Key words: dark energy, Etherington's identity, General Relativity, Hubble's law, luminosity distance, redshift.
АНОТАЦІЯ. Доведено, що тотожність Етерінгтона є паралогізмом. Ця тотожність ґрунтується на уявному релятивістському уповільненні плину власного часу галактики в (1+z) разів у системі відліку просторових координат та часу (СВ) спостерігача, в якій тотожністю фактично ігнорується наявність релятивістської анізотропії світності зірок, що швидко віддаляються від нього. Етерінгтон не взяв уваги той факт, що Всесвіт є однорідним лише в супутній в розширному Всесвіті СВ (ССВРВ). І, отже, він необачно зробив «коктейль» з явищ і особливостей, що є властивими двом різним СВ. Показано, що згідно загальної теорії відносності (ЗТВ) лінійній залежності Хаббла можуть підкорятися лише поперечні метричні відстані – поперечна супутня відстань і подібна до неї відстань за кутовим діаметром. Поперечна супутня відстань належить ССВРВ і визначається за червоним зсувом z довжини хвилі випромінювання. Відстань за кутовим діаметром належить СВ спостерігача розширення Всесвіту і визначається за червоним зсувом частоти хвилі випромінювання. Світимісна ж відстань не є поперечною метричною відстанню. Томуто, її залежність від червоного зсуву і не є лінійною. Взято до уваги, що стала Хаббла, як і стандарти довжини, і стала швидкості світла, є принципово незмінною величиною в жорстких СВ. Емпірично знайдено її точне значення.
Ключові слова: темна енергія, тотожність Етерінгтона, Загальна Теорія Відносності, закон Хаббла, світимісна відстань, червоний зсув.
1. Introduction
Only two known solutions of equations of GR gravitational field can be juxtaposed to expanding Universe. Those are: Schwarzschild solution when the value of cosmological constant is Λ=3H^{2}/c^{2}, which corresponds to the local representation of the process of Universe expansion, and Friedman solution when Λ=0 (Λ≠0 in ΛCDM model), which corresponds to the global representation of the process of Universe expansion.
According to Schwarzschild solution and Einstein hypothesis distant galaxies are falling free on the “event horizon” constantly moving along the geodesic lines of spacetime continuum (STC) of their observer. They fundamentally cannot reach that pseudohorizon of the past because it belongs (at any moment of observer’s time) to infinitely far cosmological past (in coordinate cosmological time) as well as to infinitely distant objects of the Universe in its background Euclidean space (Zel’dovich&Grishchuk, 1988). And this is, of course, related to the conformity (Penrose, 1968) of these two infinities that are mutually compensated in the gravithermodynamic FR (GTFR) of Schwarzschild solution (Danylchenko, 2020). Exactly in this background Euclidean space of the Universe, where physical vacuum rests (Danylchenko, 2004), according to Weyl hypothesis (Weyl, 1923; 1930) galaxies perform only small peculiar moves. And standards of length are evolutionally decreasing together with all objects of matter in this space.
Friedman solution due to negligibly small values of average density of mass in the Universe (comparing to 3H^{2}/4πG) and pressure in the outer space (comparing to 3H^{2}c^{2}/4πG) is the special case of the Schwarzschild solution in the background Euclidean space of the Universe: namely in the FR of physical vacuum (Danylchenko, 2004) of identical comoving FR in the expanding Universe (CFREU) when the value of gravitational radius of astronomical object, from which the observation of Universe expansion is performed, is negligibly small. In contrast to Schwarzschild solution that includes pseudohorizon of events in the equations of Friedman solutions (as well as in the equations of Schwarzschild solution in background Euclidean space) event horizon (on which the speed of light is equal to zero) is absent. This denotes the absence of the Hubble radial motion of galaxies and, thus, the absence of relativistic effects in the space of Friedman solution. Galaxies in this space perform only small peculiar moves while distances between them are increasing in this space due to mutually proportional decreasing of the dimensions of both length standards and all material objects in this space. This, of course, requires the constant renormalization of nonnormalized spatial parameters to align them with the new values of length standard.
Thus, there fundamentally cannot be any radial motion of objects in Friedman solution because of the absence of singular surface of event horizon in this solution. Therefore, Doppler effect and other relativistic effects related to motion are not applicable for this solution.
2. Imaginary Etherington's Paradigm
Luminosity of astronomical objects of fast moving galaxies is isotropic only in their intrinsic FRs. However, this luminosity is also considered as isotropic in the intrinsic FR of any far observer during the astronomical photometric calculations. Therefore, relativistic transformations of angular coordinates are ignored in those calculations (Danylchenko, 2008; Weisskopf, 1972). Thereby, distances to galaxies are not determined by those calculations in the GTFR (Danylchenko, 2020), comoving with the matter of observer’s planet. They are, in fact, determined in the CFREU. Only in CFREU the luminosity of all galaxies is isotropic and the Universe itself is uniform. However, the imaginary Etherington’s identity (Etherington, 1933) for uncorrected luminosity distance D_{L} and for imaginary value of angular diameter distance ^{i}D_{A}, that corresponds to it, in the calculations is also taken into account:
Etherington’s identity is based on the imaginary relativistic dilation of intrinsic time of the galaxy by (1+z) times (Hogg, 2000). That time dilation (inherent to GTFR) is actually absent in CFREU when using the CTMHS. The primary frequency of radiation of the galaxy is the same as the frequency of identic to it radiation in nearby vicinity of observer in CFREU by CTMHS. That frequency is only progressively decreasing in “ontogenesis” (in the process of propagation of that radiation) together with decreasing of velocity of light in CFREU in accordance with metrically homogeneous scale of cosmological time (CTMHS) (Danylchenko, 2004).
Such imaginary time dilation by (1+z) times takes place in CFREU by physically homogeneous scale of cosmological time (CTFHS). The velocity of light does not change during its propagation when using the CTFHS, in contrast to CTMHS. The frequency of radiation that is lesser by (1+z) times corresponds to “phylogenesis” (to the process of the emission of that radiation). The infinitely far future becomes finite when using the exponential CTFHS.
Thus, we are dealing with the Etherington’s paralogism. This paralogism is caused by the mixing of observations in two different FRs – in CFREU and in GTFR. The Universe is observed in CFREU, as uniform (monotonous), with the single for all its objects cosmological time and without the presence of global relativistic effects. Consequently, the relativistic time dilation on the astronomical objects moving away from each other in the expanding Universe, which is observed in the GTFR of each of the objects, is imaginary (fictive) for CFREU (and, therefore, for the global perception). The Universe is nonuniform (not monotonous) in GTFR. And not only relativistic time dilation on far astronomical objects, but also relativistic anisotropy of their luminosity is observed in the GTFR. That relativistic anisotropy of luminosity was ignored by Etherington in contrast to relativistic time dilation. We go deeper into the cosmological future, the rate of physical processes increases according to CTMHS. That is, for sure, similar to the imaginary increasing of the rate of physical processes while deepening into cosmological past, caused by the use of the exponential scale of the cosmological time (CTES). This CTES is currently used in cosmology. Infinitely far cosmological past imaginarily becomes finite by that CTES.
Thus, we are dealing with the Etherington’s paralogism. This paralogism is caused by the mixing of observations in two different FRs – in CFREU and in GTFR. The Universe is observed in CFREU, as well as in Friedman solution, as uniform (monotonous), with the single for all its objects cosmological time and without the presence of global relativistic effects. Consequently, the relativistic time dilation on the astronomical objects moving away from each other in the expanding Universe, which is observed in the GTFR of each of the objects, is imaginary (fictive) for CFREU (and, therefore, for the global perception). The Universe is nonuniform (not monotonous) in GTFR. And not only relativistic time dilation on far astronomical objects, but also relativistic anisotropy of their luminosity is observed in the GTFR. That relativistic anisotropy of luminosity probably was ignored by Etherington in contrast to relativistic time dilation. Of course, Etherington could consider these relativistic effects (inherent to Schwarzschild solution only) as applicable for Friedman solution without understanding that the Hubble radial motion of objects of matter is absent in this solution.
The similar imaginary effect of mutually observed time dilation in two inertial FRs (IFRs) takes place in the clocks paradox in Special Relativity (SR). This is due to the fact that events at different points are not simultaneous events in the observer's IFR, although they are simultaneous events in the IFR of the observed moving body. And such resultant time dilation becomes true only for the observer that transits from one IFR to another IFR that moves in opposite direction in order to make remeeting possible. In the case of mutual observation of time dilation for two distant galaxies that are mutually distancing only in GTFR and resting in CFREU such difference between these galaxies is absent. That is why time dilation is fictive (seeming) for both distant galaxies.
It is worth to mention, that Lorentz transformations in SR are only the transformations of increments of the coordinates and not of the increments of metrical intervals (segments) (Danylchenko, 2020). That is, apparently, why relativistic dilation of only coordinate time, and not metric time, takes place in distancing galaxies when observations are performed in GTFR of that galaxies. Intrinsic time dilation in distancing galaxies, which is defined based on the redshift of radiation spectrum, is just the imaginary phenomenon. That time dilation is the similar to such imaginary phenomenon as the movement of the Sun across the earthly sky. And, of course, it is the similar to the phenomenon of Universe expansion in people’s world “from nothing” and “into nowhere”. That is why relativistic decreasing of the quantity of radiation quanta, which are registered by observer, is determined in its GTFR by the (z+1) factor, and not by (z+1)^{2} factor, which is declared by unreliable Etherington’s identity.
So, nowadays Etherington’s identity is only the imaginary Paradigm. The real astronomic identity should, of course, be taken instead of it:
This identity, in fact, connects the luminosity distance D_{L} with corrected photometric distance (Schwarzschild radius) in GTFR r=D_{A}. This corrected photometric distance is used in the Schwarzschild solution of GR gravitational field equations.
According to imaginary Etherington’s identity (paralogism) only imaginary (wrong) value of transverse comoving distance to the galaxy is determined nowadays in astronomical photometric calculations:
It is (1+z)^{1/2} times smaller than the right (real) value of transverse comoving distance to the galaxy:
And, therefore, it is (1+z)^{1/2} times smaller than the radial coordinate R=^{r}D_{M} of the galaxy in Euclidean space of CFREU in the moment of registration of its radiation (Danylchenko, 2004). And it is also (1+z)^{1/2} times bigger that the Schwarzschild radius of the galaxy in GTFR:
This radius is equal to radial coordinate R_{0} of the galaxy in CFREU in the moment of radiation emission. And, therefore, it is identical to corrected photometric distance to the galaxy in GTFR and is equal to the right (real) value of angular diameter distance ^{r}D_{A}. That is because of:
3. Imaginary Dark energy
Equations of GR gravitational field, in fact, describe the isolated from outer world states of matter and of its STC. Spatial distribution of the mass of matter in those equations specifies how the STC should be curved, while the STC specifies in what spatially inhomogeneous thermodynamic state matter should be.
Consequently, the external gravitational influence on that isolated matter and on its STC is not taken into account in those equations. That external influence can be reflected in the tensor of energymomentum due to the normalization (calibration) of gravitational constant that is the part of the expression for the Einstein’s constant:
where: ^{u}v_{cos} is the coordinate velocity of light in the outer space of Universe, G is the Newton’s gravitational constant.
It can be reflected in the tensor of spacetime curvature only using the normalization of cosmological Λpart. That is because in contrast to coordinate velocities of light that are defined by the tensor of energymomentum:
the constant of the velocity of light c (which is used in the spacetime curvature tensor) cannot be normalized. It is the spatiallytemporal invariant.
It is obvious, that the increment of logarithm of Hubble’s parameter defined by the Λpart may be connected by certain proportionality coefficient m with the increment of gravitational potential of outer space:
And, probably, this increment can be also connected by proportionality coefficient n with the increment at the distant point j of GTFR of gravitational Hubble’s potential:
Then, evolutional change of Hubble’s parameter can be defined by the following empirical dependency:
The dependency of the increment of metrical value of comoving distance ^{r}D_{M} to distant galaxy in CFREU on the increment of redshift z of radiation spectrum will be the following:
Dependencies of luminosity distance D_{L} to supernovas of type Ia on the redshift z of their radiation spectrum have been modeled (Riess et al, 1998; Semiz & Çamlibel, 2015; Dempsey, 2016; Soloviev, 2016) based on the results of astronomical observations of supernovas of type Ia (Perlmutter et al, 1999; Riess et al, 1998). According to graphs of that dependencies (q.v. Fig.) evolutionary change of Hubble’s parameter is almost not observed (k=0). That is because in case we use the most suitable values of Hubble constant the values of luminosity distance ^{g}D_{L} shown on graphs (q.v. Table) are very slightly different from their calculated values:
Figure: Dependencies of distances to astronomical objects on the redshift of radiation of astronomical objects z:
a) luminosity distance D_{L} (red solid line) to those objects (Soloviev, 2016) and metrical transverse comoving distance ^{r}D_{M } (blue dotted line) to astronomical objects in CFREU, as it is justified here;
b) graphical MD (straight) and ΛCDM (curve) models, and the onesigma confidencelevels. The inset shows the right end, magnified (Semiz & Çamlibel, 2015).
Table: Dependencies of distances to astronomical objects on the redshift at different values of H [km/sMpc].
H 
D, Gpc 
z 

0,2 
0,4 
0,6 
0,8 
1,0 
1,2 
1,4 

62,164 
^{r}D_{M} 
0,96 
1,93 
2,89 
3,86 
4,82 
5,79 
6,75 
^{r}D_{A} 
0,80 
1,38 
1,81 
2,14 
2,41 
2,63 
2,81 

D_{L} 
1,06 
2,28 
3,66 
5,18 
6,82 
8,58 
10,5 

62,295 
^{r}D_{M} 
0,96 
1,92 
2,89 
3,85 
4,81 
5,77 
6,74 
^{r}D_{A} 
0,80 
1,37 
1,80 
2,14 
2,41 
2,62 
2,81 

D_{L} 
1,05 
2,28 
3,65 
5,17 
6,81 
8,57 
10,4 

a) ^{g}D_{L} 
1,03 
2,25 
3,65 
5,2 
6,9 
8,65 
10,5 

65 
^{r}D_{M} 
0,93 
1,85 
2,77 
3,69 
4,62 
5,54 
6,46 
^{r}D_{A} 
0,77 
1,33 
1,73 
2,05 
2,31 
2,52 
2,69 

D_{L} 
1,01 
2,18 
3,50 
4,95 
6,52 
8,21 
10,0 

b) ^{g}D_{L} 
1,00 
2,16 
3,50 
4,955,0 
6,46,8 
8,28,8 
9,911,0 
Thus, teams of astronomers leaded by Perlmutter and Riess indeed confirmed (with high precision) the linearity of the dependence of redshift of radiation wavelength of distant galaxies on transverse commoving distance to them. And this their achievement is not at all less than attributed to them “discovery” (in reality – false one) of accelerated expansion of the Universe.
It is taken into account that the Hubble constant, like the length standards and the constant of the velocity of light, is a fundamentally unchangeable quantity in the rigid FRs. And this follows from the condition of continuity of spatial continuum in rigid FRs. (Danylchenko, 1994). The most corresponding to astronomical observations value of Hubble constant is the value determined by the following empiric dependencies of it on the well known physical constants and characteristics:
where: Λ is the cosmological constant, N_{Dn}=1,5(t_{p}ν_{Bn})^{2}= 3πchm_{n}^{2}/G=0,999885•10^{40} is the neutron large Dirac number, α=e^{2}/cħ is the fine structure constant, ν_{Bn}=m_{n}c^{2}/2πħ is the de Broglie wave frequency of the neutron, t_{p}=(c^{5}ħG)^{1/2} the Planck time, ħ=h/2π is the DiracPlanck constant, e is the electric charge of the proton and electron, m_{n} is the mass of neutron.
However, the value of Hubble constant H=(π^{4}α/8N_{DH})ν_{BH}=62,16420 [km/sMpc] (Λ=1,35457·10^{52} [m^{2}]), that corresponds to the de Broglie wave frequency of hydrogen atom ν_{BH}=m_{H}c^{2}/2πħ=2,270262·10^{23} [s^{1}] (m_{H}=1,67375·10^{27} [kg], N_{DH}=1,5(t_{p}ν_{BH})^{2}=1,001292·10^{40}), only for small distances guarantees slightly worse correspondence to the data of graphical extrapolation of the results of astronomical observations. It is possible that Hubble constant took “hydrogen” value only after spontaneous transformation of quark or neutron medium of the Universe into hydrogen medium. However, of course, it was impossible before that to metrically characterize its continuous protomatter and, therefore, it is senseless to characterize it by “neutron” Hubble constant. Therefore, the final choice of one of these two close values of Hubble constant can be done based on the more precise results of astronomical observations.
It is obvious that supposed need in the presence of dark energy in The Universe is based not only on the taking into account the imaginary (fictive) dilation of the time on distant astronomical objects (postulated by Etherington’s identity), but also on the wish to have the linear dependence of redshift of radiation spectrum z on luminosity distance D_{L} to those objects. In fact, according to GR (Danylchenko, 2004; 2008) the redshift is linearly dependent only on the transverse comoving distance D_{M}:
and on the angular diameter distance:
Moreover, the supposed dark energy could not be a certain physical entity at all. It could be just the effect of ubiquitous negative feedback. The deceleration of evolutionary selfcontraction of matter in CFREU could take place in the distant past due to the presence of this negative feedback. Thus, evolutionary decrease of the velocity of light in CFREU using CTMHS in the distant past would also be decelerated. This deceleration, of the outer space course, could have been the greater the smaller the coordinate velocity of light ^{u}v_{cos} in the outer space in GTFR had been in distant past.
However, it is quite probable that Hubble’s parameter is indeed unchangeable in time, as we had to make sure of it here. It even can be a spatiallytemporal invariant alike the proper value of the velocity of light. The value of Hubble’s constant can be precised after the more accurate processing of results of astronomical observations.
4. Conclusions
The Hubble constant is a fundamentally unchangeable quantity similar to the length standard and to the constant of the velocity of light. Therefore, the law, discovered by Hubble, is immutable. The dark energy and the Etherington’s identity are paralogisms.
The author is very grateful to professor Zhuk A.I. for his critical remarks that helped to take a fresh look on the Friedman solution.
References
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Danylchenko P.: 2004, in Gaugeevolutional interpretation of special and general relativities, Vinnytsia: O.Vlasuk, 3378.
Danylchenko P.: 2008, in Introduction to Relativistic Gravithermodynamics (in Russian), Vinnytsia: Nova knyga, 106.
Danylchenko P.: 2020, in Foundations and consequences of Relativistic Gravithermodynamics (in Ukrainian), Vinnytsia: Nova knyga, 5.
Dempsey A.: 2016, (Re)Discovering Dark Energy and the Expanding Universe: Fitting Data with Python.
Etherington I.: 1933, Philosophical Magazine, Issue 7, 15, 761.
Hogg D.W.: 2000, Distance measures in cosmology.
Penrose R.: 1968 “Structure of spacetime”, New York, Amsterdam: W.A. Benjamin, Inc.
Perlmutter S. et al.: 1999, The Astrophysical Journal, 517, 565.
Riess A.G. et al.: 1998, The Astronomical Journal, 116, 1009.
Semiz I. and Çamlibel K.: 2015, What do the cosmological supernova data really tell us?
Soloviev V.: 2016, in Spacegid.com (in Russian).
Weisskopf V.: 1972, “Physics in the twentieth century: Selected Essays”, Cambridge, Massachusetts, and London: The MIT Press.
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Weyl H.: 1930, Philos. Mag., 9, 936.
Zel’dovich Ya., Grishchuk, L.: 1988, PhysicsUspekhi (in Russian), 155, 517.
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