So, in what sense does the application of the Einstein clock adjustment procedure to sound signals lead to nonsensical results, and why?
As we have seen, it leads for example to the result that walking speeds become infinitely fast in some frames of reference, and that, by speeding up a little bit more, it becomes possible to walk into the past in such frames. The concepts of one-way speed and causality thus break down. In purely mathematical terms, people would even be able to cause serious causality paradoxes by walking into the past and then turning round and arriving back home before they left. Of course, this is not actually possible because it would be physically impossible to move back at a sufficiently high speed - ultimately because the mathematical "plane of simultaneity" defined by such a clock adjustment procedure is not actually a plane of simultaneity at all. The concept of simultaneity thus breaks down.
It therefore seems that the Einstein synchronization procedure cannot be applied to sound signals. But what about the idea that synchronization is just a matter of definition? In other words, prior to deciding on a synchronization procedure, how can we know that, from the point of view of an observer moving relative to a body of air, the one-way speed of sound is not equal in different directions?
First, we can use a limited concept of one-way speed that is independent of any synchronization procedure: the one-way speed of a first signal sent out from A is higher, equal to or lower than that of a second signal sent out from A at the same time if the first signal arrives in B before, at the same time as or after the second, respectively. Clocks in A and B do not need to be synchronized to make such observations. The concept of one-way speed in a single direction is thus well-defined in comparative terms - lower, equal or higher - regardless of any synchronization procedure.
Now consider an observer A who is stationary with respect to a body of air. This observer will find empirically that different signals sent out from A, including sound signals, exhibit the same relative one-way speeds as defined above regardless of the direction in which they are sent out. An observer B moving relative to the same stationary body of air, however, will find that sound signals sent out from B in the "forward" direction travel more slowly compared to the same range of other signals sent out from B, while sound signals sent out in the "backward" direction travel faster compared to the same range of other signals. The second observer thus has good reasons to conclude on empirical grounds that the one-way speed of sound is not equal in the forward and backward directions even prior to defining any synchronization procedure.
But there is more. Our theoretical knowledge of how sound waves propagate, too, tells us that, for an observer travelling relative to a body of air, the one-way speeds of sound in the forward and backward directions cannot be equal. We know, for example, that sound consists of outwardly propagating vibrations of air molecules. This means that, for an observer who is stationary relative to a body of air, the conditions in which sound propagates are symmetrical in all directions, while for an observer moving relative to a body of air the conditions in which sound propagates are not symmetrical. We can therefore expect the one-way speed of sound to be equal in different directions for the first observer and unequal for the second.
And there is even more. For any signal, if we believe, assume or stipulate that the conditions in which it propagates are symmetrical for a given observer, then it is difficult to see how they can also be symmetrical for a second observer who is moving relative to the first. This seems fundamental to me and it goes to the heart of my difficulties in getting to grips with the special theory of relativity.
I have no particular problem with the idea that acceleration may lead to the slowing down of clocks or even the contraction of measuring rods. The empirical finding that the two-way speed of light is always measured as c, if indeed there is such an empirical finding, is somewhat surprising but may be explicable precisely as a result of the effects of acceleration on clocks and measuring rods - in fact, if there are such effects, we may choose to correct for them by adjusting the rate at which clocks that have undergone acceleration tick, resulting in the two-way speed not always being measured as c.
But what I cannot yet fathom is on what grounds generations of physicists have seen fit to declare that, if the conditions in which light propagates are deemed to be symmetrical relative to a source of light, the one-way speed of light is deemed to be the same in all directions even for an observer who is moving relative to that source of light. Or why they would want to adjust clocks in such a way that it is made to be the same in all directions for such an observer.
What I have endeavoured to show thus far is that, while of course physicists are free to adjust clocks in this way, they are not free to declare that such a clock adjustment procedure is a synchronization procedure. And that if it isn't the concepts of simultaneity, one-way speed and causality may be undermined. Ultimately, then, it may turn out that special relativity is certainly a usable theory in that it successfully predicts the outcome of experiments, but that it relies on a clock adjustment procedure which is not a synchronization procedure and that therefore the concepts of one-way speed, simultaneity and causality have limited applicability in the framework of that theory.
I think it is now time for a few propositions relating to the role played by definition or stipulation in the principle of the constancy of the speed of light for every observer in the special theory of relativity.
P1: The constancy of the two-way speed of light in every coordinate system in which the laws of mechanics hold is an empirical matter once the rate at which clocks in different such coordinate systems tick has been set.
P2: The constancy of the one-way speed of light in every direction in every coordinate system in which the laws of mechanics hold is an empirical matter once a synchronization procedure for clocks in different locations in different such coordinate systems has been decided on.
P3: Not every clock adjustment procedure is a clock synchronization procedure.
P4: A clock adjustment procedure resulting in clock settings under which a signal sent from a location A may arrive in a location B "before it was sent" is not a synchronization procedure.
P5: A clock adjustment procedure that relies on sending a signal from location A to B and back and sets the signal arrival time in B to be in the middle between the signal sending and arrival times in A is a synchronization procedure only if the conditions in which the signal propagates are symmetrical in both directions.
So, is the Einstein clock adjustment procedure using light signals a synchronization procedure? On the face of it it is difficult to see how it can be, but once again, the answer depends on a number of empirical and theoretical issues which I have yet to look into in any detail over the months and years to come.
Before I embark on this quest, I think it's high time I started reviewing what the physics literature has to say on the role of definition or stipulation in the principle of the constancy of the speed of light for every observer - in my next blog post.
