Variable light velocity **
AN INTERPRETATION OF COSMOLOGICAL MODEL
WITH VARIABLE LIGHT VELOCITY.
**
Jean-Pierre PETIT.
Observatoire de Marseille
For scientific correspondence: Chemin de la Montagnère, 84120, Pertuis. France.
Modern Physics Letters A, Vol. 3, n°16, nov 1988, p.1527
ABSTRACT: A cosmological model with variable c, h, G is proposed. The characteristic lengths of physics (Compton, Jeans, Schwarzschild) are assumed to vary like R(t). Both the worlds of light and matter obey the same law R » t²/³. The Planck constant varies like t, the gravitational one like 1/R, while the Planck length varies like R. The particle masses follow m ~ R. The Hubble law still applies. The redshifts come from the secular variation of the Planck constant.
1 - INTRODUCTION
...Since 1930, the constancy of several "constants" of physics has been questioned by many authors [1,2,3,4]. Accurate laboratory measurements show that these values appear quite constant in our current space-time field, which is very small compared to the entire space-time, although Van Flandern [6] has claimed observational evidence of the variation of the gravitational constant G. As far as we can see, the extension of the constancy of the light velocity, and other so-called "fundamental constants" over the overall cosmic scale is still a debated hypothesis. The purpose of this paper is to examine some of the consequences of a model in which the "constants" (especially the light velocity) are assumed to vary with time.
- THE POSSIBLE SECULAR VARIATION OF c
...Milne [1] was the first to propose an attempt of this type. He suggested that the observed redshifts are due to a secular change of the Planck constant, and not to the classical Doppler effect. If the energy of the traveling photon remains constant, the apparent decrease of the observed frequency would be only due to the linear increase of h with the cosmic time t. In addition, Milne [1] suggested a decrease in time of the gravitational constant G.
...Similarly, Fred Hoyle [2] argued against the assumption of the constancy of the mass content of the universe. He also suggested a secular change of G and continuous creation of matter. Dirac [3,4], starting from an hypothesis about the variation in time of some large numbers, built with characteristic physical quantities (like the ratio of the electromagnetic force to the gravitational force), arrived at a variable G and continuous creation of matter. Later, Canuto and Hisieh [8], Lodenquai [5], and Julg [7] explored some consequences of Dirac's initial idea. But, surprisingly, no one contested the absolute constancy of c.
In the field equations, the so-called Einstein constant c is determined by identification to the Poisson equation, which gives:
(1)
...The quantity c must be an absolute constant with respect to the four dimensions, for the field equation to be divergenceless. But once the aforementioned identification refers to a steady situation, it does not imply the absolute constancy of G and c. A cosmological model could, in principle, be constructed with G and c varying with cosmic time (which will be defined later), provided that the ratio G/c² remains an absolute constant.
In the following, we will analyze the effects of a secular variation of the speed of light.
- PROPOSAL OF GAUGE RELATIONS
The Robertson-Walker metric, based on the assumptions of isotropy and homogeneity, leads to the following system:
(2)
(3)
...In this system, k is the sign of the curvature, p the pressure, and r the energy-matter density. In the classical model, we define the cosmic time t from the chronological variable x°, by x° = c t, where c is considered as an absolute constant. In addition, the wavelength of the photon varies like R.
Let us now consider the less restrictive following condition:
(4)
d x° = c(t) dt
which represents an alternative interpretation of the chronological parameter x°. We will now relate the main physical constants to R, considered as a gauge parameter:
(5)
(6)
m (particle mass) » R
(7)
h » R³/²
(8)
G » 1/R
...Referring to relation (1), note that G/c² = constant. In addition, if V is the relative velocity of a given element, for example the random velocity of a galaxy in a cluster, or the velocity of a free particle in a cloud, we assume that V follows the secular variation:
(9)
V » R⁻¹/²
If we assume that the number of particles is conserved, the matter density r obeys:
(10)
r » 1/R²
...As a consequence, we can express the cosmic evolution through a gauge process, i.e., the Compton wavelength, the De Broglie wavelength, the Schwarzschild length, and the Jeans length all vary like R.
In addition, our model still considers mc² = constant and:
(11)
...The classical model preserved the masses, assumed to be constant, but not the total matter-energy, due to the variation of the cosmic background energy. In our scenario, it is the opposite: the energy-matter is constant in time, not the masses. In addition, it should be noted that the quantity Gm²/R, which can be considered as a characteristic gravitational energy, is conserved.
...Since the energies are conserved in our model, the momentum defined as mVi varies like R¹/². It is only constant if we define it as ruic.
Finally, the Planck length varies with time like R(t), the Planck time varies like t, and the gravitational forces like 1/R(t).
- THE EVOLUTION EQUATION
Introducing (4) into the system (2), (3), we obtain the following equations:
(12)
(13)
The use of the following equation of state
(14)
leads to:
(15)
...In the case where R = a tm, the parameter b disappears from (15). From (5), Rc² = Roco² is an absolute constant, Ro and co being the present values of the gauge parameter R and the speed of light c. The only possible value for k is -1, which means that, in our model, the curvature is negative. The evolution then becomes:
(16)
Here, unlike the classical models, light and matter obey the same law of evolution. Moreover:
(17)
...If we know to, the age of the universe, and co**, the present value of the speed of light, we can derive the present value of the gauge parameter of the universe Ro = (3/2) co to, using:
(18)
The consequence is that the horizon is found to be identical, at any time, to the gauge factor R(t).
- GAUGE INVARIANCE OF SOME FUNDAMENTAL EQUATIONS
...Let us first take the Vlasov equation, referring to collision-free fluids. f (r,V,t)** ** is the velocity distribution function, which depends on the position vector r, the velocity vector V, and the time t. Y is the gravitational potential, so that - m ¶ Y/¶ r is the force acting on the particle of mass m.
(19)
Introduce non-dimensional variables, such that:
t = t* t ; f = f* x ; ** V** = V* w ; r = R* z ; Y = ( Gm/R*) j
The equation (19) becomes:
(20)
...Introduce the previous gauge relations G* » 1/R* , m* » R*. The dimensional analysis of the equation (2O) gives V* » 1/(R*)¹/² and:
(21)
R* » t*²/³
These relations can be interpreted as gauge relations and related to the solution (16). Consider now the Schrödinger equation:
(22)
Introduce:
t = t* t , r = R* z , h = h* h , m = m* m , U = U* u.
The dimensional analysis of the equation (22) gives:
(23)
i.e. R* » t*²/³. Now let us write the Maxwell equations, referring to empty space:
(24)
(25)
and write:
E = E* e , B = B* b, r = R* z , t = tt, c = c w
We get:
(26)
(27)
Combining with c* » 1/R¹/², we find R » t²/³.
- CONCLUSION.
...In this paper, we have derived some of the implications of allowing the fundamental constants to vary with time. This can only be done with the addition of some further gauge constraints. Following Milne's suggestion [1], the classical interpretation of the redshift in terms of the Doppler effect has to be replaced by another one taking into account the secular change of the Planck constant. The fundamental parameters R and c are related to each other by some gauge relation. The particle masses vary like R, while the energy-matter and the gravitational energy are conserved.
...This model predicts that the cosmological horizon L(t) should be identical to R(t), which would justify the overall homogeneity of the universe. The curvature of space should be negative and the gauge relationship between R and t should be R ~ t²/³.
...The Planck constant would vary like t, and the gravitational constant G like 1/R, such that the Planck length would vary like R, as well as the Planck time would vary like t. The gravitational force would vary like 1/R.
REFERENCES:
[1] E.A. MILNE: Kinematic Relativity Oxford 1948.
[2] F.HOYLE & J.V.NARLIKAR: Cosmological models in conformally invariant gravitational theory. Mon. Notices Roy. Astr. Soc. 1972 155 pp 3O5-325.
[3] P.A. DIRAC: 1937, Nature, **139,**323
[4] P.A. DIRAC: 1973 Proc. Roy. Soc. London , A333, 4O3
[5] V.CANUTO & J.LODENQUAI: Dirac cosmology, Ap.J. 211 : 342-356 1977 January 15.
[6] T.C.VAN FLANDERN: Is the gravitational constant changing ? Ap.J, 248 : 813-816
[7] A.JULG. Dirac's large numbers hypothesis and continuous creation. Ap.J. 271 : 9-1O 1983 August 1
[8] V.CANUTO & S.H. HSIEH: The 3 K blackbody radiation, Dirac's large numbers hypothesis, and scale-covariant cosmology. Ap.J., 224 : 3O2-307, 1978 September 1
[9] ADLER R. BAZIN M. SCHIFFER M.: Introduction to general relativity. Mc Graw Hill 1965.
[1O] SOURIAU J.M.: Géométrie et relativité. Hermann ed, France, 1964
