The motional dependence of clock rates and the related problem of synchronization are analyzed. It is shown that any possible motional effect on observed clock rates can be experimentally determined by the use of unsynchronized clocks alone, and that in consequence this result can be utilized to synchronize a system of clocks without light signals and without any assumptions whatever regarding the properties of space. Thus one is not free, as special relativity has claimed, to establish a time scale by defining the synchronization procedure.

It has long been believed that the synchronization of spatially separated clocks cannot be accomplished without the use of, and without certain assumptions concerning, light signals or their various equivalents. The first statement of this belief appears to have been made by Einstein, who, in the presentation of his special theory of relativity, categorically denied the possibility of synchronizing a clock at some point A with an identical clock at some spatially removed point B "...unless we establish by definition that the 'time' required by light to travel from A to B equals the 'time' it requires to travel from B to A"⁽¹⁾ [emphasis original]. On this basis, a light signal emitted from the location of a clock C_A at time t_A and returned to A at time t'_A via a reflection at B, can be used to "synchronize" the clock C_B to C_A by setting t_B = (t_A + t'_A)/2 at the instant when the signal is reflected at B.

A little reflection reveals that the SRS gives rise to a rather unusual situation. In the first place, the SRS incorporates the equality of one-way times into the time scale not as an assumption subject to test, but rather as the explicitly stated conviction ("...cannot be defined..."¹) that one cannot do otherwise; in this context, no physical meaning can be attached to one-way times, and all experiments purporting to measure them will always reduce to a trivial retrieval of the information stored in the SRS⁽³⁾. The measurement of one-way times is in this way excluded by the SRS.

In the second place, and because of the first, the SRS compels a view of time intervals between events occurring "at the same place," i.e., measured by a single clock, as being intrinsically different from time intervals between events occurring "at different places." The former time intervals are therein simple quantities requiring only one physical system, clocks, for their specification, whereas the latter become a complex of clocks and light signals, bound up together in an inseparable manner by the SRS. It is emphasized that this difference in viewpoint is required only if it be true that the one-way time assumption cannot be tested. If this impossibility is genuine, then one can only conclude that a signal's one-way times are indeed operationally meaningless, and that there exist, in fact, two species of "time." All the foregoing conclusions remain unchanged if any alternate synchronization means be utilized that is based on presumed spatial properties; one will always return to the same difficulty.

The fact is, however, that the above claims pertaining to the SRS are not valid; the limitations which ensue therefrom are not due to the nature of things, as is commonly believed, but to the method. There is a means of synchronizing clocks, in which nothing need be assumed about either signal propagation or the structure of space. It will be shown in this connection that a synchronization can be effected without the use of any physical systems except clocks, and furthermore that this synchronization is fixed by the very nature of clocks. Thus the freedom to define the synchronization process, which relativity claims, does not exist; on the contrary one must, if a synchronous array is to be established, conform to whatever requirements are imposed by clocks themselves.

The naturalness of the synchronization here proposed will be seen to follow from the fact that a clock C_M, moving between two similar clocks C_A and C_B that are at rest with respect to each other, can provide information that reveals any motional effect on the observed rate of C_M, and thence a synchronization of C_A and C_B. Thus by an empirical procedure using only clocks, it becomes possible to synchronize an extended system and to do so to any desired accuracy. This course is the only defensible one, since the use of time as a fundamental physical variable is contingent upon its being independent of all else save timekeeping devices, without recourse to signals, lengths, or direction, and without invoking "symmetries" of any sort. For this reason, the method requires only that clocks can be constructed that will "beat at the same rate" when separated but relatively at rest. Within this scheme then, it becomes possible to measure the signal propagation characteristics of a given locality, if any, and to erect a time scale that depends only on clocks, while the measurement of the one-way speed of light reduces in principle to a trivial procedure.

The effect of a clock's motion upon its rate is considered next in Sect. II. Section III then deals with the synchronization process, and the results obtained in II and III are discussed in Sect. IV.

The key to the synchronization scheme to be discussed in Sect. III lies in a property of clocks that has heretofore apparently lain unrecognized: if observed clock rates are indeed dependent upon motion, such dependence can be experimentally determined using only clocks, and, most importantly, clocks that are not synchronous. This can be accomplished by the following procedure.

Imagine that a clock is moving, in a manner specified below, between two points A and B that are fixed with respect to each other (see Fig. 1), and at which points clocks CA and CB, identical to the moving clock CM, are located. Let all points fixed with respect to CM be referred to as the frame F', while F denotes all points fixed relative to CA and CB. In addition, let all time measurements be identified in accordance with the following list of symbols.

Figure 1. Various times pertaining to coincidence of CM with A, and with B

(i)	tA denotes time readings on CA.
(ii)	denotes time readings on CM.
(iii)	TB denotes time readings on CB (not yet synchronized to CA).
(iv)	tB = TB + is the as yet unknown synchronous time at B, with = const.
(v)	is the transit time of CM over AB as read on CM, i.e., = B - A.
(vi)	f() is the as yet unknown transit time, in F, of CM over AB; i.e., f() = tB - tA. It will be called the derived transit time (DTT) corresponding to the measured transit time in F'.

The method of transport of C_M from A to B is to be such that if AB were divided into n equal segments (where n is an integer), then C_M would note identical time intervals between coincidences with the end-points of each  . This is recognized as being a prescription for a "constant velocity" process, without, however, appeal to the as-yet-undefinable concept of velocity; it will be referred to here as uniform motion. It is assumed that means are provided for producing such motions, and also for varying while retaining uniformity.

Let C_M now be sent on a pass from A to B along AB. Upon passing A, let C_M be set to the time on C_A, which reads, say, t_A = t_A¹. Then when C_M reaches B, the following relations describe the various times pertaining to the second coincidence:

_B¹ = t_A¹ + ₁,

t_B¹ = t_A¹ + f(₁) = T_B¹ + ,

T_B¹ = t_B¹ - = t_A¹ + f(₁) - .

Of these three times, only _B¹ and T_B¹ are observable, for t_B¹ is connected to the known t_A¹ by the unknown derived transit time f(), or, alternatively, to T_B¹ by the likewise unknown synchronization constant .

It is now easy to show that the difference 

(1)

between  the above two observable times will reveal the motional effect on the observed rate of C_M, if any such exists. The most effective way to do so is as follows: Let two clocks C¹_M, C²_M , be sent simultaneously from A towards B such that they do not arrive simultaneously at B. Then

where ₂ -₁ is the observed difference in transit times shown by the moving clocks, and T_B² - T_B¹ is the difference in arrival times as observed on C_B. Whereupon, if ₂ -₁ = T_B² - T_B¹, i.e.,₂ -₁ = 0, there is no motional effect whatsoever. If, however, it is found that ₂ -₁ T_B² - T_B¹, i.e.,₂ -₁ 0, then the rates of the moving clocks are indeed affected by their motions. More generally, multiple sets of _i -_j can be obtained which, because of the rate information they yield, will be called rate discriminants.

These simple considerations show that the determination of whether clock rates are affected by their motion can be made without a pre-existing synchronization. But in addition--and most usefully, in the event that _i -_j 0, it is self-evident that the values obtained can be utilized to yield a quantitative relationship between the rates of moving and stationary clocks, that is, to obtain the function f() or, equivalently, a table of values connecting the two rates. Clearly then, the presently interesting case would be the second, for which ₂ -₁ T_B² - T_B¹; this, according to relativity, is what one would find.

It is most important to note at this point, that no appeal to light signals has been made anywhere in the specification of the observable quantities_i. So that, should this latter situation indeed occur, the variation of need not have, and in fact ought not have, any connection with the velocity of light, nor could it depend on any of the spatial assumptions contained in the SRS. Rather, such variation, if any, reveals instead some property intrinsic to clocks themselves (possibly affected by the actual spatial properties of the locality), a property that manifests itself upon their transport from one place to another. As will be shown next in Sec. III, the -variation, if it occurs, can subsequently be used to obtain f() and thereby to synchronize C_A and C_B.

Before presenting the proposed synchronization technique, it will be necessary to discuss the central question of what, exactly, constitutes the synchronization of two separated clocks. This question can be resolved by a development of the result found in Sec. II that it is possible to determine, using only clocks, whether a clock's motion has any effect on its rate as observed in F. Assume that this motional effect does indeed exist. Then the inequality ₂ -₁ T_B² - T_B¹ must hold; this, together with the known behavior of clocks, is utilized to infer the form of the DTT function f() (see Appendix) to within some arbitrary constant. This constant is then evaluated using the experimental fact that the rates of "stationary" and "moving" clocks become very accurately equal as for fixed AB. Thus the meaning of "synchronous" is embodied in the requirement that over that range of wherein the rates in F and F' approach equality, the measurement of the rate of C_M by C_A and C_B in combination shall yield this asymptotic result .

Of course, one might contend that if the rates for the large- limit become equal, it ought to be sufficient to simply synchronize by means of a clock that is moving "slowly" enough; consequently, the trouble of obtaining the functional connection between t_B and might appear superfluous. However, it will be seen that it is the time interval , rather than the synchronous time, that has fundamental meaning in physical descriptions (Sec. IV, paragraph five). Thus, anticipating that result, the importance of f() emerges: once it is known, measurements of t_B - t_A made by a synchronous, stationary array can be used to invert f() and thus obtain .

The determination of the DTT, f(), will of course depend on the ability to make very precise measurements of time intervals, a process which, if not yet within reach, presents in essence no impediment; all experiments proposed will be seen to be in principle possible.

Proceeding now to the method for obtaining f(), let the clock C_M be sent from A to B as specified in Sect. II. Upon passing A, let C_M again be set to the time on C_A,_A = t_A. When C_M subsequently passes B, the time on C_M will be _B = t_A + by virtue of relation (v) in Sect. 2, while the synchronous time at B is given, at this instant, by

(2)

where f is now written to include a possible dependence on spatial properties relevant to the locality of AB; these properties are symbolically represented by S.

Consider now this function f(, S): for present purposes, f(, S) is assumed to be analytic, but this is not essential, since a table of measured values can always be used instead, as mentioned. Surprisingly, the known behavior of clocks is sufficient to show (see Appendix) that if analyticity holds, f can be represented by an expansion of the form

(3)

It is stressed that this form for f(, S) follows directly from the known behavior of clocks. Equation (2) therefore becomes

(4)

with the a_n to be obtained by the experimental procedure given below. In part, these coefficients are the expansion equivalents of the symbolic quantities S of Eq. (2), and as such will contain all information appropriate to the locality of AB that is of temporal significance. The problem now is the evaluation of the a_n; if these can be obtained, then f() is determined and Eq. (4) yields directly the synchronous time at B. As it stands, however, Eq. (4) is useless: the only available information is and t_A, and this is not sufficient. But note that if, as before, two clocks, C_M¹ and C_M² are sent simultaneously from A in such a way that they do not arrive simultaneously at B, then the following additional information can be obtained.

Now, while it is true that T_B¹ and T_B² are related to the synchronous arrival times t_B¹ and t_B² in an as yet unknown way, nevertheless, by virtue of relation (iv) in Sect. 2, T_B² - T_B¹ t_B² - t_B¹, and so these last observations, together with Eq. (4), yield

With ₁,₂ and  T_B² - T_B¹  known by virtue of (a) and (b) above, Eq. (5) can now be used to evaluate as many of the a_n as one pleases, so that f(, S) can be known to any desired accuracy. That is, one can make a number of separate observations with two clocks sent simultaneously from A, the 's being varied on each pass. The resulting data can then be used, in conjunction with Eq. (5) suitably cut off at n passes, to solve the resulting system of n linear equations in the n a_n; the system will always have a solution, by virtue of the linear independence of the various powers of .

It is interesting to note that the observational procedure just above yields exactly the information needed to obtain the a_n: the individual values ₁ and ₂, and the difference T_B² - T_B¹.

The dependence of the a_n on =_B - T_B (Eq. (1)) is also evident in Eq. (5). For it is easy to see that the motional effect is completely contained in the ensemble of rate discriminants

Hence, if _i -_j = 0 for all i and j, then it is clear that the a_n would all be zero, and Eq. (3) would give f() - 0, which is the formal condition for a classical, absolute time with respect to all frames of reference. If _i -_j 0, the individual -values may of course be directly utilized to solve for the a_n in exactly the same way as above; in fact, this latter method may be preferable, since it is operationally simpler to obtain sucessive 's than to simultaneously send two clocks from A to B, a procedure that is nothing more than the measurement of _i - _j with a single pass.

It can thus be said that there exists a means for obtaining a synchronization of spatially separated clocks without the use of any pre-emptive assumptions⁽⁴⁾. Furthermore, the clocks can be made synchronous to any accuracy by evaluation of a sufficient number of the expansion coefficients of Eq. (5). Because these a_n's are experimentally obtained at a given locality, they will embody whatever temporal properties appertain thereto, and anything of physical consequence to clocks or clocklike devices will indeed emerge in those coefficients.

It is noted that the relativistic expression analogous to Eq. (4) is of the same form as Eq. (4). The usual time dilation expression for an SRS system is, in terms of the present notation, 

where v is the transit speed of C_M and c is a constant, the speed of light. With t_B - t_A =t and v = AB/t = /t, there results = t(1 - ²/c² t²)^½.For > /c, this can be inverted to give 

where ß _n^1/2 = (-1)ⁿ⁺¹(|2n-3|!!)/2ⁿn! are the binomial coefficients for exponent ½. It is seen that Eq. (4) and Eq. (6) are identical, with a_2n = 0, a_2n-1 =ß_n^1/2(/c)²ⁿ. The speed of light c enters in the SRS, however, not as a consequence of the properties of the locality as measured, but as imposed⁽⁵⁾.

It has been shown that a synchronization of clocks can in principle be accomplished by the use of clocks or clocklike devices alone. Thus, it is possible to disentangle the meaning of synchronous time at separated points from electromagnetic signals propagating between such points. Moreover, the proposed technique is not tied to pre-emptive assumptions about properties of space or radiation; it is in essence completely empirical.

Since this disentanglement is possible, it becomes legitimate therefore to consider time a "fundamental" variable in physical descriptions, because a synchronous array of clocks is shown thereby to be a simple entity, complete in itself, rather than a synthesis with other elements. The intrusion of extraneous elements was seen to be an objectionable feature of the SRS. For, questions of validity of the theory aside, two obviously distinct and independent physical systems, clocks and light signals, were fused and confounded, with the result that possibly significant physical properties simply vanish from measurability in the context of the theory. For example, the common belief that the SRS incorporates isotropy and homogeneity of space and time has no operational presence in the theory. The customary association of special relativity with spatial homogeneity and isotropy arises from the insistence of the mind on a prior and absolute synchronization being somehow achievable, i.e., if the clocks were already synchronous, and if space were homogeneous and isotropic, then exactly the same situation as in relativity would result. But such "conceptualizing" is pointless: within the relativistic scheme, no amount of experimentation could ever reveal, say, a possible anisotropy. One would always find that light propagates "isotropically," but only because the relevant clocks had been synchronized to give that result. So within the SRS, isotropy could not be said to be a property of space, but rather of the SRS itself.

A part of this section will now be devoted to an examination of some characteristics of a synchronous array of clocks, including an exposition of what is manifestly the natural temporal description of events "not occurring at the same place."

It is first of all noted that the measurement of the time interval between two events has an elementary and unambiguous physical significance only if both events occur at the locality of the same clock. That is, no timekeeping device exists or can be fashioned that can be used to actually make time measurements of events not occurring at the location of the device. The main obstacle to an appreciation of this fact is the notion of a "time" that somehow pervades all of space, or, in the case of relativity, a given reference frame. This notion is obviously--as mentioned above--no more than a construct of the mind, for in nature there exists only an ordering of events as witnessed at the location of an observer. What is commonly called the "time" between any two such events is thus no more than the quantification, in terms of a chosen cyclic device (clock), of the interval between the events, and this again can only be realized if both occur at the location of the cyclic device. The utilization of signals from events occurring at places removed from the clock is a pointless procedure, since one obtains thereby not the times of the events, but the arrival times of the corresponding signals at the location of the clock, a procedure fraught, obviously, with the unknown properties of the intervening space.

It appears, therefore, that no counterpart exists in nature for the intuitively appealing notion of a universal "time" for an extended reference frame. Only by the additional procedure of synchronization can one arrive at a physically useful method for measuring the "time" at different locations.

In the face of this situation, there suggests itself the natural and simple procedure of measuring the time between events by moving a clock such that the events all occur at the location of the clock. In this fashion, the pertinent time intervals ("proper times," since they are proper to the measuring clock) are described in an absolute and unambiguous way without even the need, at this stage, for the use of synchronous stationary clocks. True, the assumption is here implicit that all clocks, whatever their nature or construction, will record the same time at every point over the traversal (for example, any two or more different types of clocks, sent together over the path, would all show the same time at every point on the path). But this assumption is not at all restrictive, for one may simply discard from use as "clocks" any device or group of devices that shows a departure from the behavior of some other (arbitrarily chosen) group; in the last analysis, even one cyclic physical system is sufficient, much as one might decide that a particular material is best suited for use as a length standard. It is known, of course, that all commonly employed time-keeping devices give very good agreement in the above respects, at least for ordinary transport velocities and accelerations.

In addition, an observer moving with the clock C_M past a series of points notes one other absolute effect that is independent even of the concept of timekeeping: the sequential ordering of the coincidences of C_M with these points. It is obvious that no possible motional effect can reorder events and that an observer moving with C_M need not refer to timekeeping devices at all in order to tabulate or describe this ordering, although it may be convenient to do so.

Clearly then, a clock moving with an object or mass-point in a physical process enjoys a privileged position with respect to specification of the process: it alone has an unambiguous temporal description of the process, which in the present case happens to be uniform motion relative to a given coordinate system F. The synchronization of clocks stationary in F then makes possible the transfer, to these clocks, of the temporal information which initially only C_M possesses.

Summarizing the preceding considerations, and because of them, the view is put forth here that the natural mode of description of a point moving relative to F is that given by: (1) its locations relative to F as a function of (2) the time intervals shown by a clock moving with the point.

The difficulty with operationally using this proposed mode of description is of course obvious. It is very awkward and difficult, in most cases, to append a clock to a moving system under study, and this underscores the desirability of disposing an array of clocks, stationary in F, from which the time intervals in the moving system can be obtained. With this in mind, the dual utility of f() now becomes evident: first, it makes possible the synchronization of C_A and C_B, and it can thereafter be inverted to obtain from time measurements made in F.

In this way, is obtained in a completely empirical--albeit indirect--fashion, exactly as if one were actually reading it from C_M. However, while one ordinarily works exclusively in terms of the time in F, it is important to bear in mind the derivative nature of this time scale and its total dependence on the temporal information initially provided (through the synchronization procedure) by the moving clock C_M. This whole procedure might of course seem both superfluous and trivial in the absolute, pre-relativistic scheme, since no motional effect and no distinction between "times" exists there. Nevertheless, the stationary time scale is in that case no less dependent on C_M; it is impossible, even there, to measure time except at the locality of a clock nor to effect a synchronization except by using C_M.

In conclusion, it is briefly remarked that a great deal could probably be inferred, on the basis of physical arguments, about the coefficients a_n appearing in Eq. (4). However, it is considered best to leave their evaluation completely to experiment, since any other approach can lead, by way of some subtle misconception, to the introduction of physically unwarranted constraints. These and related matters will not be considered further, inasmuch as the primary goal has been to demonstrate the attainability of an empirically-based time scale⁽⁶⁾.

It was stated in Sect. III that, as a consequence of the known behavior of clocks, the function f(, S) must have the form

This result is obtained in the following way.

Suppose, as in Sect. III, that identical clocks C_M¹, C_M² sent simultaneously from A arrive at B with elapsed times ₁, ₂ on the C_M's, and at times T_B¹, T_B² on C_B. Then it is experimentally noted, without exception, that T_B² - T_B¹ =₂ - ₁ to extremely high accuracy (a discrepancy, if any exists, is beyond detection at the present time, as is well-known). Thus, evidently, ordinary processes utilizing clocks always take place over a range of motions wherein the rates of moving clocks are imperceptibly affected, or else not affected at all. Expressing this experimental result in differential form, there results dt_B = d (recall that dt_B dT_B). Hence taking t_A = 0 as the time on C_A when the moving clocks passed A, one obtains t_B =   where is a constant and (, S) results from integration over that range of  where presumably dt_B d. This function must have the behavior (, S) 0 for ordinary motions (any possible constant term is included in ) if the observed T_B² - T_B¹ ₂ - ₁ is to hold for such motions.

The departure of f(, S) from f = , which may be expected to occur outside the range of ordinary motions, is represented by (, S). Because of its behavior, (, S) may be expressed in terms of the complete set ^-n, with n1 guaranteeing the observed asymptotic value. Consequently,

This investigation was done in 1966 while the author was a doctoral candidate in physics at the University of Miami, and during the fall of that year was published privately by distribution among the physics faculty and graduate students. Subsequently, it was presented at a meeting of the American Physical Society at Miami Beach in November 1968, where the proposal elicited scant interest. Still later, attempts at a more extensive publication under non-academic circumstances had to be shelved for the time then being because of other undertakings by the author. As it happened, that "time being" turned into a lapse of some thirty years, during which time L.C. Hawkins, who was from the beginning familiar with the calculation, would periodically urge the author to give it the wider exposure now undertaken.

The author wishes to also acknowledge the efforts by P. E. Jindra of the Florida Solar Energy Center, whose many valuable suggestions have contributed to a more lucid and compact presentation of the calculation.

1. Einstein, A., Ann. D. Phys. 17, 891 (1905); A. Einstein et. al., The Principle of Relativity, Dover Publ., Inc., New York).

2. See, for example, Moller, C., The Theory of Relativity, Oxford University Press, 1962.

3. Sama, N., Bull. Amer. Phys. Soc. 1380 (Nov. 1968).

4. That is, assumptions such as those regarding the velocity of light in the SRS--assumptions which, once incorporated into the time scale, pre-empt any possibility of their verification.

5. Whatever factors may justify the SRS notwithstanding, certain observations must be made as to the logical structure upon which the kinematics of the special theory is erected. The starting point for the SRS is Einstein's assertion that "...a common 'time' for A and B ... cannot be defined at all unless we establish by definition..." etc. [emphasis original]. That foundational assertion is incontrovertibly in error, as has been shown here.

Nevertheless, in view of the as yet insurmountable experimental difficulties of determining the DTT function f() developed herein, the assumptions regarding the propagation of light that lead to the SRS would have to be seen as a laudable hurdling over those difficulties had those assumptions been properly framed. So the problem is not that the assumptions were made as a stop-gap measure, but that they were made exclusionary by the assertion "...cannot be defined at all...," and worse, that Einstein was--and seemed to have remained--unaware of the error in the claim by which he made them so.

And most disturbing of all to ponder: that defective claim has since been canonized by a continuing succession of reputedly great scientists, who also, it seems, have blithely disregarded Einstein's putting the cart before the horse by invoking a dynamical process--the propagation of light--as the basis for a system of kinematics.

6. What has been established here regarding time scales should be kept in mind vis-a-vis broader aspects of the SRS. As is well-known, the interpretation given to the transformation of coordinates that proceeds from the SRS has led to much being made about the reduction of time to a variable "on an equal footing" with spatial variables. Indeed, such discussions have at times taken on an almost mystical aura, a phenomenon apparently originating with Einstein's onetime mathematics professor and mentor, H. Minkowski, who, in the address Space and Time, given at the 80^th Assembly of German Natural Scientists and Physicians at Cologne, 21 September, 1908, declared, "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows...."