Scale Relativity, Non-differentiability and Fractal Space-time

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1456 Special scale relativity log-Lorentzian dilation laws, invariant scale limit under dilations It is with special scale relativity that the concept of space-time-djinn takes its full meaning However, this has only been developed, until now, in two dimensions: one space-time dimension and one for the djinn A complete treatment in ve dimensions remains to be made The previous comment, according to which the standard fractal laws (in constant fractal dimension) have the structure of the Galileo group, immediately implies the possibility of generalizing of these laws Indeed, we know since the work of Poincar [POI 05] and Einstein [EIN 05] that, as regards motion, this group is a particular and degenerated case of Lorentz group However, we can show [NOT 92, NOT 93] that, in two dimensions, assuming only that the law of searched transformation is linear, internal and invariant under re ection (hypotheses deducible from the only principle of special relativity), we nd the Lorentz group as the only physically acceptable solution: namely, it corresponds to a Minkowskian metric The other possible solution is the Euclidean metric, which correctly yields a relativity group (that of rotations in space), but is excluded in the space-time and space-djinn cases since it is contradictory with the experimental ordering found for velocities (the sum of two positive velocities yields a larger positive velocity) and for scale transformations (two successive dilations yield a larger dilation, not a contraction) In what follows, let us indicate by L the asymptotic part of the fractal coordinate In order to take into account the fractal to non-fractal transition, it can be replaced in all equations by a difference of the type L L0 The new log-Lorentzian scale transformation is written, in terms of the ratio of dilation between the resolution scales [NOT 92]: ln(L /L0 ) = ln(L/L0 ) + ln 1 ln2 / ln2 ( / ) + ln ln(L/L0 )/ ln2 ( / ) 1 ln2 / ln2 ( / ) (1434) (1433)

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The law of composition of dilations takes the form: ln ln( / ) + ln = ln ln( / ) 1+ ln2 ( / ) (1435)

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Let us specify that these laws are valid only at scales smaller than the transition scale (respectively, at scales larger than it when this law is applied

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Scaling, Fractals and Wavelets

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to very large scales) As we can establish on these formulae, the scale is a resolution scale invariant under dilations, unattainable, (we would need an in nite dilation from any nite scale to reach it) and uncrossable We proposed to identify it, towards very small scales, with the space and time Planck scale, lP = ( G/c3 )1/2 = 1616 05(10) 10 35 m and tP = lP /c, which would then own all the physical properties of the zero point while remaining nite In the macroscopic case, it is identi ed to the cosmic length scale given by the inverse of the root of the cosmological constant, LU = 1/2 [NOT 93, NOT 96a, NOT 03] We have theoretically predicted this scale to be LU = (27761 00004) Gpc [NOT 93], and the now observed value, LU (obs) = (272 010) Gpc, is in very good agreement with this prediction (see [NOT 08] for more details) This type of log-Lorentzian law was also used by Dubrulle and Graner [DUB 96] in turbulence models, but with a different interpretation of the variables To what extent does this new dilation law change our view of space-time At a certain level, it implies a complication because of the need for introducing the fth dimension Thus, the scale metrics is written with two variables:

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2 d 2 = d 2 (d ln L)2 /C0 ,

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