First, let's see how Mansouri and Sexl approach the issue. They begin by assuming that Einstein's clock adjustment procedure has been applied in a first frame of reference Σ. The authors then consider a second frame S whose origin moves at v < c with respect to Σ, as measured with standard rods and clocks in Σ. They assume that the relationship between coordinates (X, T) in Σ and (x, t) in S is linear and they express that relationship in the following form:
t = aT + εx
(1)
x = b(X - vT)
It is striking that, rather than opting for a standard linear transformation in which t and x are represented in terms of T and X:
t = AT + BX
(2)
x = CT + DX
Mansouri and Sexl choose to present t as a function of T and x. (1) and (2) are equivalent in the sense that a, ε, b and v can be expressed in terms of A, B, C and D and vice versa. The reason why the authors use (1) rather than (2) is that the factors a, b, v and ε have special significance. As Mansouri and Sexl point out, a and b represent "the time dilatation and length contraction factors", which may depend on v and are a matter of empirical fact, while "arbitrary values of ε can be achieved" by choosing a suitable clock adjustment mechanism.
To see why b can be interpreted as a "length contraction" factor, consider the following diagram:
Figure 1
In this diagram, clocks in Σ have been Einstein-adjusted, so in Σ time coordinates are well-defined in any location. In S, by contrast, distant clocks have not yet been adjusted, so we can only specify a time for a single clock. We choose that clock to be the one located at x = 0 and set it so that (x, t) = (0, 0) in S and (X, T) = (0, 0) in Σ coincide.
Now, it may be found empirically that any x differs from the X with which it coincides by a particular factor. Assuming that movement at v relative to Σ affects measuring rods at any location in S equally compared to identical measuring rods in Σ, this length contraction/dilation factor is the same for any pair of coinciding coordinates x, X in Figure 1 and also for any coinciding lengths l in S and L in Σ. Inserting T = 0 in (1), we obtain x = bX, so b is that length contraction/dilation factor.
To see that a is the time dilation factor, consider the following diagram, which shows the situation of Figure 1 after the Time T1 has passed in Σ:
Figure 2
It may be found empirically that t1 differs from T1 by a particular factor. Assuming that movement at v relative to Σ affects clocks equally at any location and time in S, this time contraction/dilation factor is the same for any single clock in S passing a series of clocks of identical construction in Σ. Inserting x = 0 in (1), we obtain t1 = aT1 , so a is that time contraction/dilation factor.
And that's all in the relationship between S coordinates and Σ coordinates that is a matter of empirical fact. How we set the time on clocks in S in locations other than x = 0 is a matter of choice, preference, decision-making, convention... and that choice is expressed mathematically in the value of ε, which may in fact be chosen to be a function of x as well as v. In terms of the factors occurring in (2) it can be expressed as ε = B/D, while b = D, v = -C/D and a = BC/D - A.
As Mansouri and Sexl point out, the value of ε in special relativity is -v/c2. This value can be called "relativity's conventionality factor". It was chosen by Einstein because it ensures that the one-way speed of light is the same for every observer in every frame of reference, which in turn ensures that the laws of physics take on a particularly simple form.
But what if a different value is chosen, say ε = 0 for any reference frame moving at any speed v relative to Σ? Mansouri and Sexl state that this particular case leads to a theory maintaining absolute simultaneity. What they mean by that is that, in the framework of this theory, two events that have the same time coordinate in a frame S moving at v relative to Σ also have the same time coordinate in any other frame S' moving at v' relative to Σ. In other words, if two events have coordinates (t1, x1) and (t2, x2) in S and (t1', x1') and (t2', x2') in S', then t1 = t2 if and only if t1' = t2'. This follows from ε = 0 in (1) by first transforming t1 and t2 into T1 and T2 in Σ and then T1 and T2 into t1' and t2' in S'.
Mansouri and Sexl add that a theory in which the conventionality factor is set to ε = 0 is "equivalent to special relativity" in the sense that if the outcome of an experiment can be predicted correctly using special relativity, then it can also be predicted correctly using a theory in which ε = 0, and vice versa. This is no doubt true because the conventionality factor merely expresses how we have chosen to adjust distant clocks and that does not in and of itself tell us anything about the world.
Nevertheless, Mansouri and Sexl dismiss the choice of ε = 0 as an "unsuitable" convention. Their reasons for this conclusion and the implications for the role of convention in the light speed principle will be the subject of my next post.
