Bennett's Clock Paradox

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    I claim that General Relativity merely extends Special Relativity, as Einstein did. But Einstein's version of General Relativity differed so much from Special Relativity in its mathematical expression that the two theories appear virtually unrelated. To avoid repeating that impression, I want to develop General Relativity in a way that retains the mathematics of Special Relativity as far as possible. For that plan to succeed we need something that begins to tie non-inertial frames into inertial frames; that is, we need something that ties accelerated motion into the Relativity of uniform motion. A very special clock paradox fills that need perfectly.

    Jack Bennett of Greenwich, Connecticut, has devised a remarkably clever clock paradox that stands with the more familiar clock paradox, the paradox of the twins, as an aid to understanding Relativity. Indeed, Bennett has given us the more profound of the two paradoxes and I make that claim for two reasons. First, whereas the more familiar clock paradox arises from neglecting to account properly for the temporal offset in the time equation of the Lorentz Transformation, Bennett's clock paradox arises from a perfectly reasonable description of Reality: Bennett has not neglected something obvious in devising his paradox, so in resolving it we learn something significant about the nature of Reality. And second, the resolution of Bennett's clock paradox yields the very temporal offset that resolves the other clock paradox.

    Let's return to our strange little world in which light flies at a speed of one hundred miles per hour and imagine a train accelerating out of the Fresno station. Again we ignore the effects of gravity, assuming especially that those effects do not include the bending of light rays. The train includes only one long passenger carriage and before it leaves the station two timekeepers have carefully synchronized the clocks at the ends of the carriage. As in our previous experiments, the train leaves the station and accelerates up to 86.6 miles per hour. After reaching that speed the train passes a surveyor who snaps a picture of the passenger carriage just as the carriage's midpoint passes her camera. In the developed picture both of the carriage's clocks are visible, their readings clearly legible. Our derivation of the Lorentz Transformation predisposes us to expect that the readings will differ from each other, that the clocks will be out of synchrony. But will that, in fact, be the case?

    Howard Hayden, who described Bennett's thought experiment, thinks not. "How, pray," Hayden asks, "can two identical clocks undergo exactly the same acceleration from exactly the same state of rest to exactly the same final velocity but with different results?" The law of causality, to which Hayden refers, tells us that from the same causes we may expect to see only the same effects. By that law we should expect the surveyor's picture to show both of the train's clocks telling the same time after the train's acceleration has ceased. That statement and Hayden's stated belief that Special Relativity has a few problems with causality imply that Hayden and, presumably, Bennett see Bennett's clock experiment as a proof that the temporal offset term that appears in the time transformation equation of the Lorentz Transformation gives us an incorrect description of Reality and, thus, that clocks synchronized in one inertial frame are necessarily synchronized in all other inertial frames.

    That seems a reasonable conclusion to reach as a result of Bennett's imaginary experiment, but you should notice that I described the experiment only from the frame of an observer standing by the track. By now we should know that we will see the full truth only if we also examine the experiment from the viewpoint of observers aboard the train. At first such an examination seems unnecessary: after all, the train's crew will certainly see what the observers by the track will see; specifically, they will see both of the train's clocks beginning and ending their accelerations at the same times. But does existence make the tacit assumption behind that statement true to Reality? In order to answer that question we must attend to the facts 1) that the observers standing by the track occupy an inertial frame and 2) that the train's crew occupies a noninertial frame.

    The noninertial frame differs from the inertial frame in that the train's acceleration generates a temporal distortion between the clocks at the carriage's front and rear. We can explore that distortion by imagining that the timekeepers that tend the clocks communicate with each other via beams of monochromatic light. The timekeeper at the front of the carriage sends a pulse of light with frequency fb to the rear and in the time the light takes to make the trip the train gains an increment of forward speed, so the pulse comes to the timekeeper in the stern Doppler-shifted to a frequency fb'. Because no actual motion comes between the timekeepers and their clocks, the timekeepers will see no dilation of the time elapsed on the clocks and in the emission of the light, so the Doppler shift will have the purely classical form; that is, D+ = 1+v/c. In this case, the velocity in the formula must be the velocity that the train gains as the pulse goes from the front to the rear and that is simply the product of the train's acceleration, as measured aboard the train, and the pulse's transit time.

    How shall we calculate the time that elapses as a pulse of light travels from the front to the rear of an accelerating railroad carriage? We know that light travels at the same speed in all inertial frames, so it must remain unchanged in a frame that slides across the continuum of inertial frames between two states of uniform motion; that is, in an accelerating frame light travels at its standard 299,792.458 kilometers per second. We thus describe the time that elapses between the emission and the absorption of the pulse by dividing the distance between the carriage's front and rear, represented by x, by the speed of light. We have for the Doppler factor, then,

(Eq'n 1)

in which "a" represents the acceleration of the train.

    The timekeepers can then calculate

(Eq'n 2)

In that equation the subscript a refers to the forward clock and the subscript b refers to the clock at the rear of the carriage. Again the timekeepers note that the carriage's front and rear clocks do not move relative to each other and infer that they have no basis for attributing the change in frequency to the Doppler effect: that interpretation belongs exclusively to observers in inertial frames. Logic compels the timekeepers to infer that the frequency change reflects a difference in the rates at which their respective clocks tick off time. Because a uniform frequency simply expresses the inverse of a uniformly counted time, the timekeepers report to the train's conductor that

(Eq'n 3)

That is the degree to which the two clocks will be out of synchrony with each other.

    Given that the train must achieve a speed V relative to the track, the conductor calculates the time to achieve that speed as V/a. No, that can't be right, the brakeman objects; it's too simple to be a relativistically correct formula and it implies that this train could accelerate to a speed exceeding that of light. That brakeman seems to be right on that latter point: if the train accelerates at one mile per hour per second, it will reach the speed of light in one hundred seconds. That calculation is correct, the conductor replies, but bear in mind that we count those seconds on the train's clocks. If the train could survive the environmental hazards of a runup to the speed of light (such as a stretch of bad track), the last infinitesimal fraction of a nanosecond of that acceleration would correspond, via time dilation, to an eternity on any clocks that might still exist in whatever's left of the world. No, the formula is correct, but we must take care not to read into it things that are not actually there.

    So at the end of the train's acceleration the two clocks display the times

(Eq'ns 4)

Or is it

(Eq'ns 5)

The answer to that question would seem to depend upon which clock was running the locomotive's throttle. Let's say, for the sake of the discussion, that the clock at the rear of the carriage controls the throttle. Equations 4, then, give us the description that we want to use of the time elapsed on the two clocks. A quick examination of those equations tells us immediately that when the train's clocks have been resynchronized by the train's timekeepers, those clocks will display to observers alongside the track the temporal offset required by the Lorentz Transformation. But before the timekeeper can carry out that process, we must resolve a new version of the twin paradox.

    Please note, by the way, that Bennett gave a correct description of the situation at the end of the train's acceleration: both clocks appear in perfect synchrony to observers standing by the track. But to observers aboard the train the clocks have gone out of synchrony and must be resynchronized. When that resynchronization process has been carried out, the clocks will appear synchronized to people aboard the train and will appear to trackside observers to be out of synchrony by the amount specified in the Lorentz Transformation.

    Now let's ask, with the conductor, How long did the train actually accelerate? The timekeeper at the front of the carriage will claim that it did so for more time that the conductor specified and the timekeeper at the rear will claim that the acceleration time was just what the conductor ordered. Does that mean that when the engineer closes the throttle the front of the carriage travels faster than the rear? No, of course it doesn't. It merely means that we have neglected to notice that there is a spatial distortion within the carriage to match the temporal distortion. That spatial distortion, like the temporal distortion, is reflected in the Doppler shift of light passing between the two train's clocks.

    You can discern the distortion in the effect it has upon the process of measurement by its distortion of the unique standards that make measurement possible. In measuring the time between two events an observer, in essence, counts the number of times that a stable, uniquely defined oscillation, such as the rate of vibration of the electromagnetic field in the hydrogen alpha emission, repeats between the events. In the accelerating train the timekeeper at the rear sees that the hydrogen alpha from the front has been blue-shifted, its frequency raised higher than that of the hydrogen alpha used by the rear clock, so he infers that the front clock counts more time between two given events than the rear clock counts, all in accordance with Equation 3. In like manner, the timekeeper at the rear knows that the wavelength of the hydrogen alpha from the front has been shortened relative to the hydrogen alpha used by the rear clock, so the timekeeper at the front measures between any two given points separated in the direction of acceleration a distance greater than the distance that he would measure. If the distance measured by the timekeeper at the front is represented by la, then the timekeeper at the rear has

(Eq'n 6)

in which we must now see the "x" as the distance between the two train's clocks measured when the train is not accelerating. If the two events that the timekeepers measure are defined by a feature on the track passing two marks on the carriage's chassis, then the ratio of the distance between the marks and the interval between the passings is the average speed of the train between those two instants. Combining Equations 3 and 6 in the appropriate ratio shows that the speed will be the same for both timekeepers. The different times displayed in Equations 4 thus reflect the fact that the timekeepers will measure different accelerations in their respective parts of the carriage. And the resemblance to the older twin paradox, in which one observer's clock runs faster than the other observer's clock, is merely superficial: there is no paradox aboard our accelerating train because the timekeepers do not disagree over whose clock is running fast.

    So when the train's acceleration ceases, the train's clocks appear synchronized to observers alongside the track and appear out of synchrony in accordance with Equations 4 to the train's crew. The timekeepers subsequently resynchronize the train's clocks by subtracting the overcount xV/c2 from the clock at the front, thereby shifting the clocks out of synchrony in the track's frame by just the temporal offset required by the Lorentz Transformation. Thus, for two events that occur at the train's front and rear clocks, the observers alongside the track will calculate in their own coordinates

(Eq'n 7a)

(Eq'n 7b)

That result would seem to give us sufficient proof and verification of the proposition implicit in my stated disagreement with the conclusion that Bennett drew from his analysis of this imaginary experiment. But I want to pursue this analysis further, if only to ensure that I have not neglected some feature that would invalidate my result. I also want to pursue it because it leads us to an interesting problem.

    So far we have considered only the clocks aboard the train. Now let's consider the clocks alongside the track and ask how they look from the train. Will they automatically display the correct temporal offsets?

    To make the analysis more transparent, let's assume that the track is oriented toward a rapidly pulsing strobe light and that the clocks measure time by counting the pulses flying by them. We have so spaced the clocks along the track that two consecutive pulses in the train trigger neighboring clocks simultaneously; that is, the distance between neighboring clocks is equal to the distance between the pulses, the pulse train's analogue of wavelength. If the frequency at which pulses pass a given clock is represented by "F", then the distance between pulses is L = c/F. The time that elapses on any given clock between any two events is equal to the number of pulses that pass the clock divided by the frequency of the pulses; that is, T = N/F.

    The train's crew has in that mechanism a straightforward way in which they can determine the difference, if any, between the times elapsed on any two of the trackside clocks between two given events that are simultaneous in the train's frame. Let the two events be defined by the timekeepers at the front and rear of the train carriage photographing two of the trackside clocks at the same time on their resynchronized clocks. Let the trackside clocks in question be clocks that are separated by a distance X as measured by observers alongside the track. In order to determine whether there is a temporal offset displayed between the clocks in the photographs, the timekeepers must determine the number of pulses (n) between the clocks in their frame and subtract from that number the number of pulses (N) between the clocks in the track's frame. The temporal offset is then calculated by dividing the difference by the track frame frequency of the pulses; that is

(Eq'n 8)

In the track's frame we have the number of pulses between the clocks at a given instant as being equal to the number of pulses that pass one clock while one pulse goes from one clock to the other; that is,

(Eq'n 9)

That's the number of pulses that fit between two clocks a distance X apart. In the train's frame that distance becomes

(Eq'n 10)

and the frequency of the pulse train is Doppler shifted to

(Eq'n 11)

The number of pulses that fit between the two clocks in the train's frame is simply

(Eq'n 12)

With those equations in mind the timekeepers can evaluate Equation 8 and obtain

(Eq'n 13)

which is just the temporal offset required by the Lorentz Transformation.

    That offset comes about because even though the distance between the clocks is contracted in the train's frame, the Doppler shift contracts the distance between pulses by an even greater factor. By putting extra pulses between the two clocks, the Doppler shift has pushed one clock into the future relative to the other. More generally we say that if we move toward some object, that object exists in the future relative to the temporation it occupies in the frame in which it is at rest. That proposition raises a very interesting question about the nature of time as we experience it.

    Suppose that a rocketship is scheduled to travel to Alpha Centauri at a speed of 86.7% of lightspeed (Lorentz factor 2) and that the people who have settled the planets of Alpha Centauri have scheduled an election for a day one month after the rocketship is launched. If the rocketship takes less than one day to achieve its full speed, then at the end of that day the ship's crew must acknowledge that the election has been decided over three-and-a-half years earlier while space workers that the rocketship passes believe that the election has not yet been decided. But now suppose that the rocketship decelerates to a full stop (one of the crew forgot their toothbrush, so the ship has been stopped to let them buy one at the convenience store at the heliopause) and then reaccelerates to Lorentz factor 2. In the rocketship's frame the election has been decided, then undecided, and then redecided: was the outcome the same the second time?

    If we say, "Yes, it must have been", then the temporal offset term in the Lorentz Transformation necessarily entails that Reality be absolutely, perfectly deterministic. That entailment would seem to be falsified by Werner Heisenberg's indeterminacy principle (and it is indeed indeterminacy, not uncertainty, according to Heisenberg himself) and by the Born interpretation of the wave function as representing a probability distribution.

    If, on the other hand, we say, "No, it might have been different", then the temporal offset must entail something like Hugh Everett's many-worlds interpretation of the quantum theory. To get a feel for what that proposition means, I contemplated the idea of stop-and-go traffic on the Santa Monica Freeway convulsing the histories of entire civilizations across the full length, breadth, and depth of the Great Galaxy in Andromeda (at a distance of two million lightyears a delta-vee of fifty-five miles per hour creates a temporational displacement of 216 days) and then asked whether I and my history are equally phantasms to the inhabitants of one of that galaxy's worlds.

    That is a serious problem, but it takes us beyond the scope of this little essay. We shall confront it later and in doing so, I believe, we shall discover between Relativity and the quantum theory a relationship more intimate than that presented by Dirac, Klein and Gordon, and Proca. For now I will say only that, while Relativity has come through unscathed, the current interpretations of the quantum theory will likely not survive the journey of Jack Bennett's imaginary train.


1. Howard Hayden, "Forced to Conclusions", Galilean Electrodynamics, Vol. 5, No. 5, Pg 82, Sep/Oct 1994.


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