Einstein’s Train Thought-Experiment, Fast Neutrinos (and One-Way Measurements of the Speed of Light)

My previous, somewhat rambling, post attempted to explain the recent experimental observation of neutrinos apparently travelling faster than light. I concluded that an adjustment needs to be made to the clocks used to measure the neutrino time of flight to compensate for the absolute motion of the experiment, i.e. in relation to the cosmic microwave background radiation (CMB). Observations suggest the Earth is moving at ~600kps (~1/500th light speed) relative to the CMB.

This post attempts to prove that conclusion by a thought-experiment.

Let’s remind ourselves of Einstein’s train thought-experiment (figures from Wikipedia, copied here purely for convenience):

What Einstein demonstrates here is the relativity of simultaneity. Events (say light emitted at the centre of the carriage reaching the two ends) which are simultaneous in one frame of reference (top, from within the train) are not simultaneous in another frame of reference (bottom, from the platform).

I’d like to extend this slightly.

There’s even a little story. Xavier, in his hyperspeed train, has lured Bob to the train platform under the pretext of performing a relativistic experiment. He hopes to distract Bob for long enough to send Agent Zero to shoot Alice, who is also standing on the platform:

Consider a scenario where Xavier is at the centre of the train, and Bob is standing on the platform, each with their atomic clocks. Bob’s colleagues are also standing on the platform with their atomic clocks at the precise points where there are clocks on the train. Clearly, if set correctly, each pair of clocks would, for an instant, show the time in the other frame of reference – T1(train) would be the same as T1(platform) and so on.

Xavier doesn’t just want a pretext for his evil scheme. He also wants to perform an experiment. He wants to impress Bob by displaying Bob’s correct time on clocks at the ends of his train when he passes Bob on the platform. Bob’s clocks are synchronised by signals from T1(platform) so reflect light speed transmission delays in Bob’s frame of reference. Unfortunately for Xavier, he doesn’t know how fast his train will be moving relative to the platform at the moment he passes Bob.

How can Xavier get his clocks to show the same as Bobs’? There are two possibilities. He could either send a timing signal to each of the clocks at the ends of his train or he could walk the clocks from his position in the centre to each of the two ends. In the first case, the clocks would show the same as T1(train) adjusted for the transmission delays at the speed of light in the train. But we know from Einstein’s thought-experiment that these clocks will not show the same times as those on the platform. In the second case, the clocks would initially show the same as T1(train) but to show the same time as the corresponding clocks on the platform would have to be adjusted to reflect light transmission delays in Bob’s frame of reference. The adjustments would be different for the two clocks (the clock at the front of the train would have to be adjusted by more than the transmission time for light in the train and that at the rear by less).

It is evident from Einstein’s thought-experiment that neither method will yield the times on Bob’s clocks on the platform (Xavier would need to know the exact speed of his train relative to the platform and apply some equations from special relativity).

Signals sent at light speed to the clocks at the front and rear of the train will set them too early and late, respectively, since Bob sees events at the front of the train later than those at the rear. Similarly, walking the clocks from the centre of the train would ensure T1(train)=T2(train)=T3(train), but if adjustments are made to set T2(train) and T3(train) to be in synchrony like T2(platform) and T3(platform), T3(train) at the front would be behind T3(platform) and T2(train) at the rear would be ahead of T2(platform).

Xavier would either have to assume the speed of light was different in each direction or that time at the front of the train was delayed relative to time at the rear (and middle) of the train.

Let’s consider further what would happen if, on seeing Bob on the platform, Xavier told Agent Zero to set off in his near-light speed ship within the train (did I mention it was a big train?) in the direction of travel of the train, towards Alice, who Xavier knows is standing further along the platform in order to help Bob with the experiment.

Now, Bob is nobody’s fool, and sees Zero setting off (did I mention that the train is transparent?) and transmits a message to Alice: “Duck!”. She does so just before Zero arrives at T3(platform) (before he reaches the end of the train at T3(train)) and he misses. Obviously if Zero had reached Alice before the message from Bob, Zero would have travelled faster than light in Bob’s frame of reference. This is not possible, says Einstein.

Similarly, Bob might retaliate by sending Charlie in a similar near-light speed ship to shoot Yolanda at the rear of Bob’s train (which, again, he would reach before he came to the end of the platform). In this case, Xavier would be able to warn Yolanda.

The critical point is that both Bob and Xavier see light travelling at the same speed in both frames of reference. Apologies if you knew this already, but I enjoyed telling the story.

Not shown on my diagram is that Bob would have the same problem with clocks if he was also moving (did I mention the platform is a movable construct in space? – thought-experiment technology really has moved on since Einstein’s day!). If he passed Harry’s Bar on the Intergalactic Highway, he would have the same issues as Xavier (and if Harry also tried to shoot Alice in the same way – after all, he is simmering with jealousy after she ran off with Bob – Bob would also be able to warn Alice). Bob would have to assume either that clocks on his ship were delayed or that light travelled at a different speed along and against his direction of travel.

It follows that it is impossible to measure the one-way speed of light in any frame of reference since you either have to assume there is an error between all clocks in line of motion (proportional to distance) or that light travels at different speeds in different directions.

Unless, of course, you know that your frame of reference is stationary or exactly how you are moving in relation to a stationary frame of reference.

Now, Big Bang theory* tells us that the cosmic microwave background radiation (CMB) was all generated simultaneously. The CMB was generated throughout the Universe and what we see now is what’s reaching us from those regions in all directions that happened to be 13.7 billion light-years away at the time of the Big Bang. Clearly, therefore, the anisotropy (or redshift and blueshift) of the CMB is due to the motion of the Earth rather than that of the CMB itself.

The CMB therefore represents a stationary frame of reference.

As an aside, note that if neutrinos did travel faster than light, they’d eventually (in around 500 trillion years or so, based on the magnitude of the superluminal neutrino velocity reported for the CERN-OPERA experiment) reach a region of space where the Big Bang was still in progress, potentially interfering with their own creation. I don’t think the Universe would allow that, somehow. Alice would be miffed.

I therefore propose that if we adjust the fast neutrino timings to allow for the motion of the experiment, relative to the CMB, during the time the atomic clock was** moved between the GPS at San Grasso and the point where the neutrinos were detected, we will determine the true speed of neutrinos (which I predict will be that of light, c, within the limits of experimental error).

Note that the GPS timings and distances in the fast neutrino experiment do not contribute to the error, since all these are distorted in the same way (i.e. they are consistent within our frame of reference, or, to put it another way, systematic errors cancel out) due to the motion of the Earth (I may put up another post on just this point).

* Just in case anyone assumes I’m an unreconstructed Big Bangian, I want to point out that I’m not entirely comfortable with Big Bang theory, but we can explain the difficulties of one-way lightspeed measurement and, by extension, fast neutrinos, within the paradigm. Obviously any replacement theory would have to explain all the Big Bang predictions (and more – but “more” doesn’t include fast neutrinos, since, just to be crystal, that phenomenon can already be explained) just as Einstein explained Newton’s.

** Postscript (15/10/11): It is incorrect to suggest that the time when the clock was moved (or clocks were synchronised) is important. My brain must have shorted at around the point I came up with that idea last weekend. In fact, the fibre time-delay calibration procedure (which uses two techniques based on the same misconception, one of which involves a transportable caesium clock) always gives the same result for the time-delay (excepting small measurement errors) based on an implicit two-way lightspeed measurement. The (one-way) neutrino flight time is on average greater than that expected based on the fibre time-delay calibration procedure (in this particular experiment, probably due to the N-S component of the CERN-OPERA neutrino flight-path), but dependent on the experiment’s orientation with the CMB when they are sent and detected (the change in orientation, as opposed to the motion, of the planet during their flight time is seriously insignificant). More detail on this point is included in the paper I have subsequently drafted on the erroneous superluminal neutrino result.  (Also made one unrelated wording change in the text, to clarify that the 500 trillion year estimate is based on the reported CERN-OPERA superluminal neutrino flight time).