If, as Kevin Brown says in his book on relativity, Einstein's clock adjustment procedure using light signals leads to the same clock adjustment as Newton's clock adjustment procedure using objects, it is worth investigating whether or not Newton's procedure is a synchronization procedure, in other words whether or not it ensures that events with equal time coordinates are simultaneous.
From a purely logical point of view, it is clear that in general Newton's procedure cannot be a synchronization procedure because I have already shown, here and in subsequent posts, that in general Einstein's procedure is not a synchronization procedure.
But it would be nice to be able to see that directly, by examining Newton's procedure more closely.
In Newton's procedure - or more precisely the procedure which is implicit in his laws of mechanics - two clocks A and B in a first spatial inertial system S1 are adjusted to show the same time when identical objects projected in opposite directions from the mid-point between A and B using identical acceleration mechanisms arrive at those clocks, provided the conditions in which the objects are accelerated and in which they travel are symmetrical in both directions. Events with equal time coordinates in S1 are then simultaneous by virtue of the very meaning of simultaneity.
The crucial question is how we can then adjust clocks in a second spatial inertial system S2 that moves relative to the first. According to Newton, we can do that by accelerating the clocks, the objects and the accelerating mechanisms used in S1 so that they come to rest in S2 and performing the same procedure in S2 to adjust distant clocks in that system.
Let's assume that in both frames the procedure is performed in "empty space", notably in the absence of any medium such as air which might disturb the symmetry of the conditions in which the objects are accelerated or travel.
But even in the absence of any medium such as air, after the clocks, the acceleration mechanisms and the objects have been accelerated in one particular direction, are the conditions in which the procedure is then performed in S2 still symmetrical in opposite directions?
Newton evidently thought they were. And it's easy to see why. Newton didn't know that "empty space" is in fact filled with all manner of fields which surround any object. To him, the objects used in S2 would be just the same objects as they were in S1 and the symmetry of the conditions of any further acceleration and movement in opposite directions would not have been disturbed by their acceleration from S1.
That is why he thought that, as measured by Newton-adjusted clocks in S1, two identical objects at rest in S2 would undergo the same acceleration if they were projected in opposite directions using identical acceleration mechanisms. Today we know that this is not the case.
And that does not come as a surprise given that today we know that all objects are surrounded by, for example, electromagnetic fields. We also know that disturbances in such fields need some time to travel over any closed path.
Let's take the example of a simple electron whose field is spherically symmetrical in S1 and let's see how that field is affected if the electron is accelerated locally, for example as the result of a collision.
Qualitatively, and without adjusting any clocks, any field disturbance can be expected to propagate symmetrically around the electron's original position of rest, but the movement of the electron in one particular direction then creates asymmetrical conditions in the field surrounding it.
Quantitatively, let's assume that at a time t1 in S1 an electron at rest in S1 and surrounded by a spherically symmetrical field undergoes such a sudden acceleration to a speed v.
The field is represented here by a series of concentric spheres surrounding the electron. At the time t2 > t1 in S1, the electron moves at v and information about its acceleration is spreading outwards through the electron's field. Each of the spheres surrounding the electron starts to move at v when the information about the acceleration reaches it, leading to the following situation at the time t2:
As the information spreads further, the area around the electron in which the sphere density is asymmetrical grows:
At the boundary between areas of different density, shown in red, an electromagnetic signal propagates away from the electron (for details on how this happens, see Lesson 4 in this document by Daniel V. Schroeder). Any further sudden acceleration of the electron would give rise to a new electromagnetic signal. And the light speed law appears to be that any acceleration information always traverses the same number of spheres in the same amount of time. This means that locally in S1 the speed of an electromagnetic wave is independent of the speed of the light-emitting body.
The sphere model illustrates the fact that if the field around an electron, and thus around any object containing electrons, is symmetrical, then the field around the same electron accelerated locally to a speed v is no longer symmetrical. And for that reason Newton's clock adjustment procedure applied in a second frame S2 moving at a speed v relative to S1 is not a synchronization procedure.
At least not locally. Over cosmic distances, Newton's or Einstein's procedure may well be applicable to sets of clocks that move at a speed v relative to one another. The reason for that is that clocks that are very distant from each other may be moving relative to each other as a result of certain kinds of acceleration which disturb the local fields around them only negligibly. As far as I understand it, cosmic expansion and gravity are examples of such locally non-perturbative accelerations:
Locally, however, all particles or objects are subjected to the same gravitational or cosmic expansion influences. Therefore, if they move relative to each other locally then that must be the result of previous local accelerations which have disturbed the electric fields surrounding them.
My investigation into whether or not Newton's clock adjustment procedure is a synchronization procedure has led me to develop my sphere model of electricity and light propagation a bit further in this post.
After these flights of fancy, it is time to test my thoughts against the best of the literature on simultaneity in Newton's and Einstein's theories. And that leads me straight back to Kevin Brown's book Reflections on Relativity. In my next post.
In his 1905 article "On the Electrodynamics of Moving Bodies", Einstein explained how clocks can be adjusted "in a coordinate system in which Newton's equations of mechanics hold" (p. 892).
There is a problem with this formulation since, if we have not already adjusted any distant clocks in a given frame of reference, we have no way of knowing whether or not Newton's laws hold in that frame, not to mention the fact that in special relativity Newton's laws do not precisely hold anyway.
So, what kind of coordinate system was Einstein really trying to identify? The answer given by Kevin Brown in his book Reflections on Relativity (2010) is that he was trying to identify "inertial coordinate systems", defined by Brown as systems "in terms of which inertia is homogeneous and isotropic, so that free objects move at constant speed in straight lines, and the force required to accelerate an object from rest to a given speed is the same in all directions" (p. 27).
This quote requires some unpicking.
First, what is "inertia"? According to Brown, the "principle of inertia" is the idea that "in the complete absence of external forces, an object would move uniformly in a straight line, and that, therefore, whenever we observe an object whose speed or direction of motion is changing, we can infer that an external force... is acting upon that object". Brown says this principle was first formulated in the 17th century and represents "the most successful principle ever proposed for organizing our knowledge of the natural world" (p. 22).
Of course, Brown's explanation of the principle of inertia introduces some new concepts that may be deemed to be in need of explanation, such as "uniform motion", "straight line" and "force". However, it seems to me that these can in turn be explained without resorting to any laws of mechanics and even without first adjusting distant clocks: "straight lines" can be equated with lines of sight in the absence of very massive bodies, such as stars; "uniform motion" can be established using signals emitted at regular intervals from a moving body; and instead of "in the absence of external forces" we could say "in the absence of specific sets of conditions, such as collisions, known to stop objects from moving uniformly and in straight lines".
So, a "spatial inertial coordinate system" could be defined as a series of markers which are arranged in straight lines at regular spatial intervals, established for example by means of standard rods, and relative to which objects that are not subjected to any force move uniformly in straight lines.
What is missing from this definition is time coordinates or, what amounts to the same thing, a procedure to adjust clocks in distant points.
As Brown points out, such a procedure to adjust distant clocks is implicit in Newton's "laws" of mechanics, which thus turn out to be not pure laws of nature but in part the result of a particular clock adjustment choice.
This choice was to use identical mechanisms to propel identical objects in opposite directions from the mid-point between two clocks at rest in a spatial inertial coordinate system and to set those clocks to the same time when the objects arrive there.
Once this has been done, the "speed" of such an object propelled by such a mechanism can be measured using those two clocks and the distance between them. Knowledge of that speed can then be used, together with knowledge of the distances between other clocks, to adjust any other clocks at rest in the same spatial inertial coordinate system and thus to establish a full inertial coordinate system. It is found empirically that the time coordinates in such a coordinate system do not depend on the kind of object or acceleration mechanism used to adjust distant clocks.
It is then clear that in such a coordinate system inertia is "isotropic", in other words if identical objects at rest in the system are accelerated by the same kind of mechanism in different directions, they reach the same speed. This is guaranteed by the way we have adjusted our clocks. And this clock adjustment convention, or choice, or decision, was written into Newton's second "law", which states that the "change of motion is proportional to the motive force impressed, and is made in the direction of the right line in which that force is impressed" (quoted in Brown, p. 23).
Newton's clock adjustment procedure may well seem perfectly plausible even to a modern mind: if we use identical mechanisms to propel identical objects in opposite directions along paths of equal length, then we may surely assume that they arrive at the end of those paths at the same time, by virtue of the very notion of simultaneity. So it seems that we are entitled to adjust clocks using Newton's procedure. Indeed, it would appear that we are obliged to do so if we want to synchronize our clocks, in other words if we want equal time coordinates to express a relationship of simultaneity.
But then Brown says on page 32 that Newton's procedure is empirically found to lead to the same clock adjustment as Einstein's light signal procedure, and I have previously stated that in general Einstein's procedure is not a synchronization procedure. Was I wrong? My next post will tell.
In their article on "A Test Theory of Special Relativity: I. Simultaneity and Clock Synchronization" (1977), Reza Mansouri and Roman U. Sexl raise the question of whether there are any meaningful alternatives to applying Einstein's clock adjustment procedure in every uniformly moving frame of reference. In particular, they consider whether there might be a preferred "ether" frame in which it would be sensible to apply Einstein's procedure while all clocks in every other frame would be adjusted based on clock readings in the preferred frame.
The authors begin by suggesting that the cosmic background radiation defines a system of reference which is a candidate for a possible "ether frame", but they do not explain how or why that might be the case.
Later in the article, the authors refer to a frame Σ in which Einstein's clock adjustment procedure and the slow transport clock adjustment procedure agree as an "ether frame". In the slow transport procedure, clocks in different locations are adjusted to the time shown by a single clock that is "slowly" moved to those locations. The slowness of the clock has to be established either by an existing set of clocks that have already been adjusted somehow, or by using the clock's "self-measured" speed: the distance it has traversed divided by the time that has passed as shown by the clock itself.
Once again it is not clear in what sense such a reference frame would constitute an "ether frame" if by "ether frame" the authors mean a frame in which Einstein-adjusted clocks are synchronized.
What is more, any suggestion that slow clock transport provides an independent way of adjusting distant clocks is somewhat undermined later in the article when the authors show that slow transport clock adjustment is equivalent to Einstein clock adjustment if and only if the empirically determined time dilation factor in Σ is given by its relativistic value. A slightly modified version of their proof can be found here.
In a nutshell, Mansouri and Sexl end up without any method of identifying a "preferred" system of reference in which we can be sure that Einstein-adjusted clocks have been synchronized at least locally.
They nevertheless consider what happens if we arbitrarily choose a frame Σ in which clocks are Einstein-adjusted and then adjust clocks in all other uniformly moving frames by comparing them with clocks in Σ, a procedure the authors call "external synchronization".
Specifically, they consider in some detail what happens if we adjust clocks in all other frames by setting them to zero when they pass a clock in Σ which shows zero:
In this diagram, the time in Σ is T = 0 everywhere and all clocks in S take over that time from clocks in Σ as they pass them.
Mathematically, this amounts to setting the conventionality factor ε in the transformation equations between coordinates X, T in Σ and coordinates x, t in any frame S moving at v relative to Σ to ε = 0, yielding
t = aT
x = b(X - vT)
where a and b are the empirically determined time dilation and length contraction factors.
This transformation establishes "absolute simultaneity" according to the authors, by which they mean that events that have the same time coordinates in Σ also have the same time coordinates in S. It is important to note that the authors use the word "simultaneity" in a purely technical sense here to refer to equal time coordinates as measured by clocks which have not necessarily been synchronized.
Mansouri and Sexl observe that, if clocks are adjusted in this way, for an observer at rest in Σ moving clocks are slow and moving measuring rods shrink. Seen from a system S that moves relative to Σ, on the other hand, clocks in Σ are fast and measuring rods elongated, purely as a result of the way in which the clocks used to measure time have been adjusted. What is more, superluminal speeds are not ruled out and light moves at different speeds in opposite directions in all frames except Σ. These results, which can be formally deduced from the transformation equations, do not contradict special relativity because they depend on the way in which clocks have been adjusted.
So, is the choice of the conventionality factor just arbitrary and is any ε as good as any other? Mansouri and Sexl don't think so. With reference to special relativity's twin paradox, they point out that in the case of ε = 0 the explanation for the twins' difference in age varies depending on whether the unaccelerated twin is at rest in Σ or in some other frame S. The fact that the end result is the same "appears as a fortunate coincidence without deeper reason", and this "shows how the choice of unsuitable conventions can destroy the internal symmetry of a physical theory".
What the authors suggest here is that applying Einstein's clock adjustment procedure in all frames, which amounts to choosing ε = -v/c2, helps to reveal the "deeper reason" why the twin experiment has the same outcome regardless of which frame is the rest frame of the unaccelerated twin. This "deeper reason" is presumably the assumed equivalence of all uniformly moving frames of reference.
Against this, I would argue that neither Einstein clock adjustment in all frames nor the choice of ε = 0 in an arbitrarily chosen frame in which clocks have been Einstein-adjusted can reveal any "deeper reason" for the twin experiment outcome because both of these adjustment choices are just that: choices, decisions or conventions. In particular, it turns out that the special relativistic equivalence of the behaviour of moving clocks and measuring rods in all frames is not a fundamental fact of nature but an artifact created by a particular choice of clock adjustment.
It is true that this choice makes for a theory with great "internal symmetry", and maybe that's what makes it attractive to many. But, I would suggest, any investigation into the "deeper reason" for the twin experiment outcome would have to explore different avenues, such as the effects of different kinds of acceleration on matter or fields.
Despite this little niggle, it is clear that Mansouri and Sexl's article offers much greater insights into the role of convention in special relativity than any of the textbooks I have discussed in this blog. Like Einstein, the authors are perfectly clear about the fact that clock adjustment decisions are conventional. They go further by isolating what I have called the "conventionality factor" in the transformation equations between a first frame in which clocks have been Einstein-adjusted and other frames in which clocks are adjusted by some other means. They show that this factor ε represents an arbitrary element in the transformation equations as opposed to the empirically determined time dilation and length contraction factors a and b. And they show that this factor is the same for Einstein clock adjustment and slow clock transport adjustment if and only if a is given by its relativistic value.
All this is fascinating stuff. But there is one fundamental issue which the authors fail to address: which, if any, of the various clock adjustment procedures they discuss actually synchronizes distant clocks, and over what kind of distance? The authors vaguely suggest that there might be a preferred frame, possibly determined by the cosmic background radiation, in which Einstein's clock adjustment procedure can be deemed to synchronize any distant clocks. But they do not even hint at why or how that might be the case.
In previous posts I have explained why in general clocks in special relativity are not synchronized, why this leads to conceptual mayhem, and why the problem cannot simply be resolved by redefining "simultaneity". The issue can be illustrated by means of two clocks A and B spaced a few metres apart. I adjust the clocks by setting clock A to 1 o'clock and then walking over to clock B and setting clock B to 2 o'clock. I can then walk from clock B to clock A "arbitrarily fast" even if I take extended breaks, and I can even arrive at A "before I set off" from B, thus "reversing cause and effect", according to the clock readings on the two clocks. All this is, of course, a result of the fact that the two clocks have not been synchronized. So far none of the texts I have discussed addresses the basic distinction between clock adjustment procedures, which can be arbitrary, and clock synchronization procedures, which must meet additional criteria.
Enter Kevin Brown, who writes about the role of convention in relativity repeatedly and in depth in his recent book Reflections on Relativity (2010). What is more, Brown carefully develops his thoughts by charting the historical journey from Galileo and Newton to Einstein while at each stage discussing the logical structure of the theories set out by those scientists from a modern point of view.
Discussing Brown's analysis of the role of convention in relativity will therefore take some time. I'm going to make a start in my next post.
