3. On the nature of the fundamental ratio : a gravitational coupling constant
It may be useful at that step to be more specific about the nature of the new fundamental constant, even though a more detailed analysis will be given in a forthcoming work (Nottale in preparation). The meaning of can be anticipated from a comparison with the quantum hydrogen atom. Indeed, it is well known that, on its fundamental level, the average orbital velocity of an electron is given by , where is the fine structure constant, i.e. the coupling constant of electromagnetism.
In the macroscopic case considered here the problem is purely gravitational, but w still gives the average velocity of the fundamental level. Let us demonstrate that plays the role of a gravitational coupling constant. The fine structure constant appears in the expression of the Coulomb force when the square of the electric charge is expressed in terms of quantum units, i.e.:
In Nottale 1996b, the value of w was determined to be km/s from several different quantization effects ranging from the scale of the solar system to extragalactic scales, in agreement with Tifft & Cocke (1984) own precise determination (144.9 km/s) from the Tifft effect of redshift quantization. This value corresponds to an inverse coupling constant .
The question of a theoretical prediction of the value of this constant might reveal to be a difficult one, owing to the fact that there is still no theoretical understanding, in the standard model of elementary particles, of the value of the electromagnetic coupling constant itself (however, see Nottale 1996a). A full discussion of this problem will be considered elsewhere.
However, one can already remark here that its solution is expected to involve connections between local and global scales, i.e. it might be related with Mach's principle. Recall that, in other contributions, new dilation laws having a log-Lorentz form have been introduced (Nottale 1992), that lead to re-interpret the length-scale of the cosmological constant and the Planck length-scale as impassable, respectively maximal and minimal length-scales, invariant under dilations of resolutions (see e.g. Nottale 1993 , 1996a): i.e., they would play for scale transformations of resolutions a role similar to that of the velocity of light for motion transformations.
Their ratio defines a fundamental pure number, . The logarithm of this ratio has been found to have the numerical value , i.e. from an analysis of the vacuum energy density problem (Nottale 1993 , 1996a). This value corresponds to for a Hubble constant km/s.Mpc) and it has been corroborated by recent indirect measurements of the cosmological constant using SNe I (Garnavich et al. 1998; Perlmutter et al. 1998; Riess et al. 1998).
Moreover, one is also lead, in the scale-relativistic framework, to give a new interpretation of gauge invariance as being invariance in the resolution space. The universal limit on possible scale ratios thus implies a quantization of coupling constants (this amounts to defining a wave in scale space). This allows one to set new fundamental relations between coupling constants and Compton lengths over Planck length ratios, that typically write (Nottale 1996a):
When it is applied to the electron structure, which is upper limited in scale by its Compton length and lower limited by the Planck length-scale, this method yields a relation between the electromagnetic coupling as it is defined in the electroweak theory, , and the electron mass in Planck mass unit:
This relation is satisfied within 0.3% by the experimental values of the fine structure constant and of the electron mass. Recall that this method also allows one to suggest a solution to the hierarchy problem between the GUT and electroweak scale (WZ). Indeed we have suggested, in the minimal standard model reformulated in the special scale-relativity framework, that bare couplings are given by the critical value , so that one can define a fundamental scale given by
which is nothing but the electroweak scale ( GeV).
We can now apply the same reasoning to gravitation in the new framework. Indeed, contrarily to what happens in the classical theory, the equation of motion (Eq. 2) can be shown to be gauge invariant. If the potential is replaced by , where the factor GMm ensures a correct dimensionality, then Eq. (2) remains invariant provided is replaced by , with related to by:
which is the previously established relation for . Therefore, gauge invariance allows one to demonstrate the form of the coefficient that we obtained from dimensional considerations. The advantage of this result is that it will be generalizable to gravitational potentials different from the Kepler one.
Finally, similarly to the electromagnetic case, we can interpret the arbitrary gauge function , up to some numerical constant, as the logarithm of a scale factor in resolution space. In the special scale-relativity framework, such a scale factor is limited by the ratio of the maximal cosmic scale over the Planck scale, i.e. . This limitation of in the phase of the wave function implies a quantization of its conjugate quantity , following the relation:
The numerical constant k remains to be determined. A possible suggestion is that , which yields a predicted value of and km/s, in good agreement with it precise observational determinations. Reversely, from such a relation, if it was confirmed, a precise measurement of w would provide one with a new way of determining the cosmological constant.
© European Southern Observatory (ESO) 2000
Online publication: September 5, 2000