**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.

Check out this paper: http://arxiv.org/ftp/arxiv/papers/1110/1110.2832.pdf

I particularly like the abstract.

Another favourite of mine is by Max Tegmark, the famous poet: http://adsabs.harvard.edu/abs/1996ApJ…470L..81T

Hi Ted,

For simplicity, van Elburg assumes “that the velocity vector v of the satellite is strictly parallel to the baseline and points in the same direction as a vector pointing from the photon source to the photon detector”. This doesn’t seem like a reasonable approximation to me: it completely fails for a satellite directly overhead, which isn’t heading towards the detector at all!

Second, the “common-view” time transport protocol OPERA uses takes the difference in arrival times of the same satellite signal according to the local clocks at CERN and Gran Sasso, using that to find the offset between the two clocks. For this, they only depend on the difference in propagation times from the satellite to CERN vs. to Gran Sasso, so errors in position for distant satellites, where both sites have approximately the same motion wrt the satellite, will largely cancel out.

Third, they use multiple GPS satellites simultaneously. Since the projections of the satellite velocities are going to be different for different satellites, any significant error should show up as an inconsistency in the offsets calculated from the different satellites, as well as a discernible variation in the time offset as the satellites move.

The OPERA folks didn’t roll their own system for synchronizing their clocks–they used well-studied protocols developed by NIST and other national standards bodies, and had the setup calibrated by the Swiss Metrology Institute and independently verified to about 1 ns by the German Metrology Institute. Metrologists worry about things like the variations in the propagation delay from ionospheric and tropospheric conditions, they certainly know that the geometrical delay has to be calculated using the position at the time signal was emitted.

Further, from the report the German institute wrote, they used standard software developed by the Royal Observatory of Belgium, which calculates the propagation delay using observation and navigation data directly supplied by the satellite–so the natural way to calculate the satellite position automatically does the right thing.

As Chad says, “experimenters aren’t complete idiots”, but van Elburg’s theory seems to rely on a lot of people being complete idiots!

-dan

Dan Riley — You’re quite right. I was totally wrong about this. I’ve replaced the original version of the post with a new one that says pretty much the exact opposite. Sorry about that!

Dear Ted,

I had traced, in details, the misconceptions

related to the intrinsical proper time reasoning

throughout the arXiv:1110.2685 in

http://www.technologyreview.com/blog/arxiv/27260/

and in

http://vixra.org/abs/1110.0047

My best regards,

Armando V.D.B. Assis.

My reading of Elburg’s paper is that he is calculating the time of flight between CERN and the opera detector, not the time of flight between the satellite and a ground based location. So your analysis only makes sense if you assume the satellite is at CERN at the time of emission. I don’t think Elburg claims that equation 13 in your paper (equation 4 in his paper) is wrong, just that if the opera people incorrectly project the satellite’s measurements back to the baseline they won’t get that equation.

Equation 5 in Elburg’s paper, tau_o = Gamma * tau_s, is how he understands the opera/GPS adjustment is performed; Elburg doesn’t seem to be claiming that Gamma*tau_s is a reasonable measure of time of flight. He seems to be suggesting that this is the very mistake the GPS and/or the opera team has encountered.

Louis3 is right. I misread the paper yet again. I really haven’t been covering myself in glory here!

But I don’t think you’re right in your second paragraph. Note the word “should” in the sentence before equation (5). He’s clearly claiming that tau_o is what the should have done, whereas tau_b is what they did do.

Despite my misreading of the procedure, I think that what I wrote is right in broad outline (although wrong in detail). Specifically, equation (5) is not the correct procedure, because you can’t apply time dilation at that stage in the argument. If you apply the full Lorentz transformation at that stage, then the correctly-calculated tau_o will be identical to tau_b. It has to be — if it weren’t, then the speed of light wouldn’t be invariant!

So I think I stand by the following statements, despite my misinterpretation:

1. van Elburg claims that the OPERA group *did* use equation (4).

2. van Elburg glaims that the OPERA group *should have used* equation (5).

3. His equation (5) is based on an incorrect calculation.

4. If one correctly uses the Lorentz transformation in place of equation (5), then the procedure is correct, and tau_o (eq. 5) comes out the same as tau_b (eq. 4).

But given my track record so far, I don’t blame you if you don’t believe me!

Elburg is ambiguous in his use of the word “should” so I’m not sure what to believe on that point; you may have understood him correctly. For Elburg’s equation 4, he seems to be saying that opera incorrectly assumes that the GPS way of projecting satellite time to ground time will be equivalent to the right equation, equation 4. Equation 4 is just the distance between CERN and opera divided by c and Elburg doesn’t seem to say that it should not be used (if it is used correctly). Elburg is ambiguous in his use of the word “should”; he seems to mean that equation 5 is how GPS adjust things so that “should” be used to explain the discrepancy. Bit english is something of a screwball language and it would be nice if Elburg re-worded that section so that his meaning is clearer.

Could you comment on this – I think the problem is in the “common view” method – it assumes the two clocks are synchronized if they are synchronized to the satellite clock (by the time – transfer method), but that means the clocks are not synchronized in the earth’s reference frame.

I’ve reread Elburg’s paper with Louis3’s correction in mind, and I now believe Elburg’s objection is more obviously wrong than I had first thought. As I understand the time transport technique they used, OPERA does not synchronize their clocks to the satellite clock, nor do they need to project the satellite clock to the baseline. According to the protocol, the only uses they should make of the satellite time is to (1) calculate the position of the satellite at the time it transmitted a signal, and (2) verify that the receivers at the two sites see the same signal from the same satellite. For the actual clock synchronization, the satellite time drops out of the equation to determine the offset between the two clocks, so the time in the satellite frame is irrelevant to the clock synchronization.

Please take a look at my analysis if you have a chance

http://arxiv.org/abs/1111.1922

Can anyone tell me how can the scientist be sure that neutrino is from their source?