Note: The original version of this post was completely, embarrassingly wrong. I replaced it with a new version that says pretty much the opposite. Then Louis3 in the comments pointed out that I had misunderstood the van Elburg preprint yet again, but, if I’m not mistaken, that misinterpretation on my part doesn’t fundamentally change the argument. I hope I’ve got it right now, but given my track record on this I wouldn’t blame you for being skeptical!
If you’re reading this, you almost certainly know about the recent announcement by the OPERA group of experimental results showing that neutrinos travel slightly faster than light. I didn’t write about the original result here, because I didn’t have anything original to say. I pretty much agreed with the consensus among physicists: Probably something wrong with the experiments, extraordinary claims require extraordinary evidence, Bayesian priors, wait for replication, etc.
Recently, there’s been some buzz about a preprint being circulated by Ronald van Elburg claiming to have found an error in the OPERA analysis that would explain everything. If you don’t want to slog through the preprint itself (which is short but has equations), this blog post does a good job summarizing it.
van Elburg’s claim is that the OPERA people have incorrectly calculated the time of flight of a light signal between the source and detector in the experiment. (This is a hypothetical light signal, used for reference — no actual light signal went from one place to the other.) He goes through a complicated special-relativity calculation involving switching back and forth between an Earth-fixed (“baseline”) reference frame and a reference frame attached to a GPS satellite. I don’t understand why he thinks this complicated procedure is necessary : the final result is a relationship between baseline-frame quantities, and I don’t see why you can’t just calculate it entirely in the baseline frame. But more importantly, his procedure contains an error in the application of special relativity. When this error is corrected, the discrepancy he claims to have found goes away.
As a mea culpa for getting this completely wrong initially (and also for the benefit of the students in a course I’m teaching now), I’ve written up a critique of the van Elburg preprint, in which I try to explain the error in detail. I find it cumbersome to include equations in blog posts (maybe I just haven’t installed the right tools to do it), so I’ve put the critique in a separate PDF document. I’ll just summarize the main points briefly here.
van Elburg calculates the time of flight between source and detector in the following complicated way:
- He relates the satellite-frame source-detector distance to the baseline-frame distance via Lorentz contraction.
- He calculates the flight time in the satellite frame (correctly accounting for the fact that the detector is moving in this frame — which is what he claims OPERA didn’t do).
- He transforms back to the baseline frame.
At the very least, this is unnecessarily complicated. The whole point of special relativity is that you can work in whatever inertial frame you want, so why jump back and forth this way, rather than just doing the calculation in the Earth frame? In fact, I originally (incorrectly) thought that he’d done the calculation correctly but in an unnecessarily cumbersome way. It turns out that it’s worse than that, though: his calculation is just plain wrong.
The main error is in his equation (5), specifically when he writes
This is supposed to relate the time of flight in the satellite frame to the time of light in the Earth frame. But the time-dilation rule doesn’t apply in this situation. It’s only correct to calculate time dilation in this simple way (multiply by gamma) if you’re talking about events that are at the same place in one of the two reference frames. The standard example is two birthdays of one of the two twins in the twin paradox. When you’re considering two birthdays of the rocket-borne twin, you’re considering two events that are at the same place in the rocket frame, and the multiply-by-gamma rule is fine.
But in this case the time intervals under consideration are times of flight. That means that they’re time intervals between one event at one place (radio wave leaves the source) and another event at another place (radio wave arrives at detector). To properly relate time intervals of this sort in two different frames, you need the full machinery of the Lorentz transformation. If you use that full machinery to convert from satellite frame to Earth frame, you find that the time of flight comes out just the way you’d expect it to if you’d done the whole calculation in the Earth frame to begin with. (Of course it had to be that way — that’s the whole point of the principle of relativity.)
Now if the OPERA people had done their analysis the way van Elburg does (jumping back and forth with wild abandon between Earth and satellite frames), and if when they were in the satellite frame they had calculated a time of flight without accounting for the detector’s motion, then they would have been making an error of essentially the sort van Elburg describes. But as far as I can tell there’s no credible evidence, either in this preprint or in the OPERA paper, that they did the analysis this way at all, let alone that they made this error.
So this explanation of the OPERA results is a non-starter. Sorry for originally stating otherwise.