VARIABLE SPEED OF LIGHT COSMOLOGICAL MODEL:
INTERPRETATION OF REDSHIFTS.
Jean-Pierre PETIT.
Marseille Observatory
Physics Letters A, Vol.3, n° 18, dec. 1988, p.1733
ABSTRACT:
c
The model with variable speed of light c, gravitational constant G, and Planck constant h, presented in reference [1], is extended to electromagnetism. Entropy is found to be proportional to Log t and, in a space-entropy representation, the metric is conformal to a flat metric. A new gauge relation is proposed, based on geometric considerations, corresponding to a Rydberg constant varying as R. The Hubble law remains valid. The age of the universe remains unchanged while its radius is found to be half of the Mattig value. Complete decoding of the redshift becomes possible. The distances of the sources are very similar. The volumetric power densities of distant quasars could have been greatly overestimated, while the increase in their absolute magnitude, as obtained by classical theory, could be due to the secular variation of c. Assuming that the electron-proton mass ratio varies as R, we obtain a fine structure constant a, a Bohr radius and a ratio of electromagnetic to gravitational force that behave as absolute constants.
1 - INTRODUCTION
Several authors have tried to develop models with physical constants varying in time [2, 3, 4, 5, 6, 7, 8, 9]. None have questioned the speed of light c, always considered as an absolute constant. If one wants to preserve the form of the conservation equations, the Einstein constant
c = - 8 p G / c 2 must be considered as an absolute constant. Under these conditions, if one wants to maintain the speed of light c as an absolute 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 continuous creation of matter.
In a previous article [1], we showed that a variable speed of light could lead to a consistent model if both G and the Planck constant h followed appropriate gauge relations. Thus, continuous creation of matter was no longer necessary. The extension of the Robertson-Walker metric to a configuration with variable speed of light, introduced in the field equation, gave a complete set of gauge relations. Let us recall the main characteristics of the model described in [1]:
(1)
(2) m (mass of particles) » R
(3) h (Planck constant) » R 3/2 » t
(4) G (gravitational 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/R 2
(8) mc 2 = constant
Here is a brief digression on entropy.
- TIME OR ENTROPY?
The relativistic formulation of the velocity distribution function is:
(9)
where m is the rest mass, T the temperature, n the number density and K2 a Bessel function.
If b = (<v 2 > 1/2 / c) << 1, we obtain the classical Maxwell-Boltzmann distribution function:
(10)
Let us now calculate the entropy per baryon, defined by:
(11)
where k is the Boltzmann constant. We have n » R -3, m » R and R » t 2/3, T = constant (see reference [1]), such that:
(12)
Then:
(13) s = k Log t + H(b)
In the model, b is a gauge invariant, so that s » Log t.
In classical cosmology, the universe is isentropic. It could seem somewhat paradoxical that such a huge change in time is accompanied by almost no change in entropy. In the new model, entropy increases with time. Note that the Big Bang singularity corresponds to s = - ¥.
Let us now define entropy by:
(14) s = 3/2 k Log t
Returning to the Robertson-Walker metric
(15)
We obtain:
(16)
In the representation {entropy, space variables}, the metric is conformal to a flat metric. From a cosmological point of view, entropy (which, on the other hand, is invariant under Lorentz transformation) could be a better choice than time.
Moreover, if we describe the universe in a phase space (position plus velocity), we find that the characteristic hyper-volume associated R 3 c 3 varies as t.
- REDSHIFT AND ROBERTSON-WALKER METRIC:
Consider a radiating object, say a nebula N1, which can be considered as a particle. Suppose its light is observed on a nebula N2 placed at the origin of comoving coordinates. The nebula N1 is characterized by the value of its time-independent distance marker z, defined by the relation:
(17)
The light emitted at time t1 is observed on N2 at time t2 with t2 > t1.
The distance between N1 and N2 is R(t) dz and is time-dependent, but dz is not. The light propagates along a null geodesic:
(18)
ds 2 = (dx°) 2 - R 2 dz 2 = 0
Consider the light emitted by N1 at a corresponding value x°1 + Dx°1 of the chronological parameter. It will be received at x°2 + Dx°2, where it will be determined by the relation:
(19)
Consider Dx°1 as the equivalent of the period of a physical phenomenon, for example, the emission of radiation on N1, and Dx° as being short compared to the equivalent of the travel time of N1 to N2 (in terms of the chronological parameter x°). The periodic phenomenon will appear, as seen from N2, as having a "period" Dx°2, such that, according to the above relation, the increase of the integral q will be zero. This, by a simple calculation, gives:
(20)
Now introduce the cosmological time t defined previously by dx° = c dt and note R(x°1) = R1, R(x°2) = R2, c(x°2) = c2, c(x°1) = c1, we obtain:
(21)
- THE PROBLEM OF ELECTROMAGNETISM:
We can no longer assert that the energy Ei(t1), emitted by the atom at time t1, would be identical to the corresponding emission energy Ei(t2) of a similar atom at time t2, under laboratory conditions. The emission of light is an electromagnetic phenomenon. Everyone knows that the classical description of the field, applied to a four-dimensional space-time, does not take into account the electromagnetic phenomenon. To obtain a complete description of the universe, gravity and electromagnetism should be integrated into a common geometric framework. Unfortunately, this has not yet been done in a satisfactory way, so our work loses some of its consistency. Suppose, for example, that the Rydberg constant (the ionization energy of hydrogen) obeys simply the following hypothetical gauge relation:
(22)
E i » R g
(Note that this is a completely arbitrary hypothesis). Explore the consequences of this hypothesis on the decoding of the redshift. Later, we will try to relate it to possible gauge relations.
- THE REDSHIFT PHENOMENON:
In the classical description, the redshift z is due to the Doppler effect, plus an additional effect of special relativity. The index 1 refers to the emitter and the index 2 to the receiver. For a given spectral line, let E1 = h1 n1o be the emission energy and E2 = h2 n2o be the corresponding emission energy in current laboratory 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 = l1O. n2 will be the received frequency measured, with l2 = c2 / n2 and l2o = c2 / n2o. The energy of any radiative process will follow the general gauge law assumed (22).
We can define the redshift z:
- As the ratio between the wavelengths:
(23)
(24)
(a)
(b)
(c)
we obtain:
(25)
Note that, for g = 1, we recover the classical model.
- As the ratio between the frequencies:
(26)
we obtain the same result.
- As the ratio between the energies.
(27)
same thing. The classical relation suggests the choice g = 1.
- THE HUBBLE LAW AND THE ROBERTSON-WALKER METRIC:
Expand the function 1/R(t) in a series with respect to
(28)
we obtain:
(29)
In R'2 and R"2, the prime denotes the derivative with respect to t.
In particular, at first order:
(30)
Then, expand the following expressions:
(31a)
(b)
(c)
Referring to the first order:
(32)
As a first approximation, the astronomer measures d2 @ R2 z, so that:
(33)
This is nothing more than the redshift law of Hubble, which remains valid in this variable speed of light framework. From the measurement of d2, c2 and z, we can deduce the so-called Hubble constant, that is, the age of the universe.
Take R = 3/2 c t (see reference [1]).
(34)
The age of the universe corresponds to:
(35)
A value g = 1 gives the value of the standard model.
- REDSHIFT AND DISTANCE ESTIMATION:
Return to the Robertson-Walker metric, which provides:
(36)
In the classical approach, take the Einstein-de Sitter model (k = 0). We obtain:
(37a)
With R = a t 2/3, we have:
(b)
from which
(38)
If z is small: d2 » 3/2 ct2 z
if z tends to infinity: d2 tends to 3 c t2
Of course, the Mattig formula gives the same result:
(39a)
(b)
(c)
Return to the Robertson-Walker metric, according to our model, with k = -1:
(d)
write:
(e)
(f)
So:
(g)
Then:
(h)
For radial paths:
(i)
(j)
Log (1 + z) = Arg th u
(k)
(l)
When z tends to infinity, we recover the horizon (3/2)c2 t2, which is twice smaller than the standard value 3 c2 t2.
Note that this is completely similar to the law giving v r /c (where v r is the radial velocity) as a function of z, in the standard model.
Compare the distances, as they are given by our model and by the standard model.
(40)
They are similar for small values of z.
- THE QUASAR PROBLEM:
Quasars currently correspond to z values ranging from 0.13 to 4. The sizes of quasars are estimated from their fluctuation period T. We obtain a maximum diameter c T.
Compared to the standard approach, this model gives larger values, since c was higher in earlier times.
The volumetric power is related to the size of galaxies. Let P QSO be the absolute power emitted by a quasar and P G the absolute power emitted by a galaxy. The relative power density of the QSO, compared to a galaxy, is:
(41)
But in our model, galaxies are no longer constant in size. They grow over time. Suppose the quasar is integrated into a galaxy. The size of this galaxy will grow like (1 + z). Thus, our correction factor, compared to standard values for the power density, involves three effects:
-
Change in distance (they are a bit closer).
-
Change in diameter (due to the variation of c).
-
Change in galaxy size.
Given P QSO and P G, the coefficient (57) becomes p' = p x with:
(42)
that is:
(43)
z h x 0 1 1
0.13
1.025
0.606
The closest quasar
0.2
1.03
0.467
0.5
1.048
0.177
1
1.024
0.0463
1.5
0.985
0.0157
2
0.946
0.0063
2.5
0.912
0.00296
3
0.882
0.00152
3.5
0.856
0.000842
4
0.834
0.0005
The farthest quasar
5
0.8
0.000201
6
0.771
0.0001
7
0.75
0.0000485
8
0.731
0.0000275
9
0.71
0.0000162
10
0.7
0.0000102
We see that this correction reduces the absolute magnitude of the observed quasar, and this correction increases with z. Thus, if this model were good, the classical model would have greatly overestimated the volumetric power density of quasars. Moreover, the observed increase in the absolute magnitude of quasars could be due to the secular variation of c.
Classically, the extent of galaxies is related to the Jeans length, but the model provides no available information on the sizes of emitters such as stars or quasars. This depends on the energy emission process. Since we have not defined a possible gauge relation for the fusion coefficients, we do not yet have a model available. Anyway, quasars could grow over time, like galaxies, and observations tend to support this hypothesis, which will be examined in more detail in the next article devoted to the detailed interaction of the model with the available observations.
- ASSOCIATED GAUGE RELATIONS.
The ionization energy of hydrogen obeys: E i = 1/2 a 2 m e c 2, where a is the fine structure constant and m e the mass of the electron. We have assumed that E i » R g with g = 1, to fit the classical model, see (25), (33) and (35). Introduce the electron-proton mass ratio d = m e /m p. According to the first article [1], m p » m n » m » R. Thus, mc 2 is an absolute constant. Then:
(44)
The fine structure constant a and the electron-proton mass ratio d cannot be kept constant together. We will consider two possibilities:
9.1) Take d » constant. Then:
(45)
Introducing the gauge relations for h and c, we obtain e 2 / e o » R 3/2 and the electromagnetic force F em = e 2 /4 p e o R 2 » R -1/2.
Then:
(46)
which is similar to an old idea of Dirac [4, 5].
Calculate the Bohr radius:
(47)
9.2) Now, take a » absolute constant. Then m e » R 2, and e 2 / e o » R. The Compton wavelength of the electron h/m e c, the ratio between the gravitational force and the electromagnetic force and the Bohr radius become absolute constants.
In references [10] to [13], several authors study the possible variability of several quantities: a, a 2 (g p/g e)(m e/m p), m e/m p, where g p and g e are the gyromagnetic ratios of the proton and the electron. According to B.E.G. PAGEL [13], we have:
Effect
Quantity Approximate upper limit of variation in 3 s.
Optical doublet separation.
a
3%
Comparison of optical and 21 cm redshifts.
a 2 (g p/g e)(m e/m p) 10 -3
Comparison of hydrogen and metal redshifts.
d = m e/m p
50%
Consequently, we choose the second possibility, with a variable mass ratio. Note that in both cases, we obtain g p/g e » 1/R.
- CONCLUSION
Here, we have tried to extend the model introduced in reference [1] to electromagnetism. A gauge law has been suggested: we have assumed that the ionization energy E i (Rydberg constant) varies as R g. Local geometric considerations recommend the value g = 1, which takes into account the deionization process during cosmic evolution. The distance of a radiative source, as derived from the Robertson-Walker metric, gives results quite similar to the values of the standard model, but this new model tends to significantly reduce the estimated power density of distant sources such as quasars. Moreover, the increase in absolute magnitude with z, as derived from the classical model, could be due to the secular variation of c.
With an electron-proton mass ratio d = m e/m p that varies as R, we obtain a fine structure constant a that behaves as an absolute constant.
REFERENCES:
[1]
J.P.PETIT : An interpretation of the cosmological model with variable speed of light. Modern Physics Letters A......
[2]
E.A. MILNE : Kinematic Relativity Oxford 1948.
[3]
F.HOYLE & J.V.NARLIKAR : Cosmological models in the conformally invariant gravitational theory. Mon. Notices Roy. Astr. Soc. 1972 155 pp 3O5-325.
[4]
P.A. DIRAC : 1937, Nature, 139, 323 [5]
P.A. DIRAC : 1973 Proc. Roy. Soc. London, A333, 4O3
[6]
V.CANUTO & J.LODENQUAI : Dirac cosmology, Ap.J. 211 : 342-356 1977 15 January.
[7]
T.C.VAN FLANDERN : Does the gravitational constant change? Ap.J, 248 : 813-816
[8]
A.JULG. The Dirac's large numbers hypothesis and the continuous creation. Ap.J. 271 : 9-1O 1983 1st August
[9]
V.CANUTO & S.H. HSIEH : The 3 K blackbody radiation, the Dirac's large numbers hypothesis, and the covariant cosmology on scale. Ap.J., 224 : 3O2-307, 1978 1st September
[10]
B.E.J.PAGEL : On the limits of the past variability of the proton-electron mass ratio imposed by the redshifts of the absorption lines of quasars. Mon. Not. astr. Soc. (1977), short communication, 81P-85P.
[11]
A.M.WOLFE & R.L.BROWN & M.S.ROBERTS. Limits on the variation of fundamental atomic quantities on cosmological time scales. Physical Review Letters Vol 37, number 4, 26 July 1976.
[12]
A.D.TUBBS & A.M.WOLFE : Evidence for large-scale uniformity of physical laws. A.P.J, 236 : L1O5-1O8, 198O 15 March
[13]
B.E.J.PAGEL. Implications of quasar spectroscopy for the constancy of constants. Phil. Trans. R. Soc. Lond. 3 31O, 245-247 , 1983.
COSMOLOGICAL MODEL WITH VARIABLE LIGHT VELOCITY :
THE INTERPRETATION OF RED SHIFTS.
Jean-Pierre PETIT.
Observatoire de Marseille
Modern Physics Letters A, Vol.3 , n° 18, dec. 1988, p.1733
ABSTRACT
:
c
The 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 / c
2
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/R 2
(8)
mc
2
= constant
The following is a short digression about entropy.
- TIME OR ENTROPY ?
The relativistic formulation of the velocity distribution function is :
(9)
where m is the rest mass, T the temperature, n the number of density and K2 a Bessel function.
If
b
= (<v
2
1/2
/c) << 1 then we get the classical Maxwell-Boltzmann velocity distribution function :
(10)
Let us compute the entropy per baryon, as defined by :
(11)
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)
Then :
(13) 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 = -
¥
.
Let us define now the entropy through :
(14) s = 3/2 k Log t
Let us return to the Robertson-Walker metric
(15)
We get :
(16)
In the representation { entropy, space variables } the metric is
conformally flat
. From a cosmological point of view, the entropy (which on another hand is invariant with respect to the Lorentz transform ) could be a better choice than time.
In addition, if we describe the universe in a phase space ( position plus velocity ) we find that the associated characteristic hypervolume R
3 c 3
varies like t.
- THE RED SHIFT AND THE ROBERTSON-WALKER METRIC :
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 :
(17)
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) d
z
and is time-dependent but
d z
is not. Light travels on a nul geodesic :
(18)
ds 2
= (dx°)
2 - R 2 d z 2 = o
Consider the light emited by N1 at a corresponding value
x°1 +
Dx°1 of the chronological parameter. It will be received at
x°2 +
Dx°2, where it will be determined through the relation :
(19)
Consider
Dx°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"
Dx°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 :
(21)
- THE PROBLEM OF ELECTROMAGNETISM :
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)
E i
»
R g
( Notice that this is a totally arbitrary assumption ). Let us explore the consequences of it on the red shift decoding. Later we will try to relate it to possible gauge relations.
- THE RED SHIFT PHENOMENON :
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 E
1 = h 1 n
1o
the emission energy and E
2 = h 2 n
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
n 1 = n
1o,
corresponding to the wavelength
l 1 = c 1 / n 1 = l
1O
.
n 2
will be the mesured reception frequency, with
l 2 = c 2 / n 2
and
l
2o
= c 2 / n
2o
. The energy of any radiative process will follow the general assumed gauge law (22).
We can define the red shift z :
- As the ratio between the wavelengths :
(23)
(24)
(a)
(b)
(c)
then we get :
(25)
Notice that, for
g
= 1 we refind the classical model.
- As the ratio between the frequencies :
(26)
we get the same result.
- As the ratio between the energies.
(27)
same thing. The classical relation suggests the choice
g
= 1 .
- THE HUBBLE'S LAW AND THE ROBERTSON WALKER METRIC :
Let us expand the function 1/R(t) into a series with respect to
(28)
we get :
(29)
In R'
2
and R"
2
the prime denotes differentiation with respect to t.
In particular , at the first order :
(30)
Next, expanding the following expressions :
(31a)
(b)
(c)
Refering to the first order :
(32)
As a first approximation the astronomer measures d
2
@
R 2 z
, such as :
(33)
Which is nothing but the Hubble's red shift law, which still applies in this variable light velocity conditions. From mesurement of d
2
, c
2
and z we can derive the so called Hubble's constant, i.e. the age of universe.
Take R = 3/2 c t ( see reference
[ 1) ).
(34)
The age of universe corresponds to :
(35)
A g
= 1 value gives the standard model value.
- THE RED SHIFT AND THE DISTANCE EVALUATION :
Let us return to the Robertson-Walker metric, which provides :
(36)
In the classical approach, take the Einstein-de Sitter model
( k = o ). We get :
(37a)
With R = a t
2/3
we have :
(b)
whence
(38)
If z is weak : d
2
»
3/2 ct 2 z
if z tends to infinite : d
2
tends to 3 c t
2
Of course the Mattig's formula gives the same result :
(39a)
(b)
(c)
Let us return to the Robertson Walker metric, following our model, with k = - 1 :
(d)
write :
(e)
(f)
Let :
(g)
Then :
(h)
For radial pathes :
(i)
(j)
Log ( 1 + z ) = Arg th u
(k)
(l)
When z tends to infinite we refind the horizon (3/2)c
2 t 2
, which is twice smaller than the standard value 3 c
2 t 2
.
Notice this is completely similar to the law giving v
r
/c ( where is the radial velocity ) as a function of z, ine the standard model.
Let us compare the distances, as given by our model and the standard model.
(40)
They are similar for weak z values.
- THE QUASARS PROBLEM :
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 c
T.
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 P
QSO
the absolute power emited by a quasar and P
Gthe absolute power emited by a galaxy. The relative power density of the QSO, with respect to a galaxy, is :
(41)
But in our model the galaxies are no longer constant in size. They grow in time. Suppose the quasar is imbedded in a galaxy. The size of this galaxy will grow like ( 1+ z ). Such as our correcting term, with respect to the standard values for power density, will involve three effects :
-
Change for the distance ( they are a little bit closer ).
-
Change for the diameter ( due to the variation of c ).
-
Change of the galaxy's size.
Given P
QSO
and P
Gthe coefficient (57) becomes
p
' =
p x
with :
(42)
i.e :
(43)
z h x 0 1 1
0,13
1.025
0,606
The closest quasar
0,2
1.03
0.467
0.5
1.048
0.177
1
1.024
0.0463
1.5
0.985
0.0157
2
0.946
0.0063
2.5
0.912
0.00296
3
0.882
0.00152
3.5
0.856
0.000842
4
0.834
0.0005
The most distant quasar
5
0.8
0.000201
6
0.771
0.0001
7
0.75
0.0000485
8
0.731
0.0000275
9
0.71
0.0000162
10
0.7
0.0000102
We see that this correction reduces the absolute magnitude of the observed quasar, and that this correction increases with z. Thus, would this model would be good, the classical model would have greatly overestimated the volumic power density of quasars. In addition the observed increase of absolute magnitude of quasars could be due to the secular change in c.
Classically the galaxies' span are related to the Jeans' length, but the model does not provide any avaible information about the sizes of some emitting objects like stars or quasars. It depends on the energy emission process. As we have not defined some possible gauge relation for the fusion coefficients we have no avaible model yet. Anyway the quasars could grow in time, like galaxies and the observations tend to support this hypothesis, that will be examined with more details in the next paper devoted to the detailed interaction of the model and avaible observations. .
- ASSOCIATED GAUGE RELATIONS.
The ionization energy of hydrogen obeys : E
i
= 1/2
a 2 m e c 2
, where
a
is the fine structure constant and m
e
the mass of the electron. We have assumed that E
i
»
R g
with
g
= 1, in order to fit with the classical model, see (25), (33) and (35). Introduce the electron-proton mass ratio
d = m e /m p
. According to the first paper
[ 1 ] m p
»
m n
»
m
»
R. So that mc
2
is an absolute constant. Then :
(44)
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 :
(45)
Introducing the gauge relations for h and c we get : e
2 / e o
»
R 3/2
and the electromagnetic force F
em
=e 2 /4 p e o R 2
»
R-1/2
.
Then :
(46)
which is similar to an old idea of Dirac
[
4, 5
]
.
Let us compute the Bohr radius :
(47)
9.2) Now we take
a
»
absolute constant. Then m
e
»
R 2
. and
e 2 / e o
»
R. The Compton length of the electron h/m
e
c, 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
,
a 2 ( g p/ g e) (m e /m p)
, m
e /m p
,where g
p
and g e
are the gyromagnetic ratios of the proton and the electron. Following B.E.G. PAGEL
[ 13 ]
, we have :
Effect
Quantity Approximate 3
s
upper
limit to variation.
Optical doublet splitting.
a
3 %
Comparizon of optical and 21 cm redshifts.
a 2 ( g p/ g e) (m e /m p) 10 -3
Comparizon of hydrogen and metal redshifts.
d = m e /m p
50%
As a consequence we choose the second possibility, with a variable mass ratio. Notice that in both cases we get g
p/ g e
»
1/R.
1O) CONCLUSION
Here we tried to extend the model introduced in reference
[ 1 ]
to electromagnetism. A gauge law was suggested : we assumed the ionisation energy E i
( Rydberg constant ) to vary like R
g
. 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 = m e /m p
which varies like R we get a fine structure constant
a
which behaves like an absolute constant.
REFERENCES :
[ 1 ]
J.P.PETIT : An interpretation of cosmological model with variable light velocity. Modern Physics Letters A......
[ 2 ]
E.A. MILNE : Kinematic Relativity Oxford 1948.
[ 3 ]
F.HOYLE & J.V.NARLIKAR : Cosmological models in conformally invariant gravitational theory. Mon. Notices Roy. Astr. Soc. 1972 155 pp 3O5-325.
[ 4 ]
P.A. DIRAC : 1937, Nature,
139,
323 [ 5 ]
P.A. DIRAC : 1973 Proc. Roy. Soc. London ,
A333
, 4O3
[ 6 ]
V.CANUTO & J.LODENQUAI : Dirac cosmology, Ap.J.
211
: 342-356 1977 January 15.
[ 7 ]
T.C.VAN FLANDERN : Is the gravitational constant changing ? Ap.J,
248
: 813-816
[ 8 ]
A.JULG. Dirac's large numbers hypothesis and continuous creation. Ap.J.
271
: 9-1O 1983 August 1
[ 9 ]
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
[ 10 ]
B.E.J.PAGEL : On the limits to past variability of the proton-electron mass ratio set by quasar absorbtion line redshifts. Mon. Not. astr. Soc. (1977), short communication, 81P-85P.
[ 11 ]
A.M.WOLFE & R.L.BROWN & M.S.ROBERTS. Limits on the variation of fundamental atomic quantities over cosmic time scales. Physical Review Letters Vol 37, number 4, 26 July 1976.
[ 12 ]
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236
: L1O5-1O8, 198O March 15
[ 13 ]
B.E.J.PAGEL. Implication of quasar spectroscopy for constancy of constants. Phil. Trans. R. Soc. Lond. 3 31O, 245-247 , 1983.