I give first a brief presentation designed to facilitate a reply to Professor Dingle’s present statement and to the one he gave in 1962. So far as applicable, I use Dingle’s present notation (which is not identical with what he used in 1962).

Let Ox be a rigid rod graduated in the usual way; let similar clocks be fixed to the rod at points along the rod, and let them be synchronized by a standard procedure (that described by Dingle). [Derived from the two inertial frames in Newtonian physics whose origins are Ox and O′x′ and whose cartesian coordinates are x, y and z, and x′, y′ and z′ – PB.] If anything happens at the position x of any of the clocks, let t be the reading of that clock at the event E, say. We speak of the event E as the event (x, t). Let O’x′ be the second rigid rod in motion along Ox with uniform velocity v (v not equal to 0). Let O’x′ be graduated in the same way as Ox; let clocks similar to those attached to Ox be fixed to O’x′ at points along O’x′, and let them be synchronized amongst themselves by the standard procedure. If anything happens at the position x′ of any of these clocks, let t′ be the reading of that clock at the event E,’ say. We speak of the event E’ as the event (x′, t′).

According to the theory of special relativity, this system is possible, supposing Ox, O′x′ to belong to inertial frames [coordinate systems] K, k, say. The theory then asserts that E, E’ are one and the same event if and only if the parameters satisfy the relations:–

where a = (1 − v²/c²)^½, supposing 0 < a < 1 and supposing the zero points of the various quantities are suitably chosen [and with a being the coefficient for synchronizing clocks in the two inertial frames]. This is one way of writing the Lorentz transformation (being the one used by Dingle in his earlier paper).¹

Consider in k the particular clock B permanently fixed at O’, so that every event at B has x′ = 0. Then from (ii) for every such event

(Take, for example, the case a = ½. Equation (iii) means that if clock B reads t′ then the K clock past which B is moving past reads 2t′; at 1 o’clock by B it passes a K-clock reading 2 o’clock, at 2 o’clock by B it passes a K-clock (a different one, naturally) reading 4 o’clock, and so on.)

In the immediate operational interpretation of (iii), as just illustrated, t′ is the reading of one and only one clock and t is the reading of a different clock for each different value of t. I repeat that, so far as our discussion is concerned, every event to which (iii) applies happens to clock B.

If we next consider in K the particular clock A permanently fixed at O, then every event at A has x = 0 and from (i) for every such event

This is obviously what we expect from (iii) because now K and k have exchanged roles. In (iv), t is now the reading of one and only one clock and t′ is now the reading of a different clock for each different value of t′. Manifestly the parameters t, t′ do not have the same meanings in (iii), (iv). Every event to which (iii) applies happens to the clock B; every event to which (iv) applies happens to the clock A.

(If we do require both (iii) and (iv) to hold good we get simply t = 0 = t′, since a^² is not equal to 1. That is, (iii), (iv) are both satisfied for the unique event that happens to both clock A and clock B, namely their single mutual encounter. This is obviously entirely consistent with what has just been said.)

No particular or preferred observer is concerned in these results. If a cine-camera anywhere in any state of motion takes a sequence of pictures of clock B, each picture will show clock B with some reading t′ and, adjacent to B, a K clock reading t′/a, the K-clock being a different one in each picture. If the same, or any other camera takes pictures of A, each picture will show A with some reading t and, adjacent to A, a k-clock reading t/a, the k-clock being a different one in each picture.

I turn now to Dingle’s allegation that the theory used above “must be false.” In his present paper, this is based simply on his claim to have inferred the contradictory statements (3) and (4) of his paper from the theory. So we have to do only with the logical consistency of the theory. It may help if I enumerate a sequence of arguments; the first alone is sufficient to refute Dingle’s contention, but I hope the rest throw further light on the subject as a whole.

1. Dingle’s assertion is obviously and demonstrably wrong. Using no more than the Lorentz transformation in his algebra, he claims to derive two different values for the same quantity. But the transformation is linear and any result it gives can only be unique. It is trivially impossible for it to give two different answers to the same question. If Dingle obtains two different answers it must be because a) he has made a slip in the algebra, or b) his quantities are not well defined, or c) what he treats as the same quantity are two different quantities.

2. Dingle has not made any mistake in his algebra, but in his present paper he deals with objects to which the theory especially denies a meaning. We consider event E₀, E₁, E₂ defined and described in frames K, k as follows (these being apparently the events similarly denoted by Dingle):

Here, and in physics generally, events means something happening at a particular position at a particular instant. The crucial feature is that an observer experiences an event if, and only if, the event is part of his own history, that is the event is in his own world-line.

In Dingle’s system in his present article A and B are the only observers who experience the event E₀, or are “at” the event E₀; H and B are the only observers at E₁; A and N are the only observers at E₂. Dingle arrives at his conclusions because in practice he does not adhere to the standard concept of an event. He asserts, “the reason why A must be held to read t₁ at E₁ is that H reads t₁ at this event, and on this theory the process by which a is set in relation to H synchronizes it with H.... The reason why B must be held to read t′₂ at E₂ is....” A is not “at” E₁ in any sense admitted by the theory. It simply has no meaning whatever within the theory to speak of what a must be held to do at E₁. B is not at E₂ and it has no meaning to speak of what B must be held to do at E₂.

Just before his formula (3), Dingle proceeds to state “between events E₀ and E₁, A advances by t₁....” Because A is never at E₁, the phrase is meaningless and so Dingle’s (3) is meaningless. Correspondingly his (4) is meaningless.

3. Naturally there is an event E_1A, say, at which A reads t₁. This event has x = 0, t = t₁ and so clearly E_1A is not equal to E₁, thus corroborating what has just been said.

4. Dingle’s language requires a meaning for what the clock A reads “at” some event involving B even though A and B are not adjacent. Indeed, Dingle expressly uses this phraseology in his 1962 paper. But this restores the notion of distant simultaneity.

About the first thing that relativity theory does is to deny any operational meaning to the notion of simultaneity at two different places. Naturally, this fundamental feature in the theory is not affected in the slightest by any arbitrary conventions we may adopt for the synchronization of clocks. The latter is merely a particular way of putting the readings of two relatively stationary clocks into 1–1 correspondence with each other.

5. While Dingle’s (3) and (4) are meaningless as they stand, the quantities involved can of course be assigned operational meanings in terms of readings of the relatively moving clocks A, B. The formulae do not then tell us about the “rates” of the clocks. They become simply two different ways of putting the readings of A, B into 1–1 correspondence with each other. There are infinitely many different ways of doing this! Being no more than ways of attaching labels, there can be no question of any two these ways being “contradictory.”

6. In his 1962 paper, Dingle started from equations (i), (ii) as we have written them (but in his earlier notation) and then derived precisely our equations (iii), (iv). He then asserted, “every symbol has exactly the same meaning in both cases,” and he claimed to infer a contradiction. His assertion is false, because here he is not talking about the same thing, but two different things.

1. Dingle, H., Nature, 195:985 (1962)
2. Born, M., Nature, 197:1297 (1963)

McCrea’s reply to my disproof of Special Relativity is both gratifying and disappointing. It is good that, at long last, some comment has appeared; regrettable that this one contains nothing to the point.

One single thing only is needed to refuse the disproof, and it is essential – to show an error in the derivation of my equation (4) that does not invalidate my equation (3). This I showed with unmistakable clearness. McCrea’s only contribution is the following: “Dingle’s (3) is meaningless. Correspondingly his (4) is meaningless.” This, if it were true (it is not), would merely kill the theory in another way for (3) is Einstein’s deduction and that of his followers until now.

Because this conclusively nullifies McCrea’s rejoinder, I should leave the matter here, with a final appeal to him to agree frankly that the theory is untenable, but for the fact that the overlooking of the irrelevant bulk of the statement would, in the prevailing sate of thought be misinterpreted. Nature’s prediction that “the chances are that most people would be persuaded by what McCrea has to say” would only too probably be verified. It is the general view that Relativity is beyond the understanding of most, but must be accepted because some mathematicians, who alone understand it, endorse it. Criticism of it, being on this view merely a sign of incomprehension, can therefore be ignored if a sufficiently imposing mathematical dismissal, intelligible or not, is forthcoming.

I distinguish clearly between a) the mathematics and b) the identification of the mathematical symbols with observable quantities. I have enough mathematical insight to see that it is a waste of time to look for mathematical flaws in the theory. Hence McCrea’s argument (1) which he says is enough to refute my contribution does not touch that contention. Of course, equations (iii) and (iv) (my (1) and (2)) concern two distinct sets of events, and so they cannot contradict each other. But what McCrea has to show and has not shown, is why the physical result (3), deduced from one set, can be shown to be true, while the physical result (4) similarly deduced from the other (non-contradictory) set, must be held to be false.

I agree that my equations (1) and (2) are mathematically free from contradiction; I also agree that it is perfectly possible (but not necessary) that if the experiment were made the clocks described would give readings conforming to (1) and (2) (in which case we would have to accept Lorentz’s theory). But, what is impossible is that, in that case, with the settings H and N in relation to A and B, synchronizing the pairs AH and BN would be such as to entitle us to infer his reading of A for a point on H (thus yielding (3)) and a similarly determined reading of B for an event on N (thus yielding (4)).

McCrea’s comments on the essential point of synchronization are revealing. If Einstein’s comparison were, as he says, one of “two different ways of putting the readings A, B into 1–1 ‘correspondence’ with each other, the whole theory would be an idle mathematical fancy, and a space traveller would return either older or younger than his twin brother according to the capricious choice of ‘correspondence.’” This, which I have clearly pointed out, McCrea ignores. Here we have another equivoque of the kind which I exemplified by the oscillating interpretation of the ‘FitzGerald contraction.’ Either Special Relativity says nothing physical, or its (physical statements) are contradictory. To establish his choice regarding asymmetrical ageing, McCrea supposes that it does say something physical about ageing, but selects only one of its contradictory points. When asked to face the other point he shifts his ground and denies the theory has said anything at all.

The passage that shows the deepest misconception is this: “About the first thing that Relativity Theory does is to deny any operational meaning to the notion of simultaneity at two different places. Naturally, this fundamental feature of the theory is not affected in the slightest by any arbitrary conventions we may adopt for the synchronization of clocks.” What Einstein pointed out, and this was his great insight into the matter, was that there was no natural meaning of simultaneity; one could freely be given by definition. This released him from the previously assumed necessity to assign distant times in conformity with Galilean kinematics and enabled him to base a theory on whatever operational definition he choose. But when the observations which it requires is seen to be that one clock goes both faster and slower than the other, you cannot plead that this is only because of arbitrary definition, and if you want to regain your freedom to choose another you must repudiate the theory and start again from scratch.