Special Relativity II: A Fatal Flaw?

This post revisits some of the arguments in “Special Relativity: Answers and Questions,” and introduces some additional considerations. I quote extensively from Einstein’s Relativity: The Special and General Theory (1916, translated 1920).

Einstein begins with a discussion of the coordinates of space in Euclidean geometry, then turns to space and time in classical mechanics. In the following passage he refers to an expository scenario that recurs throughout the book, the passage of a railway carriage (train car) along an embankment:

In order to have a complete description of the motion [of a body], we must specify how the body alters its position with time; i.e. for every point on the trajectory it must be stated at what time the body is situated there. These data must be supplemented by such a definition of time that, in virtue of this definition, these time-values can be regarded essentially as magnitudes (results of measurements) capable of observation. If we take our stand on the ground of classical mechanics, we can satisfy this requirement for our illustration in the following manner. We imagine two clocks of identical construction; the man at the railway-carriage window is holding one of them, and the man on the footpath [of the embankment] the other. Each of the observers determines the position on his own reference-body occupied by the stone at each tick of the clock he is holding in his hand. In this connection we have not taken account of the inaccuracy involved by the finiteness of the velocity of propagation of light.

To get to that inaccuracy, Einstein begins with this:

Let us suppose our old friend the railway carriage to be travelling along the rails with a constant velocity v, and that a man traverses the length of the carriage in the direction of travel with a velocity w. How quickly, or, in other words, with what velocity W does the man advance relative to the embankment [on which the rails rest] during the process? The only possible answer seems to result from the following consideration: If the man were to stand still for a second, he would advance relative to the embankment through a distance v equal numerically to the velocity of the carriage. As a consequence of his walking, however, he traverses an additional distance w relative to the carriage, and hence also relative to the embankment, in this second, the distance w being numerically equal to the velocity with which he is walking. Thus in total he covers the distance W = v + w relative to the embankment in the second considered.

This is the theorem of the addition of velocities from classical physics. Why doesn’t it apply to light? Einstein continues:

If a ray of light be sent along the embankment [in an assumed vacuum], … the tip of the ray will be transmitted with the velocity c relative to the embankment. Now let us suppose that our railway carriage is again travelling along the railway lines with the velocity v, and that its direction is the same as that of the ray of light, but its velocity of course much less. Let us inquire about the velocity of propagation of the ray of light relative to the carriage. It is obvious that we can here apply the consideration of the previous section, since the ray of light plays the part of the man walking along relatively to the carriage. The velocity W of the man relative to the embankment is here replaced by the velocity of light relative to the embankment. w is the required velocity of light with respect to the carriage, and we have

w = c − v.

The velocity of propagation of a ray of light relative to the carriage thus comes out smaller than c.

Let’s take that part a bit more slowly than Einstein does. The question is the velocity with which the ray of light is traveling relative to the carriage. The man in the previous example was walking in the carriage with velocity w relative to the body of the carriage, and therefore with velocity W relative to the embankment (v + w). If the ray of light is traveling at c relative to the embankment, and the carriage is traveling at v relative to the embankment, then by analogy it would seem that the velocity of the ray of light relative to the velocity of the carriage should be the velocity of light minus the velocity of the carriage, that is c – v. (Einstein introduces some confusion by using w to denote this hypothetical velocity, having already used w to denote the velocity of the man walking in the carriage, relative to the embankment.)

It would thus seem that the velocity of a ray of light emitted from the railway carriage should be c + v relative to a person standing still on the embankment. That is, light would travel faster than c when it’s emitted from an object moving in a forward direction relative to an observer. But all objects are in relative motion, even hypothetically stationary ones such as the railway embankment of Einstein’s through experiment. Light would therefore move at many different velocities, all of them varying from c according to the motion of each observer relative to the source of light; that is, some observers would detect velocities greater than c, while others would detect velocities less than c.

But this can’t happen (supposedly). Einstein puts it this way:

In view of this dilemma there appears to be nothing else for it than to abandon either the [old] principle of relativity [the addition of velocities] or the simple law of the propagation of light in vacuo [that c is the same for all observers, regardless of their relative motion]. Those of you who have carefully followed the preceding discussion are almost sure to expect that we should retain the [old] principle of relativity, which appeals so convincingly to the intellect because it is so natural and simple. The law of the propagation of light in vacuo would then have to be replaced by a more complicated law conformable to the [old] principle of relativity. The development of theoretical physics shows, however, that we cannot pursue this course. The epoch-making theoretical investigations of H.A.Lorentz on the electrodynamical and optical phenomena connected with moving bodies show that experience in this domain leads conclusively to a theory of electromagnetic phenomena, of which the law of the constancy of the velocity of light in vacuo is a necessary consequence…..

[I]n reality there is not the least incompatibility between the principle of relativity and the law of propagation of light, and that by systematically holding fast to both these laws a logically rigid theory could be arrived at. This theory has been called the special theory of relativity….

Einstein gets to the special theory of relativity (STR) by next considering the problem of simultaneity:

Lightning has struck the rails on our railway embankment at two places A and B far distant from each other. I make the additional assertion that these two lightning flashes occurred simultaneously. If now I ask you whether there is sense in this statement, you will answer my question with a decided “Yes.” But if I now approach you with the request to explain to me the sense of the statement more precisely, you find after some consideration that the answer to this question is not so easy as it appears at first sight.

After thinking the matter over for some time you then offer the following suggestion with which to test simultaneity. By measuring along the rails, the connecting line AB should be measured up and an observer placed at the mid-point M of the distance AB. This observer should be supplied with an arrangement (e.g. two mirrors inclined at 90 ◦) which allows him visually to observe both places A and B at the same time. If the observer perceives the two flashes of lightning at the same time, then they are simultaneous.

I am very pleased with this suggestion, but for all that I cannot regard the matter as quite settled, because I feel constrained to raise the following objection: “Your definition would certainly be right, if I only knew that the light by means of which the observer at M perceives the lightning flashes travels along the length A → M with the same velocity as along the length B → M. But an examination of this supposition would only be possible if we already had at our disposal the means of measuring time. It would thus appear as though we were moving here in a logical circle.”

After further consideration you cast a somewhat disdainful glance at me— and rightly so— and you declare: “I maintain my previous definition nevertheless, because in reality it assumes absolutely nothing about light. There is only one demand to be made of the definition of simultaneity, namely, that in every real case it must supply us with an empirical decision as to whether or not the conception that has to be defined is fulfilled. That my definition satisfies this demand is indisputable. That light requires the same time to traverse the path A → M as for the path B → M is in reality neither a supposition nor a hypothesis about the physical nature of light, but a stipulation which I can make of my own freewill in order to arrive at a definition of simultaneity.”

It is clear that this definition can be used to give an exact meaning not only to two events, but to as many events as we care to choose, and independently of the positions of the scenes of the events with respect to the body of reference (here the railway embankment). We are thus led also to a definition of “time” in physics. For this purpose we suppose that clocks of identical construction are placed at the points A, B and C of the railway line (co-ordinate system), and that they are set in such a manner that the positions of their pointers are simultaneously (in the above sense) the same. Under these conditions we understand by the “time” of an event the reading (position of the hands) of that one of these clocks which is in the immediate vicinity (in space) of the event. In this manner a time-value is associated with every event which is essentially capable of observation.

This stipulation contains a further physical hypothesis, the validity of which will hardly be doubted without empirical evidence to the contrary. It has been assumed that all these clocks go at the same rate if they are of identical construction. Stated more exactly: When two clocks arranged at rest in different places of a reference-body are set in such a manner that a particular position of the pointers of the one clock is simultaneous (in the above sense) with the same position of the pointers of the other clock, then identical “settings” are always simultaneous (in the sense of the above definition).

In other words, time is the same for every point in a frame of reference, which can be thought of as a group of points that remain in a fixed spatial relationship. Every such point in that frame of reference can have a clock associated with it; every clock can be set to the same time; and every clock (assuming great precision) will run at the same rate. When it is noon at one point in the frame of reference, it will be noon at all points in the frame of reference. And when the clock at one point has advanced from noon to 1 p.m., the clocks at all points in the same frame of reference will have advanced from noon to 1 p.m., and ad infinitum.

As Einstein puts it later,

Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). Every reference body (co-ordinate system) has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event.

Returning to the question of simultaneity, Einstein poses his famous thought experiment:

Up to now our considerations have been referred to a particular body of reference, which we have styled a “railway embankment.” We suppose a very long train travelling along the rails with the constant velocity v and in the direction indicated in Fig. 1. People travelling in this train will with advantage use the train as a rigid reference-body (co-ordinate system); they regard all events in reference to the train. Then every event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. As a natural consequence, however, the following question arises:

Are two events ( e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative.


FIG. 1.

When we say that the lightning strokes A and B are simultaneous with respect to the embankment, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M of the length A → B of the embankment. But the events A and B also correspond to positions A and B on the train. Let M’ be the mid-point of the distance A → B on the travelling train. Just when the flashes of lightning occur, this point M’ naturally coincides with the point M, but it moves towards the right in the diagram with the velocity v of the train. If an observer sitting in the position M’ in the train did not possess this velocity, then he would remain permanently at M, and the light rays emitted by the flashes of lightning A and B would reach him simultaneously, i.e. they would meet just where he is situated. Now in reality (considered with reference to the railway embankment) he is hastening towards the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A. Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A. Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash B took place earlier than the lightning flash A. We thus arrive at the important result:

Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). Every reference body (co-ordinate system) has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event.

It’s important to note that there is a time delay, however minuscule, between the instant that the flashes of light are are emitted at A and B and the instant when they reach the observer at M.

Because of the minuscule time delay, the flashes of light wouldn’t reach the observer at M’ in the carriage at the same time that they reach the observer at M on the embankment. The observer at M’ is directly opposite the observer at M when the flashes of light are emitted, not when they are received simultaneously at M. During the minuscule delay between the emission of the flashes at A and B and their simultaneous receipt by the observer at M, the observer at M’ moves toward B and away from A. The observer at M’ therefore sees the flash emitted from B a tiny fraction of a second before he sees the flash emitted from A. Neither event corresponds in time with the time at which the flashes reach M. (There are variations on Einstein’s thought experiment — here, for example — but they trade on the same subtlety: a time delay between the flashes of light and their reception by observers.)

Returning to Einstein’s diagram, suppose that A and B are copper wires, tautly strung from posts and closely overhanging the track and embankment at 90-degree angles to both. The track is slightly depressed below the level of the embankment, so that observers on the train and embankment are the same distance below the wires. The wires are shielded so that they can be seen only by observers directly below them. Because the shielding doesn’t deflect lightning, when lightning hits the wires they will glow instantaneously. If lightning strikes A and B at the same time, the glow  will be seen simultaneously by observers positioned directly under the wires at the instant of the lightning strikes. Therefore, observers on the embankment at A and B and observers directly opposite them on the train at A’ and B’ will see the wires glow at the same time. (The observers would be equipped with synchronized clocks, the readings of which they can compare to verify their simultaneous viewing of the lightning strikes. I will leave to a later post the question whether the clocks at A and B show the same time as the clocks at A’ and B’.)

Because of the configuration of the wires in relation to the  track and the embankment, A and B must be the same distance apart as A’ and B’. That is to say, the simultaneity of observation isn’t an artifact of the distortion of horizontal measurements, or length contraction, which is another aspect of STR.

Einstein’s version of the thought experiment was designed — unintentionally, I assume — to create an illusion of non-simultaneity. Of course an observer on the train at M’ would not see the lightning flashes at the same time as an observer on the embankment at M: The observer at M’ would no longer be opposite M when the lightning flashes arrive at M. But, as shown by my variation on Einstein’s though experiment, that doesn’t rule out the simultaneity of observations on the train and on the embankment. It just requires a setup that isn’t designed to exclude simultaneity. My setup involving copper wires is one possible way of ensuring simultaneity. It also seems to rule out the possibility of length contraction.

Einstein’s defense of his thought experiment (in the fifth block quotation above) is also an apt defense of my version of his thought experiment. I have described a situation in which there is indubitable simultaneity. The question is whether it forecloses a subsequent proof of non-simultaneity. Einstein’s thought experiment didn’t, because Einstein left a loophole — intentionally or not — which discredits his proof of non-simultaneity. (I am not the first person who claims to have discovered the loophole.) My thought experiment leaves no loophole, as far as I can tell.

If my thought experiment has merit, it points to an invariant time, that is, a time which is the same for all frames of reference. A kind of Newtonian absolute time, if you will.

To be continued.