Hubble redshift in Einstein's universe
W. Jim Jastrzebski, file 16012.htm
Unofficial PhD program supervised by prof. Józef Namysłowski
The Institute of Theoretical Physics, University of Warsaw
Room 139, Hoża 69, 00681 Warsaw, Poland

It is shown that the observed features of Hubble redshift can be
explained within the framework of Einstein's general relativity.
The observed Hubble redshift could be attributed to thus far unnoticed
mechanism of time dilation coupled to curvature of space. Einstein's
universe regains its status as a viable model predicting cosmological
observations such as the (apparent) expansion of space and the
observed acceleration of this expansion.

Derivation of Hubble "constant" of Einstein's universe
Consider Einstein's homogeneous universe filled with dust (with a particle
of dust corresponding to a galaxy).
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 "constant"
depends on the distance between the place in deep space and the
observer.
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 = 4π G ρ / c^{2},
where G is Newtonian gravitational constant, ρ is density of dust, and
c is speed of light (which makes accidentally
Λ equal to the cosmological constant of Einstein's universe or
1/R^{2}, where
R 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
d^{2}E/dr^{2},
where r is distance travelled by photons, can be written, using
relativistic relation between mass and energy
(mass = Energy / c^{2}) as 4πGρ(E_{o} 
E_{d}) / c^{2} leading to equation
Substituting 1 / R^{2} for
Λ and solving the equation with initial
conditions E(r = 0) = E_{o} and
(dE/dr)(r = 0) =  E_{o} / R
(meaning selecting a solution that makes physical sense) one gets

E / E_{o} = exp(  r / R )
 (2)

Since in 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 / R )
 (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
(d^{2}τ/dtdr)^{2}
and the curvature of space
Λ = 1 / R^{2} as

(d^{2}τ/dtdr)^{2}  Λ = 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
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 = (E_{o}  E) / E = exp( r / R )  1
 (5)

simulating the expansion of space, with the Hubble "constant"
of this apparent expansion at
r = 0
For Einstein's universe of density ρ = 6 x 10^{27}kg/m^{3}
Hubble constant H_{o} = 70 km/s/Mpc.
After expanding the Hubble "constant", H(t),
into Taylor series around t = 0
the acceleration of this apparent expansion is approximately equal to

dH / dt =  H_{o}^{2} / 2
 (7)

which agrees within one standard deviation with 1998 observations
by the Supernova Cosmology Project team [1].
Conclusions
The analysis of Hubble time dilation (HTD) can be carried out using
Einstein's general relativity and the law of conservation of energy.
The observed HTD can be attributed to the geometry of spacetime of
the stationary homogeneous dust universe. Such a model may bo oversimplified,
however, it does reproduce the essential properties of HTD, and allows one
to estimate the measured properties of the 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) that expresses the essential transition from
Newtonian approach in which space and time are distinct, to a general
relativistic spacetime. Excelent agreement of the calculated acceleration
of apparent expansion of this model universe with the measured value
[1], lends additional support for the model.
Acknowledgments
Over the years the author has benefited from discussions with many people
too numerous to to list them here. The members of the faculty of the
Institute of Theoretical Physics, Warsaw, Poland were kind enough to spare
their time and offer critique which helped in clarifying the ideas expressed
in the above manuscript.
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