HUBBLE LAW, CONFORMAL TRANSFORMATION OF METRICS AND
QUANTUM POSTULATES (Cosmological deceleration of the course of time)
I. Taganov1
1
Russian Geographical Society (Russian Academy of Sciences). Grivtsova Str.,
10. Saint-Petersburg 190 000 Russia.
Abstract. The Robertson-Walker metric gijRW ( ) can be considered as a conformal
transformation gijRW ( )ij  e ( )gijM (t) of Minkowskian metric gijM (t) . This transformation is defined by the relation d  a&   a , where a is the scale factor. It apdt
a
pears, that the relation aa  const is a geometric form of the Planck’s and de Broglie’s quantum postulates. An important consequence of these relations is Hubble’s
1
law a& H  const (z  Ht) with conformal mapping: z  1  (1  2H ) 2 . Conformal time  “decelerates” with respect to Newtonian time t and for example,
 p  3 2 H 1 corresponds to Hubble age of the universe t p  H1 . The duration of
any physical process, measured in units of conformal time, is always bigger, than
the corresponding duration, measured in units of invariable Newtonian time. This
feature leads specifically in isotopic geochronology to systematic divergence of
Newtonian ages, estimated by the use of isotopes with different decay constants.
An analysis of the isotope geochronology data for 40 samples with the ages up to
4,5 Gyr provides reliable observational confirmation of the relation   t  H t 2
2
and enables to estimate the Hubble parameter. This evaluation, calculated not from
remote star redshifts, but from the isotopic ages of Earth and Moon rocks and minerals, is H e  53  5 km s-1 Mpc-1.
1. Deceleration of the course of time.
Cosmological doctrine of the expanding universe was formed within the framework of general theory of relativity, disregarding the relativistic quantum ideology.
It gave birth to several methodological discrepancies, focused in the unforeseen
variability of the speed of light. This contradiction with quantum physics of photons appears especially annoying, considering that all astrophysical surveys of
cosmic radiation, supporting an idea of the expanding universe, presume universal
constancy of the speed of light.
To describe the universe expansion, a cosmology uses non-stationary metrics, defining intervals:
ds2  c2d2  a()2 dr 2
(1)
1
The scale factor a( ) characterizes relative size of spatial sections as a function of
time: R()  a()R 0 . This normalized form includes an initial spatial standard R 0 .
In the space-time with interval (1) the speed of light, defined as a velocity at the
zero-interval geodesic lines (ds  0) is a variable:
dr  c
. Variability of the speed of light in the space-time with a nond
a()
stationary metric was noticed many years ago. For example, in (McVittie, 1956
[1]), for the spatially homogenous and isotropic universe with Robertson-Walker
metric and the interval of a type (1) with dimensionless coordinates (, , ) and the
curvature parameter k , had been deduced the equation:
k2 )
c(1

d 
4
for variable velocity at the zero-interval geodesic
d
a()R 0
lines. Geometrically, the speed of light can be considered a universal constant only
at the world lines with Minkowski’s metric that includes Newtonian time t and defines the interval:
ds2  c2dt 2  dr 2
(2)
The use of non-stationary metrics in cosmology is at variance with the relativistic
physics of photons, that states universal constancy of the vacuum speed of light. To
clear out this contradiction it is necessary to use in cosmology the explicit occurrence of quantum postulates. We have to suggest that in the universe with nonstationary metric the photon parameters are determined in accordance with
Planck’s and de Broglie’s relations. Then in a free space of expanding universe the
phase-velocity of light does not depend on increasing photon wave-length being
the same as the group-velocity and can be considered a universal constant. Relativistic energy E   m 02c 4  p 2c 2 of photon ( m 0  0 ) is E  pc and with the use of
Planck’s ( E  h ) and de Broglie’s ( p  h ) relations it can be transformed to:

  c  ck . This equality leads to the following relation for the initial phase- and


d
 c . An evolution of exgroup- photon velocities in free space: 0  0
k0
dk 0

 0 and
panding universe will change the photon parameters:   c  c

a 0
a
k
k 1  1
 0 . But because of the relation’s   c  ck invariance with

a 0
a

respect to uniform transformation of photon parameters, the speed of light will not
change in spite of the scale factor variation:   d  c  co n s t . Such univerk
dk
sal constancy of the speed of light results in the following geometrical relation for
velocities at the zero-interval geodesic lines:
2
dr
dt
 a dr
d
 c  const
(3)
These equalities can be transformed into the relation for the   (a prime) and t  (a
dot) derivatives of the scale factor:
a& aa  const
(4)
For non-relativistic test-particle the relation aa  const can be deduced directly
from de Broglie’s equation: P  m0 v  h . Let us consider an effective impulse

Pc of a test-particle, originated from the universe expansion increasing its wavelength: Pc  m0 vc  m0 d (ar0 ) ;  c  a 0 . Here r0  const is initial coordinate of a
d
test-particle with initial wave-length  0  const . De Broglie’s equation for such
test-particle has the form: m0 r0 da  h
, or aa  const .
d
a 0
Another interpretation of the foregoing analysis can be performed in terms of
conformal transformations. The non-stationary metric with interval (1) can be considered as a result of the conformal transformation of Minkowski’s metric (Synge,
1960 [2]): the transformation t   e

 ( )
2
d , with ()  2ln a results in
d2  e ( )dt 2 and the interval (1) transforms to:
ds2  a 2 (c2dt 2  dr 2 )
(5)
Any variable scale factor does not change the value of the velocity at radial geodesic lines: dr  c  const , because the interval (5) at the photon world lines with
dt
ds  0 corresponds to Minkowski’s metric. This conformal transformation in the
form of a relation between the scale factor time derivatives coincides with Eq. (4):
a& aa
&
a& a 2a  a2
(6)
The conformal transformation in question and equalities (3) define a connecting
equation for the time variable  of non-stationary metric with interval (1) and
Newtonian time:
d
dt
a
(7)
An analysis of this equation demonstrates that in expanding universe all characteristic intervals  increase with respect to the invariable scale of Newtonian time,
accompanying the growth of the scale factor. The time in expanding universe “delays” or its course decelerates with respect to the invariable scale of Newtonian
3
time. A course of time 1 can be defined as a characteristic inverted to any chosen standard time interval  . The growth of standard time interval is equal to the
deceleration of the course of time. A notion of “the course of time” firstly was used
probably by Einstein and Minkowski in their pioneering articles, dealing with the
relativity theory (Einstein, 1905 [3], Minkowski, 1908 [4]). Decelerating time in
Eqs. (1, 7) cannot be Newtonian time of the natural science, which is always considered and used as a uniform, isotropic and invariable continuum. If we believe
that the specific features of our world’s physics are: the universe expansion, the
constancy of the speed of light and a validity of the quantum postulates, then we
could to name this decelerating time – the “physical” time. Formulae (1) and (7)
describe an increase of the 4-dimensional space-time volume. Therefore an evolution of our universe appears not only as an expansion of 3-dimensional space, but
as an expansion of 4-dimensional space-time, where time is not Newtonian, but the
physical one.
To deduce an equation, describing a non-stationary state of space-time, it is
enough to exclude by differentiation the uncertain constant in Eq. (4): aa  a2  0 .
In the form of the equation for the cosmological deceleration parameter with physical time q  this relation appears as:
q    aa
a 2
1
(8)
In cosmology, based on Einstein’s field equations and specifically in Friedmann’s
model, the Hubble’s law, declaring a proportionality of the redshifts and the distances between radiation sources and detectors, is considered as an approximate
relation, acceptable only for minor values of redshifts. Mysteriously, Hubble law
demonstrates unusually good conformity with astrophysical data for cosmic objects
with large redshifts. This enigma has innate explanation in the cosmology, that
takes into account the quantum postulates. Eq. (4) leads to the equation: a& const
with the solution: a  Ht  const . Lemaitre’s formula
z
  0
0

a 0   0
0
 a 1
(9)
used for experimental evaluation of redshifts z , and relation R  ct for the distance R between radiation source and detector allow to transform this solution into
a standard form of Hubble’s law:
z  Ht  H R
c
(10)
Therefore Hubble’s law appears a direct cosmological consequence of Planck’s
and de Broglie’s quantum postulates, being not an approximate formula, but the
exact relation. A proof of Hubble’s law does not depend on the uniformity and
4
isotropy of the space-time. Hubble parameter H must be considered not as variable
factor, but as the universal constant.
Solutions of Eq. (8) have the forms:
  0 : a  a 0 ,a  a0  H
a   a 02  2a 0 H 
  0 : a  0,a  a0  H
a   2H 
1
1
2
2
(11)
(12)
Resolving Eq. (7) with right part from Eq. 12 for initial condition: t  0 :   0 one
can get an algebraic form of the connecting equation for physical and Newtonian
time:
  t  H t2
2
(13)
This relation reveals that duration of a process, measured by decelerating physical
time, will be always bigger, than corresponding duration, measured by a scale of
invariable Newtonian time. For example, experimental evaluations of Hubble constant: (50  80) km s-1 Mps-1 (0,052  0,083)Gyr 1 allow to compare the universe
ages in physical and Newtonian time. Standard assumption: a p  1 (henceforth the
index “p” marks the values of characteristics in our epoch; the index “e” marks experimental evaluations of characteristics) allows with the help of the Hubble’s law
in the form a  Ht to evaluate Newtonian age of universe:
t pe  He1  (12  19,2) Gyr. Substitution of these values in Eq. (13) leads to the following evaluations of the universe physical age: pe  3 H 1  (18  28,8) Gyr.
2
2. Observational evidences of the time deceleration
Characteristic feature of the foregoing analysis, leading to an idea of decelerating
time, is the use of basic quantum postulates. Therefore deceleration of the course
of physical time can be considered as a specific quantum phenomenon in cosmology, an essential supplement to the space expansion. If all the processes in expanding universe develop in decelerating physical time, then just this time must be used
in physical laws of high accuracy.
2.1 Redshifts. The redshift phenomenon can be considered not only as an evidence
of the universe space expansion, but also as an experimental confirmation of deceleration of the course of physical time. To estimate the value of redshift, usually
changing wave-lengths of photons (  a 0 ) are compared, but it is also acceptable
a
) or periods of photons (  0 ) :
to compare frequencies (  c
c
a 0
  0
 
z
 0
. Photon frequency   1 can be used as a physical
0
0
5
manifestation and a standard of the course of physical time. Therefore, one of the
possible interpretations of redshift value can be the relative deceleration of the
course of physical time for two different epochs in the universe evolution.
The use of the relations: a  z  1(9), a 0  1 and   rL allows to transform Eq.
c
(11) into the redshift – luminosity distance dependence:

z  1  2H rL
c

1
2
1
(14)
This formula with the luminosity distance rL , measured in mega parsecs, can be
transformed with the use of the relation: 5lg rL  m  25 . Here m is the “distance
modulus” – the difference between the apparent magnitude of the radiation source
and its absolute magnitude. In accordance with Eq. (14) the distance modulus is
related to the redshift via:
m  5lg{ c
2H
[(z  1) 2  1]}  25
(15)
Recently, significant progress in the field of investigation of high redshift sources
has been made by using Type Ia supernovae (Riess et al, 1998 [5]; Riess et all,
2004 [6]; Perlmutter et al, 1999 [7]). Supernovae are bright stars, and Type Ia’s in
particular all seem to be of nearly uniform intrinsic luminosity with absolute magnitude around 19,5 . Astrophysical data for supernovae with redshifts from 0,008
to 1,75, published in [5 – 7] can be used for the verification of formulae (14, 15),
deduced in the decelerating time theory. Fig. 1 demonstrates the conformity at a
confidence level 1 of the theoretical formulae (14, 15) (the curve 1) and experimental data: circles from the High-Z Supernova Team [5, 6], triangles from the
Supernova Cosmology Project [7]. The congruence of experimental data and theoretical curve 1 at Fig. 1 corresponds to the following value of the Hubble’s constant estimated with the use of formulae (14, 15): H e  (64  6) km s-1 Mpc-1
(0,065  0,005Gyr 1 ) . This estimation is very close to an estimate H e  (65  7)
km s-1Mpc-1 (0,066  0,007Gyr 1 ) from [5]. An average relative disagreement
100(me  m theor )
m 
(%) of the theoretical curve 1 at Fig. 1 and astrophysical
me
data not more than 0,7%.
It seems useful to compare the relation (14) with well-known Mattig’s formula
(Mattig, 1958 [8]): rL  c 2 [zq p  (q p  1)( 2q p z  1  1)] . For q p  1 this forHq p
mula transforms into the linear approximation: cz  HrL . Calculations with the use
of this linear relation (broken curve 2 at Fig. 1) diverge distinctly from experimental data with z  0,1 . Mattig’s formula for cosmological model with the flat
6
m
1
3
2
45
40
35
30
0
0,5
1
1,5
2
z
m
4
2
1
0
-2
0
3
2
-4
z
Fig. 1
spatial geometry ( q p  1 ) has the form: rL  2c (z  1  z  1) . Calculations
2
H
with the use of this formula (broken curve 3 at Fig. 1) conform better (in comparison with the foregoing linear relation) to experimental data. Mattig’s formula with
q p  0 coincides with (14).
2.2 Supernovae light-curves time-axis broadening. Recent measurements of the
light curves of Type Ia supernovae with redshifts up to 0, 83 discovered by the Supernovae Cosmology Project, gave evidence for light-curve time-axis broadening
of the type: w  s(1  z) (Goldhaber et al, 1995 [9]). Dimensionless light-curve
width w  
is nearly equal to (1  z) , because it was demonstrated that the
0
linear stretch factor s ; 1, which parameterizes the light-curve timescale, appears
independent of z, and applies equally well to the declining and rising parts of the
light-curve. In a conception of decelerating physical time light-curves broadening
has the same explanation, as a growth of photon periods in redshift phenomenon:
finite durations of all physical processes in the past appear longer at the present,
because of deceleration of the course of physical time. Quantum condition of the
constancy of the speed of light in expanding universe, formulated with the use of
7
finite intervals: r
 c  const , for r  ar0 necessitates   a0 . The use of

Lemaitre’s formula (9): z  a  1 transforms the last equality into the light-curve
width – redshift dependence: w  
 1  z , which precisely corresponds to
0
the observational data. A phenomenon of light-curves broadening principally differs from photon periods growth in redshifts. Photon period ph (typically
2  1015 s ) is a microscopic parameter and its growth could be explained in accordance with quantum relation ph   by the evolutional increase of wave-length.
c
The light-curve width, being typically over 2  106 s and differing from photon period by 21 orders of magnitude, is a macroscopic parameter and its evolutional increase can not be explained by the growth of some generally acknowledged interrelated space interval. Light-curve time-axis broadening is expressive observational confirmation of the cosmological increase of finite physical time intervals that is
equivalent to deceleration of the course of physical time.
2.3 Illusion of accelerating universe expansion. Under the assumption that the
energy density of the universe is dominated by matter and vacuum components, the
experimental data in [5 – 7] were converted into limits on parameters  M ,   in
Friedmann’s cosmological model. After analysis of the impressive statistics of experimental pairs (mi ,zi ) for Type Ia supernovae authors of [5 – 7] favor a positive
cosmological constant and strongly rule out the model ( M ,   )  (1,0) . The analyzed experimental data appeared inconsistent with an open universe with zero
cosmological constant. Both research teams estimated negative values of the deceleration parameter ( q p  0 ), indicating an accelerating universe expansion. All these
conclusions were made after a comparison of experimental data with Friedmann’s
cosmological model.
These results have an innate qualitative interpretation in the theory of decelerating
physical time. Using Eqs. 6, one can get a connection relation for deceleration pa& with
rameter with physical time q  and the deceleration parameter q t   aa&
a&2
Newtonian time:
qt  q  1
(16)
This formula reveals, for example, that a minor ( q   1 ) deceleration of the universe expansion in physical time appears an accelerating expansion ( q t  0 ) in
Newtonian time. The uniform expansion in Newtonian time ( q t  0 ) appears the
decelerating expansion ( q   1 ) in physical time. Such coordinate effects are similar to the kinematics modifications in the transformations between inertial and noninertial frames of reference. Uniform or moderately decelerating universe expansion in physical time has the “conformal illusion” of the accelerating expansion in
Newtonian time.
8
Besides the use of Friedmann’s cosmological model for demonstration of the accelerating universe expansion there is another indicative observational evidence of
the same phenomenon. The Hubble Key Program using Hubble Space Telescope
(Freedman et al, 2001 [10]), by studying Cepheid variables in a range of galaxies
at distances out to 20 Mpc ( z  0,1) estimated the Hubble constant as H e  (72  8)
km s-1Mpc-1 (0,075  0,008Gyr 1 ) . For remote galaxies ( z  0,3  0,6 ) emitting in
earlier epochs of the universe evolution an analysis of Type Ia supernovae data [5]
gave the lesser value of the Hubble constant: H e  (65  7) km s-1Mpc-1
(0,066  0,007Gyr 1 ) . This difference in the estimates of the Hubble constant evidently exhibits an accelerating universe expansion. To understand this illusion it is
necessary to remember that Hubble’s law (10) is an exact relation only for Newtonian time. When astrophysical processes developing in physical time are used for
estimation of Hubble constant, actually the following inexact relations are employed: z  Ht ; H* . A relation between exact value of Hubble constant H and its
approximate estimation H* can be deduced with the help of Eq. (13):
z  Ht ; H*  H* (t  H t 2 ) . Using the substitution t  z one can get the rela2
H
*
tion: H  2H
, demonstrating decrease of the approximate Hubble constant
2z
with the growth of redshifts in spectra used for its estimation. This relation reveals
how an illusion of accelerating universe expansion springs up in the calculations
disregarding a difference between physical and Newtonian time.
2.4 Riddle of Pioneers anomalous accelerations. Analyses of radio Doppler and
ranging data from distant spacecrafts in the solar system indicated that an apparent
anomalous acceleration directed toward the Sun is acting on: Pioneers 10 and 11 –
a P  (8,74  1,33) 108 cm / s 2 ; Galileo – a P  (8  3) 108 cm / s 2 ; Ulysses –
a P  (12  3)  108 cm / s 2 (Anderson, et al, 2002 [11]). It seems that the values of
registered strange accelerations appear independent of the velocities of spacecrafts
in the range: 7,2  12,2 km/s. At the heliocentric distances less than 15 AU these
accelerations are masked by accelerations from the pressures of solar radiations
and solar wind. Theory of the decelerating physical time can explain a direction
and a value of these anomalous accelerations. To calculate an acceleration of
spacecraft a in [11] an observed Doppler frequency obs (t) is compared with predicted frequency mod (t) , estimated from the theoretical modeling in Newtonian
time of a spacecraft velocity v mod (t) : obs (t)  mod (t)  0 2at ;
c
mod (t)  0[1  2vmod (t) c ] (Eq. (15) in [11]). A front of radio-wave or lowfrequency photon propagates in free space uniformly only in physical time: r  c .
To calculate in the process of modeling of mod (t) a velocity of the radio-wave in
Newtonian time c ( t ) , one must use in accordance with Eq. 13 a differentiation of
the trajectory r  c  c(t  H t 2 ) and will get: c(t)  dr  c(1  Ht) . Taking into
2
dt
9
account that for relatively short observations:
(t)
(t)  0 [1 
frequency takes the form: mod
0
2vmod
1  Ht
; 0 , transformed model
]  0[Ht 
mod
0 ] . To evaluc
ate an additional acceleration a P , corresponding to this modified frequency one
(t)
(t)  0 (2a  a P )t , which leads to: a P  cH .
can use the relation: obs (t)  mod
c
A short way to estimate this additional acceleration is the double differentiation of
2
the photon trajectory r  c  c(t  H t 2 ) : a P  d r 2  cH . This acceleration has
2
dt
the same direction as a velocity toward an observer at the Earth and for remote
spacecrafts this direction almost coincides with a direction toward the Sun. For recent evaluation of the Hubble constant: H e  (72  8) km s-1Mpc-1
(t)
(2,4  0,3) 1018 s 1 for relatively short cosmic distances [10] an estimation of additional acceleration under discussion is very close to discovered anomalous accelerations of spacecrafts: a P  cHe  (6,3  8,1) 108 cm / s 2 .
2.5 Isotopic geochronology. Aside from the redshifts in spectra of remote enigmatic stars, there is quite “earthly” evidence of deceleration of the course of time,
which is almost under our feet. Isotopic geochronology uses the processes of radioactive decay of several unstable isotopes with a long life to estimate the ages of a
variety of rocks and minerals. A basis of the isotopic geochronology methodology
consists of the following formulae:
N It  N 0I exp   It t 
t 
I
e
LIe
 It
(17)
(18)
Here N It is the current total of I-isotope atoms, providing that initial ( t  0 ) quantity of unstable atoms is N0I with the decay constant  It . The function LIe , determined
by the concentrations of isotopes in a sample, could be of different forms, but the
decay constant of the main isotope always serves as a divisor in formulae of the
type (18). A probability of the decomposition of each atomic nucleus of an unstable isotope is inversely proportional to the average lifetime of the activated state of
the isotope nucleus. In accordance with a conception of the decelerating time under
discussion, this average lifetime must be estimated not in the uniform Newtonian
time scale, but in decelerating physical time. In this case a difference between the
actual quantity of the decayed isotope nuclei and a prediction, made with the use of
the kinetic equations with Newtonian time, is the bigger, the longer a half-decay
time of an isotope. Therefore a basis of the more efficient methodology of the isotopic geochronology must consist of the kinetic equations (17, 18), rewritten with
physical time. In this case the value of the decay constant  I for physical time is
10
always less than the corresponding decay constant  It for Newtonian time. A comparison of these two sets of formulae (17, 18) with Newtonian and physical time
results in the following relation for differences in the age estimations, evaluated
with the use of two different isotopes:
1
 2
 2
te  te  te [
(2)
 (1)
 t
  (1)

 (2)


(2) (1)  1]  e 
(1) 
(2) 
 t
t
t 

(19)
This formula leads to the conclusion that the difference between the physical age
of a geological sample and the estimation of its Newtonian age is more, as the value of the decay constant of the isotope, used for analysis, becomes less. It is important that if we try to estimate Newtonian age of a certain sample (which has only one actual age), using two isotopes with different values of decay constants, we
should evaluate two different ages. Moreover, the estimation of Newtonian age,
evaluated with the help of an isotope with a lesser decay constant, will be smaller.
These enigmatic from the classical point of view, phenomena reveal themselves in
the geological sample analyses with simultaneous use of uranium-lead (U-Pb) and
rubidium-strontium (Rb-Sr) isotopic methods (Gerling & Ovchinnikova, 1970
[12]). It appears, that (U-Pb)-ages are always bigger than corresponding (Rb-Sr)ages. From the standpoint of the decelerating time conception, differences revealed
in these age estimations are perfectly normal, because the value of the decay con-10
stant of 238U (  238
(yr)-1 ) is more than 10 times bigger than the decay
t = 1,55 10
-11
-1
constant for 87Rb (  87
t = 1,42 10 (yr) ).
It is possible to find in recent articles in the field of isotopic geochronology more
than 40 independent estimates of the geological sample ages, made simultaneously
by (U-Pb) and (Rb-Sr) isotopic methods, for rocks from different sites at the Earth
and Moon (Taganov, 2003 [13, Table 5.1]). For these samples with ages 0,3 – 5
Gyr the average difference of Newtonian age estimates, evaluated by two isotopic
methods, is around 0,13 Gyr (+ 6%), that is more than 5 times bigger than the average error level in isotopic geochronology. Fig. 2 demonstrates the difference in
geological sample ages, evaluated by (Rb-Sr) (circles near line 1) and (U-Pb)
(squares near line 2) isotopic methods. The ordinates at Fig. 2 correspond to avert 238  t 87
e
age Newtonian ages: t e  e
(Gyr). The abscissas are the age estimations
2
(Gyr) in Newtonian and physical time. The constant decay values for physical
-10
-11
time:  238
yr-1 and  87
yr-1 have been estimated from a
e = 1,458 10
e = 1, 25 10
condition of the minimal total difference between the estimations of the sample
physical ages. In comparison with average difference between the estimations of
Newtonian ages average difference between the estimations of physical ages (triangles near curve 3 at Fig. 2) diminishes almost 10 times and is less than 0,014
Gyr ( 0,8% at the average and 0,25% for samples older than 2 Gyr). These values correspond to the average error level in isotopic geochronology.
11
t eI , eI
6
3
2
1
4
2
2
4
6
 te 
Fig. 2
If we assume that (U-Pb) method corresponds to index 1 and (Rb-Sr) method to
index 2 in Eq. (19), then this formula will take the form:
87
t e238  t 87
e  0,068t e  0,06e . In spite of the relatively high error level in isotopic
geochronology, average relative difference (referred to the sample age) between
experimental estimations of the ages and evaluations with the use of this theoretical formula appears not more than 4%.
Conception of the decelerating physical time gives a unique chance to estimate a
value of the Hubble constant not from an analysis of remote star redshifts, but from
geochronological data for Earth and Moon rocks and minerals. Hubble’s constant
calculated by the method of minimum variance with the use of Eq. (13) and above
mentioned geochronological data for Earth and Moon rocks and minerals, is:
H e  53  5 km s-1 Mpc-1. This value is close enough to the average estimation: 55
km s-1 Mpc-1 in reviews [14, 15]. The curve 3 at Fig. 2 corresponds to the Eq. (13)
with H e  53  5 km s-1 Mpc-1 and its average relative divergence from experimental data is not more than 3%.
2. Conclusion
Discussed evidences of a deceleration of the course of physical time have a rather
different nature. The redshift dependence on the deceleration of the course of time
could be traditionally explained as a result of the sole space expansion. But supernovae light-curves time-axis broadening and an occurrence of different Newtonian
ages of the sole geological sample, estimated with the help of isotopes with different rates of decay, demonstrate the “pure” phenomenon of cosmological deceleration of the course of physical time. These effects have no obvious connection with
any moving in the space and can not be explained by an evolution of the exclusive
space geometry. Recently discovered acceleration of the universe expansion in
Newtonian time could transform in the nearest future in a remarkable experimental
evidence of deceleration of the course of physical time.
12
Conception of the decelerating physical time, which leads to the Eq. (12), can be
summarized in a laconic geometrical description of the universe evolution:
Square of growing universe radius is proportional to universe age, measured
by decelerating physical time.
There are no formal contradictions between cosmological interpretation of Einstein’s field equations and conception of the deceleration of the course of physical
time. Eq. (8) can be considered as a condition, defining the cosmic fluid equation
of state, resulting in the deceleration of physical time. For example, the substitution
of Eq. (12) in Friedmann’s model equations leads to the following cosmic fluid
equation of state:
  4vac  3p  3kc
2
8GH
1
(20)
Here G is Newton’s constant and a connection of the vacuum energy density  vac
2
and a cosmological constant  is: vac  c 
. For a cosmological model with
8G
the flat space geometry (k  0) and   0 Eq. (20) coincides with the photon gas
equation of state:   3p .
There is an inspiring analogy between current problems of the expanding universe
doctrine and an epoch of a creation of the special theory of relativity. Results of
Michelson-Morley experiments (1887) initially had been explained by contraction
of the linear dimensions of bodies (Fitzgerald, 1893). Later, the compensating contraction of the time intervals had been introduced by Lorentz transformations (Lorentz, 1895). Finally Lorentz transformations had been interpreted by Einstein’s
hypothesis of relativity and its postulate of the constancy of the speed of light in all
inertial systems of references (1905). Today, as in the days of Fitzgerald, the redshifts and a cooling of the relict photon radiation are explained by the evolutional
growth of sole space intervals in expanding universe. Similar to the special theory
of relativity the possible approach to meet the quantum statement of the speed of
light constancy in cosmological doctrine of the expanding universe could be an
admission of the compensating evolutional growth of time intervals, which is
equivalent to deceleration of the course of physical time.
Acknowledgments
For constructive criticism and helpful discussions I thank Yu.V. Baryshev, A.L.
Gromov, A.I. Tsigan and D.A. Varshalovich, provided a number of important insights, especially on metric transformations and interpretations of light-curve timeaxis broadening, Pioneers anomalous accelerations and isotopic geochronology data.
13
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