dingle so-called-paradox

One of the fundamental puzzlers about special relativity, is the "dingle paradox", named in honor of one of its earliest and most persistent advocates, who thought it proved SR flawed. In essence, it is the puzzle of how two clocks in relative motion can each be slower than the other without a "prefered reference frame" to act as a tie-breaker.

You might try this geometrical explanation or this one based on football yardage.

Or we can look at the scenario as Dingle layed it out. We'll do that here, concentrating on showing why it is inaccurate to claim that prefered-frame theories are supported by the paradox while SR is contradicted. Here's Dingle's definition of terms from a 1967 re-presentation of his position.

    A and H are two relatively stationary, regularly running, clocks.  
    B and N are two similar relatively stationary, regularly running
    clocks, moving with uniform velocity v with respect to A and H. 
    (The distances AH and BN are independent, and arbitrary.)  A and H
    are set so that a pulse of light which leaves A when A reads T1 and
    is instantaneously reflected back from H when H reads T2, returns to
    A when A reads T3=2T2-T1.  N is set similarly in relation to B. 
                       --- Dingle, Nature, vol 216,p119

So far so good. Now he describes the events at which we read the clocks, and nails down their motions and orders

    event E0:          N                 B t'=0
                                         A t=0         H

    event E1:                        N                 B
                                         A             H

    event E2:                            N                 B
                                         A             H

                       --- Dingle, Nature, vol 216,p119

And here is a spacetime diagram of this scenario

So the point is, when you compare clocks using the AH frame, you are comparing at event E1, and using the red colorcoded coordinates. When you compare clocks using BN frame, you are comparing at event E2 and using the blue colorcoded coordinates.

Dingle's claim is that this proves special relativity wrong, and lorentzian ether theory right. But if you analyze the physical situation described above, LET provides exactly the same predictions about clock readings as the SR predictions Dingle complains about.

The SR case is simple. We can just use time dilation and see that A reads T and N reads T*gamma at E2, while B reads T and H reads T*gamma at E1. The two comparisons are inverse. And you can see that the rule is, the surrogate clock (that is, H or N used as a stand-in for A or B) always has the the larger reading.

But let's demonstrate that LET yields the same answer. We are given the clock readings of A and B are zero when they cross. What is to be determined is the clock readings of H and B at E1, and the readings of A and N at E2, using only a theory with clock slowing by gamma relative to an absolute velocity (and for simplicity, assume A and H have zero absolute velocity).

We take care to do all calculations using absolute velocity, and to calculate clock rates based ONLY on absolute gamma. We will discuss "time" or "same time" in Ether Standard Time only.

So. At the Ether Standard Time of E0, we know that A and B both read zero. But we do NOT know what N and H read at that (Ether Standard) time. For that, we must use the synchronization condition above. Since A and H are motionless in the ether, H's reading is simple: it was set to zero at an Ether Standared Time of

                 (AH-AH)/2 = 0

So H is zero also. But since B and N are in absolute motion, the the Ether Standard Time when N was set to zero is:

                 (BN/(1+v)-(BN/(1-v))/2
                 (BN-BN*v-BN-BN*v)/(2(1-v^2))
                 -(BN*v)/(1-v^2)

So N is NOT zero. Indeed, at the time of event E0, N reads ( (BN*v)/(1-v^2) )*sqrt(1-v2) = (BN*v)/sqrt(1-v^2)

Note well: no SR "relativity of simultaneity" here; for purposes of this derivation, we all know there is no simultaneity but simultaneity, and the Ether is its prophet. We've just used absolute velocity, and absolute clock slowing, and followed Dingle's recipe for setting H and N given the values of A and B.

Now we are ready to calculate the readings of H and B at E1:

       reading on H:         (AH/v)
       reading on B:         (AH/v)*sqrt(1-v^2)

       Using H as a surrogate for A we have                     (A/B) > 1

and the readings of A and N at E2:

       reading on A:         (BN/v)
       reading on N:         ((BN*v)/sqrt(1-v^2) + (BN/v)*sqrt(1-v^2))
                             ((BN/v)*v^2 + (BN/v)*(sqrt(1-v^2)^2)/sqrt(1-v^2)
                             (BN/v)(v^2+(1-v^2))/sqrt(1-v^2)
                             (BN/v)/sqrt(1-v^2)

       Using N as a surrogate for B  we have                    (A/B) < 1

Each clock "slower" than the other. One slower when compared at E1. The other one slower when compared at E2.

In short, "each clock slower than the other" is a misleadingly ambiguous way of stating a precisely defined relationship between clocks and coordinates. The dingle so-called-paradox simply does not show prefered-frame theories any more consistent than special relativity.