Evidence for many-sheeted space-time from gamma ray flares

MAGIC collaboration has found evidence for a gamma ray anomaly. Gamma rays are different energy ranges seem to arrive with different velocities from Mkn 501 (see this). The delay in arrival times is about 4 minutes. The proposed explanation is in terms of broken Lorentz invariance. TGD allows to explain the finding in terms of many-sheeted space-time and there is no need to invoke breaking of Lorentz invariance.

1. TGD based explanation at qualitative level

One of the oldest predictions of many-sheeted space-time is that the time for photons to propagate from point A to B along given space-time sheet depends on space-time sheet because photon travels along lightlike geodesic of space-time sheet rather than lightlike geodesic of the imbedding space and thus increases so that the travel time is in general longer than using maximal signal velocity.

Many-sheetedness predicts a spectrum of Hubble constants and gamma ray anomaly might be a demonstration for the many-sheetedness. The spectroscopy of arrival times would give information about how many sheets are involved.

Before one can accept this explanation, one must have a good argument for why the space-time sheet along which gamma rays travel depends on their energy and why higher energy gamma rays would move along space-time sheet along which the distance is longer.

  1. Shorter wavelength means that that the wave oscillates faster. Space-time sheet should reflect in its geometry the matter present at it. Could this mean that the space-time sheet is more "wiggly" for higher energy gamma rays and therefore the distance travelled longer? A natural TGD inspired guess is that the p-adic length scales assignable to gamma ray energy defines the p-adic length scale assignable to the space-time sheet of gamma ray connecting two systems so that effective velocities of propagation would correspond to p-adic length scales coming as half octaves. Note that there is no breaking of Lorentz invariance since gamma ray connects the two system and the rest system of receiver defines a unique coordinate system in which the energy of gamma ray has Lorentz invariant physical meaning.

  2. One can invent also an objection. In TGD classical radiation field decomposes into topological light rays ("massless extremals", MEs) which could quite well be characterized by a large Planck constant in which case the decay to ordinary photons would take place at the receiving end via decoherence (Allais effect discussed in previous posting is an application of this picture in the case of gravitonal interaction). Gamma rays could propagate very much like a laser beam along the ME. For the simplest MEs the velocity of propagation corresponds to the maximal signal velocity and there would be no variation of propagation time. One can imagine two manners to circumvent to the counter argument.
    1. Also topological light rays for which light-like geodesics are replaced with light-like curves of M4 are highly suggestive as solutions of field equations. For these MEs the distance travelled would be in general longer than for the simplest MEs.
    2. The gluing of ME to background space-time by wormhole contacts (actually representation for photons!) could force the classical signal to propagate along a zigzag curve formed by simple MEs with maximal signal velocity. The length of each piece would be of order p-adic length scale. The zigzag character of the path of arrival would increase the distance between source and receiver.

2. Quantitative argument

A quantitative estimate runs as follows.

  1. The source in question is quasar Makarian 501 with redshift z= .034. Gamma flares of duration about 2 minutes were observed with energies in bands .25-.6 TeV and 1.2-10 TeV. The gamma rays in the higher energy band were near to its upper end and were delayed by about Δ τ=4 min with respect to those in the lower band. Using Hubble law v=Hct with H= 71 km/Mparsec/s, one obtains the estimate Δτ/τ= 1.6×10-14.

  2. A simple model for the induced metric of the space-time sheet along which gamma rays propagate is as a flat metric associated with the flat imbedding Φ= ωt, where Φ is the angle coordinate of the geodesic circle of CP2. The time component of the metric is given by


    ω appears as a parameter in the model. Also the embeddings of Reissner-Norström and Schwartschild metrics contain frequency as free parameter and space-time sheets are quite generally parametrized by frequencies and momentum or angular momentum like vacuum quantum numbers.

  3. ω is assumed to be expressible in terms of the p-adic prime characterizing the space-time sheet. The parametrization to assumed in the following is


    It turns out that r=1/2 is the only option consistent with the p-adic length scale hypothesis. The naive expectation would have been r=1. The result suggests the formula

    ω2 = m0mp with m0= K/R

    so that ω would be the geometric mean of a slowly varying large p-adic mass scale and p-adic mass scale.

    The explanation for the p-adic length scale hypothesis leading also to a generalization of Hawking-Bekenstein formula assumes that for the strong form of p-adic length scale hypothesis stating p≈ 2k, k prime, there are two p-adic length scales involved with a given elementary particle. Lp characterizes particle's Compton length and Lk the size of the wormhole contact or throat representing the elementary particle. The guess is that ω is proportional to the geometric mean of these two p-adic length scales:

    ω2R2 = x/[2k/2k1/2].

  4. A relatively weak form of the p-adic length scale hypothesis would be p≈ 2k, k an odd integer. M127 corresponds to the mass scale me5-1/2 in a reasonable approximation. Using me≈.5 MeV one finds that the mass scales m(k) for k=89-2n, n=0,1,2...,6 are m(k)/TeV= x, with x=0.12, 0.23, 0.47, 0.94, 1.88, 3.76, 7.50. The lower energy range contains the scales corresponding to k=87 and 85. The higher energy range contains the scales corresponding to k=83,81,79,77. In this case the proposed formula does not make sense.

  5. The strong form of p-adic length scale hypothesis allows only prime values for k. This would allow Mersenne prime M89 (intermediate gauge boson mass scale) for the lower energy range and k=83 and 79 for the upper energy range. A rough estimate is obtained by assuming that the two energy ranges correspond to k1=89 and k2=79.

  6. The expression for τ reads as τ= (gtt)1/2t. The expression for Δτ/τ is given by

    Δ τ/τ=(gtt)-1/2Δ gtt/2≈ R2Δ ω2 = x[(k2p2)-1/2-(k1p1)-1/2] ≈x(k2p2)-1/2= x×2-79/2(79)-1/2.

    Using the experimental value for Δτ/τ one obtains x≈.45. x=1/2 is an attractive guess.

It seems that one can fairly well say that standard cosmology is making a crash down while TGD is making a breakthrough after breakthrough as the interpretation becomes more and more accurate. TGD is patiently waiting;-). Interesting to see how long it still will take before sociology of science finally gives up and the unavoidable happens.

For details and background see the chapter The Relationship Between TGD and GRT.