The explanation involves besides the induced metric also the notion of induced gauge field (induced spinor connection): here brane theorists reproducing TGD predictions are bound to meet difficulties and an instant independent discovery of the notion of induced gauge field and spinor structure is needed in order to proceed;-). Here is my comment in somewhat extended form.
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Dear Sascha,
I would be critical about two points.
- I would take Poincare invariance and general coordinate invariance as a starting point. I am not sure whether your arguments are consistent with these requirements.
- The assumption that neutrinos slow down and have gigantic maximal signal velocities initially does not seem plausible to me. Just the dependence of the maximal signal velocity on length scale is enough to understand the difference between SN1987A and OPERA. What this means in standard physics framework is not however easy to understand.
If one is ready to accept sub-manifold gravity a la TGD, this boils down to the identification of space-time sheets carrying the neutrinos (or any relativistic particles from point A to point B). This TGD prediction is about 25 years old: from Peter Woit's blog's comment section I learned that brane people are now proposing something similar: my prediction at viXra log and my own blog was that this will happen within about week: nice to learn that my blog has readers!
This predicts that the really maximal signal velocity (that for M4) is not probably very much higher than the light velocity in cosmic scales and Robertson-Walker cosmology predicts that the light velocity in cosmic scales is about 73 percent of the really maximal one.
The challenge for sub-manifold gravity approach is to understand the SN1987A-OPERA difference qualitatively. Why neutrino (and any relativistic particle) travels faster in short length scales?
- Suppose that this space-time sheet is massless extremal topologically condensed on a magnetic flux tube thickened from a string like object X2×: Y2 subset M4× CP2 to a tube of finite thickness. The longer and less straight the tube, the slower the maximal signal velocity since the light-like geodesic along it is longer in the induced metric (time-like curve in M4× CP2). There is also rotation around the flux lines increasing the path length: see below.
- For a planar cosmic string (X2 is just plane of M4) the maximal signal velocity would be as large as it can be but is expected to be reduced as the flux tube develops 4-D M4 projection. In thickening process flux is conserved so that B scales as 1/S, S the transversal area of the flux tube. Magnetic energy per unit length scales as 1/S and energy conservation requires that the length of the flux tube scales up like S during cosmic expansion. Flux tubes become longer and thicker as time passes.
- The particle -even neutrino!!- can rotate along the flux lines of electroweak fields inside the flux tube and this makes the path longer. The thicker/longer the flux tube,- the longer the path- the lower the maximal signal velocity. I emphasize that classical Z0 and W fields (and also gluon fields!) are the basic prediction of TGD distinguishing it from standard model: again the notion of induced gauge field pops up!
- Classically the cyclotron radius is proportional to the cyclotron energy. For a straight flux tube there is free relativistic motion in longitudinal degrees of freedom and cyclotron motion in transversal degrees of freedom and one obtains essentially harmonic oscillator like states with degeneracy due to the presence of rotation giving rise to angular momentum as an additional quantum number. If the transversal motion is non-relativistic, the radii of cyclotron orbits are proportional to a square root of integer. In Bohr orbitology one has quantization of the neutrino speeds: wave mechanically the same result is obtained in average sense. Fermi statistics implies that the states are filled up to Fermi energy so that several discrete effective light velocities are obtained. In the case of a relativistic electron the velocity spectrum would be of form
ceff= L/T= [1+n×(hbar eB/m)]-1/2× c#
Here L denotes the length of the flux tube and T the time taken by a motion along a helical orbit when the longitudinal motion is relativistic and transversal motion non-relativistic. In this case the spectrum for ceff is quasi-continuous. Note that for large values of hbar =nhbar0 (in TGD Universe) quasicontinuity is lost and in principle the spectrum might allow to the determination of the value of hbar.
- Neutrino is a mixture of right-handed and left handed components and right-handed neutrino feels only gravitation where left-handed neutrino feels long range classical Z0 field. In any case, neutrino as a particle having weakest interactions should travel faster than photon and relativistic electron should move slower than photon. One must be however very cautious here. Also the energy of the relativistic particle matters.
Here brane-theorists trying to reproduce TGD predictions are in difficulties since the notion of induced gauge field is required besides that of induced metric. Also the geometrization of classical electro-weak gauge fields in terms of the spinor structure of imbedding space is needed. It is almost impossible to avoid M4× CP2 and TGD.
To sum up, this would be the qualitative mechanism explaining why the neutrinos travel faster in short scales. The model can be also made quantitative since the cyclotron motion can be understood quantitatively once the field strength is known.
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For background see the chapter TGD and GRT of the online book "Physics in Many-Sheeted Space-time" or the article Are neutrinos superluminal?.
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