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One suggestion for explaining superluminal neutrinos (assuming for the sake of argument that the OPERA results should hold up) is that the neutrinos have taken a route through extra dimensions, with an appropriate metric of the form $\mathrm{d}s^2=-\mathrm{d}t^2 + f(r)\mathrm{d}x^2 + \mathrm{d}r^2$. This would be consistent with ordinary physics, but we would still have faster-than-light travel in our 4 dimensions, so what would be the consequences for causality?

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In a weird way this duplicates my question on null energy condition and causality violation, but it is phrased in a more stark and clear way, so it is probably not a duplicate. –  Ron Maimon Nov 22 '11 at 18:08
    
extra dimension counter argument...science20.com/alpha_meme/… –  Vineet Menon Nov 23 '11 at 6:54
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@Vineet: The link you provided is to crackpottery. The person who is writing it does not understand the arguments he is making. –  Ron Maimon Nov 24 '11 at 4:47
    
@RonMaimon: I don't know I'm not into extra dimension studies..but for a layman(as in me)...the argument seemed logical. –  Vineet Menon Nov 24 '11 at 4:48

2 Answers 2

Causality is violated if you have a timelike and/or lightlike closed loop: that is, if you can send a signal of any sort which returns to the same point in space and time that it was sent from.

In this case (assuming $f(r) > 0$ for all $r$) you can't do that; all the timelike and lightlike paths always point in the direction of increasing $t$. So causality is not violated.

Typically, faster-than-light travel does violate causality, but only if there is no preferred rest frame. This metric does have a preferred rest frame, the one in which the time rate is independent of $r$; that is, the one in which clocks located at different $r$ remain synchronized.

EDIT: it should be noted that I'm taking your question fairly literally here. Basically, I'm assuming that the metric you provided is valid for the entire universe, although I think that a variant that allowed for gravity would be possible and would still have the same result. As Ron points out, however, such a metric couldn't explain the OPERA results because the neutrino speed should then vary due to the rotation and motion of the Earth.

In order to fix this problem, you have to tie the metric to the surface of the Earth. What happens then probably depends on the specific mechanism used to do this, but it is reasonably likely that causality would be threatened, at least in principle - although you might need to be able to accelerate a planet to near-light speeds in order to take advantage of it!

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So in this case, we couldn't set up some sort of <a href="en.wikipedia.org/wiki/… anti-telephone</a> because the "Lorentz"-like transformation would be changed somehow? –  James Nov 24 '11 at 15:06
    
Yes. In terms of the example on the Wikipedia page you mention, it would appear to some observers that Bob received the message before Alice sent it, but the reply would always take long enough to get back to Alice that it wouldn't violate causality. –  Harry Johnston Nov 24 '11 at 21:05

Your ansatz is not relativistically invariant. A Lorentz invariant warped extra dimensions ansatz has the form

$$ ds^2 = f(r) (-dt^2 + dx^2 + dy^2 + dz^2 ) + g_{ij}(r) dr^i dr^j $$

Where $r_i$ are the coordinates away from the $x,t$ surface defining us, at $r=0$.

When you have exact Lorentz invariance for us, it is generally impossible to go faster than light in the Lorentz invariant directions, despite what some people are saying, because by boosting the configuration that lets you go faster than light in two different directions, when the configuration is asymptotically Lorentz invariant, you can make a time machine. You don't have to do this at the same point, you can translate the solution in our 3 dimensions to far away points and do the boosting there, and then traverse the loop so formed.

So generic extra-dimension shortcuts between far away points do lead to causality violations if they have asymptotic Lorentz invariance. For a simple example, supposing only that there is a wormhole linking far away points A and B which allows you to traverse the distance between A and B in such a way that you outrun a far away light ray parallel to the macroscopic displacement between A and B, this will allow you to signal back in time, by boosting a copy of the A-B configuration.

So assuming asymptotic lorentz invariance, meaning that the points A and B are far apart compared to the internal structure of the space-time, there is no way to outrun light. This must be qualified, since if electromagnetic light is forced to go along a curved path, while the neutrinos are travelling straight, the neutrinos can outrun the light. But this requires the the surface on which the light is confined violates Lorentz invariance to an AdS or dS invariance (assuming approximate constant curvature) with a violation scale set by the implied curvature required for the short-cut

$$ {1\over R} \approx {\Delta X \over D^2} \approx {20m \over (700 km)^2} \approx 4 \times 10^{-11} m $$

Which might seem small, but at an Earth-sun distance of about 100,000,000 km it corresponds to a gravitational potential of black-hole order, so it is certainly ruled out by solar system measurements. If you make a corrugated extra dimension model, where the curvature switched sign like an egg-holder, you will require corrugation size approximately planetary dimension, and it is extremely implausible that the large-scale structure would look like Lorentz invariant Newtonian orbits.

I am not sure of the best way to be completely precise about this, because you can leave a lower dimensional approximate Lorentz invariance by only violating the Lorentz invariance in far away r-points from our location. This requires large extra dimensions, of course, with the attendent impossible proton-decay and neutrino mass suppression problems, coupled now with LHC observations.

So the FTL neutrino observation is absurd--- the amount of lag must depend on the absolute velocity of the Earth through the ether, so it must depend on the absolute time of year, and on the exact velocity of the galaxy through the ether, etc. In the claimed observations, it doesn't depend on the neutrino energy, nor does it depend on the time of day or year, since the measurements come many months apart. This is a certain sign of a systematic error, and the order of magnitude suggests strongly that the error involves a neglected correction related to the rotation of the Earth.

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It seems to me that you're answering a different question than the one that was asked. This metric doesn't provide a shortcut of the sort the OPs metric did, because the lightcones are the same at all values of r. –  Harry Johnston Nov 26 '11 at 1:40
    
@Harry: the issue I had with your answer is that, while you are technically right that the light cones point forward, this is not enough to make sure that there is causality violation in general. You need to also consider boosting the solutions. These solutions pick out a rest frame, so you can boost them relative to infinity, but infinity at "r" is difficult, and there are many cases to consider. –  Ron Maimon Nov 26 '11 at 2:45
    
the OPs question presumes a given metric. In that metric there is definitely no violation of causality. This is most obviously true in the coordinate system given, but it will still be true in any other coordinate system since it is a coordinate-independent property. –  Harry Johnston Nov 27 '11 at 22:02
    
@Harry: The point is that once you have a given metric, you can boost this metric and get new metrics, and when you have far away metrics which fall off, you can (approximately) superpose metrics. So it is not enough to note that there is no causality violation in this background, you have to check that all backgrounds which you can infer from this background are also causal. The argument that going faster than light violates causility requires boosting the faster-than-light machine very fast, so it will not work with just once copy of the machine which is not allowed to move. –  Ron Maimon Nov 27 '11 at 22:09
    
Obviously, the OPs metric isn't physically realistic: it doesn't vary in space or time. There may be a solution to the Einstein field equations which is approximately of the form the OP provided over some limited area, such as the surface of the Earth, but which differs elsewhere. In particular it might take a similar form but with different preferred rest frames depending on your location. Such a solution might well violate causality. Is this what you mean by "you need to also consider boosting the solutions"? –  Harry Johnston Nov 27 '11 at 22:16

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