Conservation of energy in cosmology
W. Jim Jastrzebski, file 16419.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 has been shown that the hypothesis of strict conservation of
energy predicts many observations with about one sigma accuracy.
Except explaining the mechanism of gravitational pseudo force, done by
Einstein in 1915, verifiable directly by its value of
F = gm
and the mechanism of cosmological redshift verifiable by the
value of Hubble "constant"
H_{o} = c / R_{s} where c is
speed of light and R_{s} is the radius of
3space verifiable also by the position of minimum of
angular diameters of galaxies as function of their redshifts,
the hypothesis predicts for H_{o} = 70 km/s/Mpc
the value of acceleration of cosmic probes (as Pioneer 10 and 11)
7×10^{10} m/s^{2},
and the average density of universe
6×10^{27} kg/m^{3}.
The volume of space becomes
1.6×10^{12} Mpc^{3},
and therefore the total mass of universe 3×10^{53} kg.
The predicted acceleration of apparent expansion of space becomes
dH/dt = H_{o}^{2}/2,
and the distances to quasars several orders of magnitude smaller than
those predicted by the
hypothesis of expanding space.

Introduction
The hypothesis of expanding space requires that energy is not
conserved globally.
We show that the assumption of strict conservation of energy
makes the hypothesis of expanding space redundant since the former
predicts many observations quantitatively with accuracy better than
one sigma which strongly suggests that it might be a true hypothesis
and consequently the space is not expanding.
The following section is overly detailed to be
understood also by those astronomers and astrophysicists who don't understand general relativity but understand that the creation of energy may not exist.
Stepbystep derivation of cosmological redshift in stationary space
Consider homogeneous universe in a form of Einstein's 3sphere of
radius R_{s} (and therefore of volume
2π^{2}R_{s}^{3})
[1] filled with dust of
density ρ (with a particle of dust representing a galaxy).
Let energy conservation holds.
Let photons move through this dust interacting with it only
gravitationally via dynamical friction
[2]
[3]
[4]
[5]
[6] losing gradually their energy to this dust.
Qualitatively, to the observer at some distance from the light source
this energy transfer from photons to dust manifests itself as a change in wavelength,
simulating the tired light effect, first observed by Hubble and then worked on by Zwicky.
The relativistic interpretation of this result allows the derivation
of cosmological redshift, including discovery that Hubble's "constant"
depends on the distance between the light emitting place in deep space
and the observer [7].
Let's do
some math to find out what happens quantitatively.
It turns out that the Newtonian math suffices for this purpose.
Let photons of energy E_{o} are radiated out from an
arbitrarily placed center of coordinate system called "point
zero" in the described above homogeneous space.
When the radiated out photons leave an empty hole at point zero in otherwise
homogeneous space the space becomes inhomogeneous.
The
Newtonian "gravitational force" shows up and starts pushing
the dust away from the empty hole at point zero like an air bubble in a tank of water "pushes away" with its "gravitational force" the surrounding water.
The space around point zero gains gravitational
energy that given enough time, gets eventually converted into kinetic energy of movement
of dust particles.
This effect refers to the future though that we don't need to be concerned with since we describe only the present, namely the gravitational energies,
The future state of universe can have no influence on the present and its energies.
Let the gravitational energy of dust contained in the volume of a
ball of dust of radius r be E_{d}(r).
At distance r from point zero the gravitational force
pushing the dust particles away is equal according to Newton's math, using Einstein's relativistic relation between
mass and energy E=mc^{2}

F_{particle} = ( G / c^{2 })[ E_{o}  E_{d}(r) ] m_{particle} / r^{2}
 (1)

where
m_{particle} is the mass of a particle of dust
universe (typically a galaxy).
Integrating the mass over all dust particles in a spherical shell of dust
of radius r and thickness dr we get mass of this shell
of dust at distance r from point zero as

m_{shell} =
4 π ρ r^{2} dr
 (2)

and merging (1) with (2) the total force that is the source of
gravitational energy of dust of this shell becomes

F_{shell}(r) = ( 4πGρ / c^{2} )
[ E_{o}  E_{d}(r) ] dr
 (3)

and since F_{shell}(r) =
 dE_{shell}(r) / dr we have also

 dE_{shell}(r) / dr = ( 4πGρ / c^{2} )
[ E_{o}  E_{d}(r) ] dr
 (4)

Integrating both sides of (4) over all spherical shells between
point zero and r to get total gravitational 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

 d^{2}E_{d}(r) /
dr^{2} = ( 4πGρ / c^{2} )
[ E_{o}  E_{d}(r) ]
 (5)

and substitutig E_{d}(r) = E_{o} 
E_{ph}(r) where E_{ph}(r) is energy of photons at
distance r from their source and introducing square of
radius of stationary universe
R_{s}^{2} =
c^{2}/4πGρ
we get the final differential equation controlling the phenomenon as

d^{2}E_{ph}(r) / dr^{2} = E_{ph}(r) / R_{s}^{2}
 (6)

Solving equation (6) with initial conditions
E_{ph}_{r=}0 = E_{o} and
dE_{ph}/dr_{r=}0 =  E_{o} /
R_{s} meaning selecting a solution that makes
physical sense, we get

E_{ph}(r) / E_{o} = exp(  r / R_{s} )
 (7)

Since we know that in general relativity there is nothing else but the time dilation
and the curvature of space as the media controlling gravitation we now switch from the Newtonian math
to the relativistic interpretation of the above result, which is the
time running slower at a distance from (any) observer according to
relation

dτ/dt = exp(  r / R_{s} )
 (8)

where τ is proper time at point in deep space and t is
coordinate time at observer.
The effect might be called Hubble
time dilation in honor of its discoverer, and as distinguished
from the gravitational time dilation discovered by Einstein.
After differentiating the above equation at r = 0
we get a relation between the Hubble time dilation in deep space
(d^{2}τ/dtdr) and the square root of
curvature of space (1 / R_{s} ) as

d^{2}τ/dtdr + 1 / R_{s} = 0
 (9)

that suggests the existence of a tensor named here tentatively
H_{μν} or Hubble tensor, modifying the right side
of Einstein's field equation such that
H_{μν} + R_{μν} = 0
indicating that the spacetime is intrinsically flat as proposed by
Narlikar and Arp [8] and required by the law of
conservation of energy (allowing also to remove the cosmological
constant Λ from Einstein's field equation  a note
for mathematicians rather than grannies who might even not know
what Einstein's field equation is nor they need to know to
understand this derivation).
It follows from equation (7) or (8) equivalently, that the redshift,
produced by Hubble time dilation, is equal to

Z =
exp( r / R_{s} )  1
 (10)

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

dH / dt =  H_{o}^{2} / 2
 (12)

which agrees within one standard deviation with 1998 observations
by the Supernova Cosmology Project team [9].
The additional gain from all the above is that, "Einstein's biggest blunder of his life"
[10], the infamous "cosmological constant"
[11], a.k.a. "dark energy", falls out from Einstein's field equation.
Some numerical results
Assuming
H_{o} = 70 km/s/Mpc, as it is presently
observed, and the same lower limit of dynamical friction value for space probes (since in the above derivation there was no dependence on the velocity of photons and so it refered to any velocity, also that of cosmic probes as Pioneers 10 and 11) the deceleration of cosmic probes comes out as 7×10^{10} m/s^{2}, the volume of space of our universe is
1.6×10^{12} Mpc^{3} making room
for about 10^{12} galaxies.
The average density of universe is 6×10^{27}
kg/m^{3}, and its total mass
is 3×10^{53} kg making the mass of average
galaxy equal 3×10^{41} kg or
10^{11} solar masses.
The apparent "acceleration" of space dH/dt =  H_{o}^{2}/2 as
predicted in 1985 by Einsteinian gravitation and observed in 1998
[9].
The numerical results have been never allowed to be published though
in any scientific journal (not even in a popular science magazine as
"Scientific American") so the astronomers for whom it might be
an important news couldn't learn that the hypothesis of creation of
the universe might be wrong.
That this hypothesis not only contradicts Einstein's general relativity
(replaced by Wheeler's cerationist version of general relativity based
on an axiom of creation) but contradicts also quite unnecessarily the
principle of conservation of energy since as it has been shown above,
this principle delivers quite reasonable results, not only
qualitatively but also quantitatively.
Conclusions
The analysis of cosmological redshift can be carried out in just dozen simple steps using general
relativity and the law of conservation of energy.
The observed cosmological redshift can be attributed to the time dilation coupled to the curvature of space in the universe
which remains stationary as predicted by Einstein in 1917.
In addition to reproducing the measured properties of cosmological redshift, 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 cosmological redshift, equation (10), can be derived
directly from equation (7) obtained using Newtonian approximation,
it is equation (8), which expresses the essential transition from
Newtonian approach in which space and time are distinct, to a general
relativistic spacetime.
The additional gain from the above derivation is realization that redshift of quasars being proportional to the local curvature of space integrated over the whole distance between the quasar and the observer, as equation (9) suggests, doesn't need to reflect anymore the Hubble distance/redshift relation since the local curvatures of space don't need to be the same as the average, and the local curvature might be even many orders of magnitude greater than the agerage that reflects the average cosmological redshift.
Acknowledgments
The author expresses his gratitude to Dr Halton Arp, Dr Helmut A. Abt,
Dr Chris E. Adamson, Prof. John Baez, Prof. Tadeusz Balaban,
Prof. Michal 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. Michal Chodorowski of
CAMKPAN (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.
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