cThe model with variable c, G, h presented in reference [1] is extended to electromagnetism. The entropy is found to vary like Log t and, in a space-entropy representation, the metric is conformally flat. A new gauge relation is suggested, based on geometrical considerations, which corresponds to a Rydberg constant varying like R. The Hubble's law still applies. The age of the universe is unchanged while its span is found to be half of the Mattig's value. The complete decoding of the red shift can be done. The distances of the sources are very similar. The large volumic power densities of distant quasars could have been greatly overestimated, while the increase of their absolute magnitude, as derived from the classical theory, could be due to the secular variation of c. Assuming the electron-proton mass ratio to vary like R we get a fine structure constant a, a Bohr radius and a ratio electromagnetic force to gravitational force which behave like absolute constants.

1 - INTRODUCTION

...Several authors tried to develop models with physical constants in time varying in time [2, 3, 4, 5, 6, 7, 8, 9] . None questionned the light velocity c, always considered an absolute constant. If one wants to save the form of the conservation equations, the Einstein's constant c = - 8 p G / c² must be considered as an absolute constant. In such conditions if one wants to keep the light velocity c as an absolute c constant and a variable gravitational constant G, one must add a source term to the field equation, see reference [3] . Thus, all these theories imply a constant creation of matter.

...In a previous paper [1] we showed that a variable ligth-velocity could lead to a consistent model if both G and the Planck's constant h followed convenient gauge relations. Thus, the constant creation of matter was no longer necessary. The extension of the Robertson-Walker metric to a variable light velocity configuration and its introduction into the field equation gave a complete set of gauge relations. Let us recall the main features of the model described in [1] :

(1)

(2)
m ( particle's mass ) » R

(3)
h ( Planck'constant ) » R^3/2 » t

(4)
G ( Gravity constant ) » 1/R

(5)
R ( Characteristic length ) » t ^2/3

(6)
V ( Velocity of a free particle ) » R^-1/2 » c

(7)
r ( mass density )» 1/R²

where m is the rest mass, T the temperature, n the number of density and K2 a Bessel function.
If b = (<v²>^1/2/c) << 1 then we get the classical Maxwell-Boltzmann velocity distribution function :
(10)

where k is the Botzmann's constant. We have n » R^-3 , m » R and R » t^2/3,
T = constant ( see reference [ 1] ) , such as :
(12)

s = k Log t + H( b )

In the model b is gauge invariant such as s » Log t.

...In the classical cosmology the universe is isentropic. One could consider somewhat paradoxal that such enormous change in time goes with an almost zero entropy variation. In the new model the entropy grows with time. Notice that the BIG BANG singularity corresponds to s = - ¥ .

...Consider a radiating object, say a nebula N1, which could be considered as a particle. Suppose his light is observed on a nebula N2 placed at the origin of the co-moving coordinates. The nebula N1 is characterized by the value of its time-independent distant marker z, defined by the relation :

The light emited at time t1 is observed on N2 at a time t2 with t2 > t1 .

The distance between N1 and N2 is R(t) dz and is time-dependent but
dz is not. Light travels on a nul geodesic :
(18)

ds2 = (dx°)2 - R2 dz 2= o

Consider the light emited by N1 at a corresponding value
x°1 + D x°1 of the chronological parameter. It will be received at
x°2 + D x°2, where it will be determined through the relation :
(19)

...Consider D x°1 as the equivalent of the period of some physical phenomenon, the emission of radiation for instance, taking place on N1 , and Dx° to be short compared to the equivalent of the travel time from N1 to N2 ( in term of chronological parameter x° ). The periodic phenomenon will appear, as seen from N2 , to have a "period" D x°2 which, from the above relation, will be such that the increment of the q integral will be zero. This, by elementary calculus, gives :
(20)

Let us introduce the cosmic time t as defined before through dx° = c dt and write R (x°1) = R1 , R(x°2) = R2 , c (x°2) = c2 , c (x°1) = c1 , then we get :

...Now we cannot assert that the energy Ei(t1), emitted by the atom at the time t1 would be identical to the corresponding emission energy Ei(t2) of a similar atom, at time t2, in lab's conditions. The light emission is an electromagnetic process. Everybody knows that the classical field description, applying to a four dimensional space time does not take in charge the electromagnetic phenomenon. To get a complete description of universe gravitation and electromagnetism should be imbedded in a common geometrical framework. Unfortunately it has not been done yet in a satisfactory way so that our work will now loose somewhat its self-consistency. . Suppose, for instance, that the Rydberg constant ( ionization energy of hydrogen ) would simply obey the following hypothetic gauge relationship :
(22)

Ei » Rg

...In the classical description the red shift z is due to the Doppler effect, plus some special relativity additional effect. The indix 1 refers to the emitter and the indix 2 to the receiver. For a given specctroscopic line, call E₁ = h₁n_1o the emission energy and E₂ = h₂n_2o the corresponding emission energy in the today's lab's conditions, for the same line. The light is emitted by an atom at rest at the frequency n1 = n1o, corresponding to the wavelength l₁ = c₁/n₁ = l_1O.

n₂ will be the mesured reception frequency, with l₂ = c₂/n₂ and l_2o = c₂/n_2o . The energy of any radiative process will follow the general assumed gauge law (22).

(b)

(c)

Notice that, for g = 1 we refind the classical model.

same thing. The classical relation suggests the choice g = 1 .

In R'2 and R"2 the prime denotes differentiation with respect to t.
In particular , at the first order :

(c)

As a first approximation the astronomer measures d₂ @ R₂ z, such as :

Take R = 3/2 c t ( see reference [ 1) ).

A g = 1 value gives the standard model value.

If z is weak : d₂ » 3/2 ct₂ z

if z tends to infinite : d₂ tends to 3 c t₂

(b)

(c)

(j)

...Quasars correspond presently to z values from 0.13 to 4 . The diameters of the quasars are estimated from their fluctuation period T. We get a maximum diameter cT. With respect to the standard approach this model gives larger values, for c was larger in the earlier time.

...The volumic power is refered to the size of the galaxies. Call PQSO the absolute power emited by a quasar and PG the absolute power emited by a galaxy. The relative power density of the QSO, with respect to a galaxy, is :

Given PQSO and PG the coefficient (57) becomes p' = p x with :

...The ionization energy of hydrogen obeys : E_i = 1/2 a² m_ec², where a is the fine structure constant and me the mass of the electron. We have assumed that Ei » Rg with g = 1, in order to fit with the classical model, see (25), (33) and (35). Introduce the electron-proton mass ratio d = me/mp . According to the first paper [ 1] mp » mn » m » R. So that mc² is an absolute constant. Then :

...The fine structure constant a and the electron-proton mass ratio d cannot be kept constant together. We shall consider two possibilities ;

9.1) Let us take first d » constant. Then :

Introducing the gauge relations for h and c we get : e² /e_o » R^3/2 and the electromagnetic force F_em =e² /4pe_oR² » R^-1/2. Then :

which is similar to an old idea of Dirac [ 4, 5] .

9.2) Now we take a » absolute constant. Then me » R2. and
e² /eo » R. The Compton length of the electron h/mec, the ratio gravitational force to the electromagnetic force and the Bohr radius become absolute constants.

...In the references [10] to [13] several authors study the possible variability of several quantities : a, a²(gp/ge)(me/mp) , me/mp ,where gp and ge are the gyromagnetic ratios of the proton and the electron. Following B.E.G. PAGEL [13] , we have :

...As a consequence we choose the second possibility, with a variable mass ratio. Notice that in both cases we get gp/ge » 1/R.

...Here we tried to extend the model introduced in reference [1] to electromagnetism. A gauge law was suggested : we assumed the ionisation energy Ei ( Rydberg constant ) to vary like Rg. Local geometrical considerations recommand the value g = 1 , which takes account of the desonization process during the cosmic evolution. The distance of a radiative source, as derived from the Robertson Walker metric, gives results quite similar to the standard model'values, but this new model tends to reduce considerably the estimated density power of distant sources like quasars. In addition the increase of absolute magnitude in z, as derived from the classical model, could be due to the secular variation in c.

...With an electron-proton mass ratio d = me/mp which varies like R we get a fine structure constant a which behaves like an absolute constant.

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