Derivation of Hubble constant of Einstein's universe

Consider Einstein's homogeneous universe filled with dust. Let photons move through this dust interacting with it only gravitationally. We will assume that energy conservation holds and that Newton's approximation can be applied. With these assumptions one can readily calculate energy transfer from photons to dust. To the observer at some distance from the light source this energy transfer will manifest itself as a change in wavelength, which is exactly what was observed by Hubble. The relativistic interpretation of this result allows the derivation of Hubble redshift (HR), including discovery that Hubble's constant depends on the distance between the place in deep space and the observer [1].

Let E_d = E_o - E be the gravitational energy acquired by the dust due to gravitational interaction between dust and photons of energy E and initial energy E_o and let Λ_E = 4π G ρ / c², where G is Newtonian gravitational constant, ρ is density of dust, and c is speed of light (which makes accidentally Λ_E equal to the cosmological constant of Einstein's universe or R_E ^-2, where R_E is radius of Einstein's universe). The linear density of Newtonian gravitational force acting on dust (force per unit length), which is identically equal to d²E/dr², where r is distance travelled by photons, can be written, using relativistic relation between mass and energy (mass = Energy / c²) as 4πGρ(E_o - E_d) / c² leading to equation (see "Appendix A" at the end of this paper for step by step derivation)

d2E/dr2 = ΛE E

(1)

Substituting R_E^{- 2} for Λ_E and solving the equation with initial conditions E(r = 0) = E_o and (dE/dr)(r = 0) = - E_o / R_E (meaning selecting a solution that makes physical sense) one gets

E / Eo = exp( - r / RE )

(2)

Since in Einstein's general relativity (EGR) there is nothing else but time dilation and the curvature of space as the media controlling gravitation, EGR interpretation of the above result is that time is running slower at a distance from (any) observer according to relation

dτ/dt = exp( - r / RE )

(3)

where τ is proper time in deep space and t is coordinate time at observer. The effect might be called Hubble Time Dilation (HTD) in honor of its discoverer, and as distinguished from the gravitational time dilation predicted by Einstein. After differentiating the above equation at r = 0 we get a relation between the HTD in deep space (d²τ/dtdr)² and the curvature of space Λ_E = 1 / R_E ² as

(d2τ/dtdr)2 - ΛE = 0

(4)

and it suggests the existence of antisymmetric part of Ricci tensor in time domain, named here tentatively H_μν or Hubble Tensor (HT), such that H_μν + R_μν = 0_μν indicating that the spacetime is intrinsically flat as proposed by Narlikar and Arp [2] and required by the law of conservation of energy.

It follows from equation (2) or (3) equivalently, that the redshift, produced by HTD, is equal to

Z = (Eo - E) / E = exp( r / RE ) - 1

(5)

simulating the expansion of space, with the Hubble constant of this apparent expansion at r = 0

Ho = c / RE

(6)

After expanding the Hubble "constant", H(t), into Taylor series around t = 0 and neglecting the higher order terms, the acceleration of this apparent expansion is approximately equal to

dH / dt = - Ho2 / 2

(7)

This value agrees within one standard deviation with 1998 observations by the Supernova Cosmology Project team [1].

Conclusions

The analysis of Hubble redshift (HR) can be carried out using Einstein's general relativity and the law of conservation of energy. The observed HR can be attributed to the time dilation in the universe which remains stationary as predicted by Einstein in 1917.

In addition to reproducing the measured properties of HR, the approach outlined in this paper allows direct calculation of the important parameters of our universe such as the average density of space and the acceleration of its apparent expansion. While the formula for Hubble redshift, equation (5), can be derived directly from equation (2) obtained using Newtonian approximation, it is equation (3), which expresses the essential transition from Newtonian approach in which space and time are distinct, to a general relativistic spacetime.

The value of Einstein's cosmological constant turning out, seemingly accidentally, in eq. (1) allows to eliminate this constant from Einstein's field equation (EFE) since as it was shown in the above derivation of HR it has to add an additional energy term on the right side of EFE to provide for apparent loss of photons energy due to the Hubble time dilation as suggested by eq. (4). Then EFE instead of Einstein's version of 1917 [5],  R_μν - (R / 2)g_μν + Λ_E g_μν = 8πT_μν becomes

Rμν - (R / 2)gμν = 8πTμν + (H / 2)gμν

(8)

replacing the cosmological constant Λ_E with half of contraction of Hubble tensor correcting in this way what Einstein called "the biggest blunder of my life" [6]. And finally Einstein's field equation gets the form it started with in 1915

Rμν = 8πTμν

(9)

Acknowledgments

The author expresses his gratitude to Dr Halton Arp, Dr Helmut A. Abt, Dr Chris E. Adamson, Prof. John Baez, Prof. Tadeusz Balaban, Dr Krzysztof Bolejko, Prof. Michael Chodorowski, Dr Tom Cohoe, Prof. Marek Demianski, Dr Marijke Van Gans, Prof. Roy J. Glauber, Dr Mike Guillen, Prof. Alan Guth, Dr Martin J. Hardcastle, Dr Franz Haymann, Dr Chris Hillman, Dr Marek Kalinowski, Dr Alan P. Lightman, Prof. Krzysztof Meissner, Prof. Jozef M. Namyslowski, Dr Bjarne G. Nielsen, Prof. Bohdan Paczynski, Janina Pisera, BS, Dr Ramon Prasad, Dr Frank E. Reed, Dr Anrzej Szechter, Prof. Henryk Szymczak, Jerzy Tarasiuk, MS, Dr Michael S. Turner, Dr Slava Turyshev, Prof. Clifford M. Will, Prof. Ned Wright, and anonumous referees from various scientific journals, for the time they have spent discussing with the author the subject of the paper and related issues. The most thanks goes to Prof. Michael Chodorowski of CAMK-PAN (Copernican Astronomical Center of Polish Accademy of Science), without whose friendly critique the ideas expressed in this paper might have never developed to a legible state.

Bobliography

1. Kuznetsova N, Barbary K, et al, A New Determination of the High-Redshift Type Ia Supernova Rates with the Hubble Space Telescope Advanced Camera for Surveys, 2008, The Astrophysical Journal, Volume 673, Issue 2, The American Astronomical Society, pp. 981-998, Bibliographic Code: 2008 ApJ...673..981K,
   http://supernova.lbl.gov/
2. Narlikar, J. & Arp, H. Flat spacetime cosmology - a unified framework for extragalactic redshifts, 1993, Astrophysical Journal, Part 1 (ISSN 0004-637X) vol. 405, no. 1, The American Astronomical Society, p. 51-56, Bibliographic code: 1993 ApJ...405...51N
3. STScI-2009-08, Refined Hubble constant narrows possible explanations for dark energy,
   http://hubblesite.org/newscenter/archive/releases/2009/08/full/
4. Landau, L. D., Lifszyc. E. M., 1973, "Teoria pola", Państwowe Wydawnictwo Naukowe, Wydanie II zmiennione, p. 285, eq. (88.9).
5. Meissner, Krzysztof A., 2002, "Klasyczna Teoria Pola" ("The Classical Theory of Fields"), Wydawnictwo Naukowe PWN, ISBN-83-01-13717-7, p. 120.
6. Misner, Charles W., Thorne, Kip S., Wheeler, John Archibald, 1973, "Gravitation", W. H. Freeman and Co, New York, p. 410-411.

Appendix A: Detailed derivation of equation (1)

Let c be speed of light, G the Newtonian gravitational constant, ρ the density of Einstein's dust universe (while one galaxy corresponds to one dust particle), the gravitational energy of dust gained through the gravitational interaction of light with dust ("dynamical friction") contained in the volume of a ball of dust of radius r be E_d(r). After photons of energy E_o are radiated out from arbitrarily placed center of coordinate system called "point zero" (point with radial coordinate r = 0) in homogeneous space, making the space inhomogeneous and therefore the Newtonian "gravitational force" is pushing the dust away from point zero and at distance r from point zero this force is equal, from Newtonian equation for gravitational force and energy

Fparticle = - (d/dr)Eparticle(r) = ( G / c2 )[ Eo - Ed(r) ] mparticle / r2

(A.1)

where E_particle(r) is gravitational energy of dust particle at distance r from point zero and m_particle is the mass of a particle of Einstein's dust universe.

Integrating over all dust particles in a spherical layer of dust of radius r and thickness dr we get mass of this layer of dust at distance r from point zero as

mlayer = 4 π ρ r2 dr

(A.2).

Substituting (A.2) for m_particle in (A.1) the total force that is the source of gravitational energy of dust of this leyer becomes

Flayer(r) = - (d/dr)Elayer(r) = ( 4πGρ / c2 ) [ Eo - Ed(r) ] dr

(A.3).

Integrating both sides of (A.3) over all spherical layers between point zero and r to get total energy of dust, and differentiating both sides with respect to r to get rid of the integral on the right side of this equation, we get

- d2Ed(r) / dr2 = ( 4πGρ / c2 ) [ Eo - Ed(r) ]

(A.4).

Substitutig E_d(r) = E_o - E(r) where E(r) is energy of photons at distance r from their source and Λ_E = 4πGρ / c² we get

d2E(r) / dr2 = ΛE E(r)

(A.5)

which is identical with eq. (1).