There are two generally recognised interpretations of the LT’s in respect of the rates of two identical clocks stationary in different inertial systems in uniform rectilinear relative motion:

**3.1 Interpretation a).** That the clocks in the two systems are “actually” going at different rates by the factor [1- *v*2*/c*2]½. That is, an observer situated continuously midway between the two clocks which are both moving at the same constant speed towards, or away from, him will see that they are going at different rates relative to one another (the velocities, the distances, and the time-of-flight delay of light, all being the same for both moving clocks w.r.t the observer, - that is, any effect of time-of-flight delay of light would be the same for both clocks.) This interpretation – that the clocks are actually going at different rates – has been the opinion of many. Indeed, Einstein in [2] says “Thence we conclude that a balance-clock at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions.” [In a note it is explained that he excludes pendulum-type clocks - for which that would be expected from the reduced gravitation at the equator. Also, the statement is made some seven years prior to his formulation of the General Theory (GR), so that he is talking without reference to effects of acceleration or gravitational potential, which would both be relevant; he is concerned solely with the effect of velocity.] Again, Eddington said, [3a], “it is now known that a clock in motion goes slow in comparison with a fixed clock”, and then, [3b], “Thus a clock travelling with finite velocity gives too small a reading – the clock goes slow compared with the time-reckoning conventionally adopted.” Whereas his first quote might be interpreted as including an accelerating clock, the second quite clearly confines the consideration to the clock’s velocity.

[It is of relevance that none of their quotes make any reference to any observer who is making the judgement about the clock rates. They are expressing what we may call an “objective” concept of differing rates, based on their interpretation of the conclusions of SR. It surely follows that they must also have had an “objective” concept of two clocks going at the same rate, i.e. without reference as to who is doing the observing.]

Another who held to this interpretation was Hawking. His work generally was concerned with advanced concepts, and the matter considered here is so basic that there appear to be no appropriate references in the literature in his name. However, in his video on time travel [4] he clearly holds that that the clocks are actually going at different rates due to the relative velocity effect. [At the same time, his proposal would actually work anyway, because it involves the participants travelling round the Earth – i.e. in a circular path, which would result in the desired time dilation, - in accordance with the General Theory (GR) as described at § 4.2.2 below. But he ascribes the effect as being due specifically to the relative velocity.]

Further evidence that this interpretation is widely held is the fact that textbooks and digital sources, e.g. Wikipedia, typically continue to introduce the reader to the “clock (or twins) paradox”. Although it is then usually pointed out that it is incorrect because the “travelling” twin has to accelerate and decelerate, which precludes consideration under SR, nevertheless, the whole introduction of the concept in the first place is predicated upon the assumption that the clocks are “actually” going at different rates. Analysis that includes allowance for the necessary accelerations, e.g. Møller [5a], still concludes (by setting the accelerations to approach infinity) that there is a paradox, which results from the presumed time dilation during the periods at constant relative velocity. [The question of “how do we know which is the moving twin?” is answered by the fact that it must be the one who accelerates.]

**3.2 Interpretation b)**: That there is retardation, but that it is merely a matter of how observers, with clocks moving at constant velocity *v* relative to each other, view each other’s clock – they will each view the other’s going slower than their own by the factor [1- *v* 2/*c*2]−½. This interpretation seems also now to be the opinion of many. For example, Einstein, in his “popular” text [1b], originally published in 1952 - decades after [2] - now says “As judged from K, the clock is moving with velocity *v* ; *as judged from this reference-body* [present author’s italics], the time which elapses between two strokes of the clock is not one second, but [1 – *v*2/*c*2]−½ seconds, …..As a consequence of its motion the clock goes more slowly than when at rest”. That is rather more equivocal than his statement, from [2], given above at § 3.1 and should probably be interpreted as essentially saying “as viewed”. Similarly, in considering the length of a moving rod, he first says “The rigid rod is thus shorter when in motion than when at rest, ..”, but then again “…the length of the rod *as judged from* K’ [present author’s italics] would have been\(\sqrt {1 - {{{v^2}} \mathord{\left/ {\vphantom {{{v^2}} {{c^2}}}} \right. \kern-0pt} {{c^2}}}}\) ”, which would indicate a confirmation of the “mutually as viewed” meaning.

In fact with this interpretation there is no possibility of a twins paradox, because even though the stationary twin would then view the clock of the travelling one to be retarded all the while when moving, when he comes back and is again stationary, their clocks times would then be seen to coincide (taking it that the clocks were synchronous at the start). That is, a twins paradox would not be possible. We would have to have interpretation a) holding for a twins paradox.

## 3.3 A first consideration of clock rates: mutuality

This argument is probably the simplest and most obvious of all:

Einstein says in [2] “It is clear that the same results hold good of bodies at rest in the “stationary” system, viewed from a system in uniform motion.” That is stating a mutuality of observations. Thus, considering two clocks A and B, travelling at *v* relative to one another, they must, according to interpretation a) (and with this mutuality) be “actually” going at a rate which is slower than the other one, and each by the same factor 1/ γ = [1- *v*2*/c*2] ½. *That is quite impossible in physics.* They cannot both be “actually” going slower than the other. [Of course, the quotation says “viewed from”, and that implies interpretation b).]

## 3.4 A second consideration of clock rates: an “in between” clock

A preliminary consideration of clock rates was given above in the context of the first postulate. In this second consideration we take the case of a clock C, identical to the other two, which is continuously between the two, A and B with which we are concerned, and with respect to which each is travelling rectilinearly at *v*, with both either approaching, or receding from, it, and with the three clocks being collinear. [That is not to say that the relative speed of A w.r.t. B is 2*v*, since the relativistic addition of velocities, following from the LT’s, would be applicable.] With respect to both A and B, then, according to interpretation a) of the time relation of SR, the rate of the “in between” clock C must be [1- *v* 2*/c*2]½ = 1/γ times that of each of the other two – that is, logically, those two, A and B, must be going at the same rate.

Mathematically that means that RA= RB, where RX is the rate of clock X:

In a scenario where we consider that C is taken to be the stationary clock (which, in empty space, we are allowed to do) we would say, mathematically,

RA = RC/γ

and RB = RC/γ

which means that RA = RB. That is, the clocks are “actually” going at the same rate.

Thus, in the above context, interpretation a) is seen to be incorrect.

Therefore, interpretation a) has to be discarded and interpretation b) adopted. Thus we conclude that even SR itself predicts that the two original clocks, the stationary and the moving, will be isotachic, since their rates are related by the same factor to that of the third, the “in between”, clock. (The corollary is, of course, that the “in between” clock must be isotachic with the other two.)

There is also, according to STR, the possibility of a dynamic change in the moving clocks, namely that the masses of the elements of the moving clock are increased relative to those of the stationary one, so that the accelerations of the mechanisms will be different, leading to an altered rate. Again we can consider the “in between” clock C: even according to SR, the relation of that clock’s rate to the rates of the other two, A and B, shows that A and B must be isotachic, even including any dynamical effects that there might be according to SR.

## 3.5. A third consideration of clock rates; the rate relations

We consider the clock rate relations of SR that follow from the LT’s. For clocks A and B at rest respectively in two systems S(x,y,z,t) and S’(x’y’z’t’), that are in uniform translatory motion at *v* the relations are, by derivation from the LT’s

and dt’= dt(1 – *v*2/*c*2)½ (2)

where dt and dt’ represent the rates of clocks stationary in reference systems S and S’ respectively. According to interpretation a), then, that means that both clocks are “actually” going slower than the other; *that is impossible*. Mathematically, again, if we divide Eq. (1) by Eq. (2) we get

i.e. dt 2 = dt’2 (4)

and, again, we get the result that the rates of the two clocks are the same, – contrary to what interpretation a) says.

Equally, it is clear that (1) and (2) relations could possibly both be true if

(1 – *v*2/*c*2)½ = 1, i.e. dt = dt’ (however, this infers that *v* = 0 which is trivial, since under that condition the clocks are by definition going at the same rate).

Of course, interpreting them as meaning “as viewed from the other system” is an alternative which is self-consistent, but only with an accompaniment to the interpretation, -namely that the clocks are, objectively, isotachic. [That interpretation means, of course, always allowing for the effect of the observed rate change due to time-of-flight delay of light, with which SR is not concerned.] The “equals” signs in (1) and (2) cannot otherwise be correct - and they follow immediately from the LT’s. We would need to introduce another, different, symbol meaning “equal as viewed”, but that is not available to us.

## 3.6 A fourth consideration of clock rates: clocks being accelerated apart

In a somewhat more physical/mechanical approach, we consider two side-by-side stationary identical clocks restrained by a string from moving apart, but tending to do so by virtue of the force exerted on each by a powerful compressed spring that lies between them, exerting a force on each, directed through their CG’s. Being identical, they must be isotachic.

If the string is now cut, each clock must then experience a force of the same magnitude as the other, and since they are identical, they must experience identical but opposite (and decreasing) acceleration all the while as the spring expands (the First Law). Therefore whatever effect the acceleration has on the clock rates (due to temporary deformation or to effects as accounted for in GR), it must be of the same magnitude for both clocks. (The fact that the accelerations are in opposite directions is of no consequence. The direction of the acceleration does not affect the sense of the change in clock rate, *vide* clocks in a gravitational field, - the clocks are slowed no matter their disposition w.r.t the mass. Considering the equivalence of gravitation and acceleration, it is seen that the change of the rates of our two accelerating clocks will be in the same sense. Analogously, if we consider the origin to be at the centre of the system, then g and x (see § 4.2.1 below) change mutually, (i.e. in this situation a positive x corresponds to a positive g, and vice versa) giving the same sense of clock rate change for each clock. Therefore, at the end of the spring expansion, as the clocks fly apart at constant relative speed rectilinearly, they must still be isotachic.

Although this is only a particular example, it means that we can say there is no reason to hold that clocks in uniform relative rectilinear motion must be going at different rates.