected into a single spatial dimension, would beequivalent to a choice of Reichenbach's ε, it seems thatRynasiewicz's argument is a generalization and more completely arguedversion of the argument given by Janis that is mentioned above in thethird paragraph of Section 4.

The debate about conventionality of simultaneity seems far fromsettled, although some proponents on both sides of the argument mightdisagree with that statement. The reader wishing to pursue the matterfurther should consult the sources listed below as well as additionalreferences cited in those sources.

Bibliography

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