# Einsteinian gravitation

The Einsteinian gravitation is a contemporary theory of stationary "Einstein's universe", explaning how the relativity of time and space (notions of non existence of absolute time and absolute space) are working in "Einstein's universe".

The theory which except explaining the mechanism of gravitational force as inertial force pushing masses towards their most probable positions of minimum energy [1] in which it reveals its quantum nature, explains also why due to relativity of time in curved space the universe looks as if it were expanding, and expansion were accelerating with $dH/dt=-H_o^2/2$, where dH / dt is a change of Hubble constant with distance from observer converted to time t, Ho is Hubble "constant" near the Earth (at t = 0).

The Einsteinian gravitation explains why the value of Hubble "constant" H(t) (which in reality is an exponential function), Ho equal roughly 70km / s / Mpc implies density of space of universe as $6\times 10^{-27}kg/m^3$.

According to Einsteinian gravitation the Hubble constant Ho = c / RE, where c is speed of light in vacuum, and RE is "Einstein's radius of universe" equal roughly 13 billion light years as implied by the above value of Hubble "constant" should be consistent with minimum of angular diameters of galaxies between redshifts Z=1 and Z=2.

Also the Einsteinian gravitation, not allowing the expansion of universe, does not allow the Big Bang as a way of creation of the universe. It sugests that the universe is stationary (with Newtonian arguments based on relativity of time, which any intelligent high school student should easily comprehend). Possibly for this reason it is not allowed to be published in scientific journals controlled by relativists.

Since the editors of scientiffic journals controlled by relativists don't want neither print nor even to put the desription of Einsteinian gravitation through peer review process I describe here in 11 steps how stationary "Einstein's universe" works.

## Contents

#### Step 1. Assumptions

The basic assumption is as in any cosmological theory, that space of our universe is homogeneous and filed up with dust of galaxies with one galaxy corresponding to one dust particle (or a few). Average density of this dust is ρ and this is rather important for further calculations. Additionally we assume that π is what it usually is, c is speed of light in vacuum, G is Newtonian gravitational constant. So the overall picture of our universe is rather simple. It is not expanding. No hidden magic.
Now light starts radiating from some arbitrary "point zero" and photons start interacting wirh galaxies through "dynamical friction of photons" (the gravitational effect known in astronomy as "dynamical friction" but in this case acting on photons instead its usual action on heavier objects like stars). The gravitational energy of our "dust" gained through this dynamical friction contained in a ball of "dust" of radius r around "point zero" is Ed(r). This number is needed later to adjust the whole energy of "dust" to make the calculations exact (approximation is causing problems resulting as usually from approximations that all astrophysicists who wanted to make approximate calculations, including me, had, so I know that it has to be done exactly).
After photons of energy Eo 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" starts pushing the "dust" away from "point zero" this force is equal, from Newtonian equation for "gravitational force"
Fgalaxy(r) = (G / c2)[EoEd(r)]mgalaxy / r2, [2]
and
Fgalaxy(r) = − dEgalaxy(r) / dr
where Egalaxy(r) is gravitational energy of "dust particle" (galaxy) gained from the action of light through the "dynamical friction of photons on this particular "dust particle" (really a galaxy, or a few) at distance r from "point zero" [3]

#### Step 2: Integrating the mass of galaxies

... of spherical shell of radius r of galaxies around "point zero" and thickness dr gives number
mshell = 4πρr2dr,
where mshell is the mass of whole thin (of thickness dr) shell of galaxies of radius r and
Fshell = (G / c2)[EoEd(r)mshell / r2

#### Step 3. Calculating sum of all forcess acting on the shell

Substituting number from step 2 for the mass of galaxy resulting in
Fshell(r) = (4πGρ / c2)[EoEd(r)]dr.
and
Fshell(r) = − dEshell(r) / dr.
and so
dEshell(r) / dr = (4πGρ / c2)[EoEd(r)]dr.

#### Step 4. Integrating both sides of above equation

$-d E_d(r)/dr =(4 \pi G \rho/c^2)\int[E_o-E_d(r)]dr$.
... over all spherical shells between "point zero" and r to get total energy of "dust" absorbed from photons, and then differentiating both sides with respect to r to get rid of integral on the right side of this equation, we get finally the starting Ed(r) where subscript d now, as before, means "volume of universe of radius r around point zero"
d2Ed(r) / dr2 = (4πGρ / c2)[EoEd(r)].

#### Step 5: Setting differential equation controlling the redshift

Substituting Ed(r) = EoE(r) where E(r) is energy of photons at distance r from their source and $1/R_E^2=4\pi G\rho/c^2$, number called in cosmology "curvature of space of Einstein's universe" (or RE = "Einstein's radius of universe"), we get
$d^2E(r)/dr^2=E(r)/R_E^2$

#### Step 6: Solving the above equation

with initial conditions E(r = 0) = Eo
and (dE / dr)(r = 0) = − Eo / RE meanig selecting a solution that makes physicsl sense) one gets solution
E / Eo = exp( − r / RE)

#### Step 7: Slowing of proper time in deep space

Since in Einsteinian general relativity (EGR) there is nothing else but the time dilation and the curvature of space as the media controlling gravitation the EGR interpretation of the above result is that the time is running slower at a distance from (any) observer according to relation
dτ / dt = exp( − r / RE),
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. It is the above equation that expresses the esential transition from Newtonian approach in which space and time are distinct, to general relativistic spacetime. It demonstrates also the essential relativity of time of Einsteinian theory.
Excelent agreement of the shown in step 11 acceleration of apparent expansion of this model universe with the value measured by the Supernova Cosmology Project team of astonomers lends strong additional support for this model.

#### Step 8. Flatness of spacetime

Differentiating the above equation at r = 0 we get the relation between HTD in deep space (d2τ / dtdr)2 and the curvature of space of Einstein's universe (1 / RE)2 as
(d2τ / dtdr)2 − (1 / RE)2 = 0
It suggests the existence of overlooked by cosmologists the antisymmetric part of Ricci tensor in time domain, named tentatively Hμν or Hubble Tensor (HT) such that Hμν + Rμν = 0, where Rμν is as before the (spatial) part of Ricci tensor, and Hμν is its anti symmetric (temporal) part, agreeing with the fact that conservation of energy (or 4-vectors in general, during their parallel transport) require the flatness of spacetime proposed already by Narlikar and Arp. It would explain also the flatnes of Tμν (stress-energy tensor) and the application of Noether Theorem in flat spacetime that implies conservation of all four components of stress-energy tensor as asserted by Feynman in 2nd volume of "Feynman lectures on physics".
So all the pieces of puzzle seem to fit the "principle of conservation of all components of 4-momentum". They contradict the speculations of authors of 1973 "Gravitation" by Misner, Thorne, and Wheeler who maintain that the spacetime is curved and that only the magnitude of stress-energy tensor is conserved (infamous Wheeler's "Momenergy"). The Einsteinian conclusion is still that the spacetime is intrinsically flat (as proven by elementary tensor calculus) and therefore the universe is not expanding and therefore it must be eternal, as once assumed by the Einsteinian gravitation.
But look at the further steps, even more interesting.

#### Step 9. Hubble redshift

It follows from results of steps 6 and 7 equivalently, that the redshift, produced by HTD, is equal to
Z = (λ − λo) / λo = (EoE) / E = exp(r / RE) − 1,
where λ is "shifted" wavelength, λo is original wavelength, etc.

#### Step 10. The same thing in form of Hubble "constant"

Such redshift simulates the expansion of space, with Hubble constant of this apparent expansion at r = 0 equal
Ho = c / RE
It is observed as about Ho = 70km/s/Mpc which implies that the radius of curvature of space of Einstein's model universe, RE is about 13Gly (13 billion light years). Very tough to measure but luckily very easy to calculate. And BTW, this value of Ho implies the density of our universe about $6\times 10^{-27}kg/m^3$, which is within about one standard deviation from what is observed.

#### Step 11. Apparent accelerating expansion of space

After expanding Hubble "constant" H(t) into Taylor series around t = 0 acceleration of this apparent expansion is approximately equal $dH/dt=-H_o^2/2$.
the above value agrees within one standard deviation (maximum available accuracy for testind results of a theory) with 1998 observations by the Supernova Cosmology Project team of astronomers.
So the "accelerating expansion of our universe" blamed on mysterious "dark energy" for which cosmologists look since 1998 is just the Taylor term of expansion series of Hubble constant which is no constant but exponential function as shown in steps 6 or 7 (first as Newtonian result and second as Einsteinian interpretation of this Newtonian result). Cosmologists also call it "cosmological constant" (the "biggest blunder of Einstein's life") which can be shown not to exist at all. It might be eliminated from Einstein's field equation in rather straight forward manner. Though I couldn't show it to Einstein for his approval since I had it only in 1986, over 30 years too late [4].

## Metric of Einstein's universe

For those who care about such things, there is a time-space (radial coordinate only) metric of Einstein's universe:

dτ2 = exp( − 2r / RE)dt2 + 2sinh(2r / RE)dtdr / cexp(2r / RE)dr2 / c2,

where τ is proper time, r is radial coordinate, t is coordinate time, RE is radius of curvature of space of Einstein's universe (Einstein's radius of universe). This metric approximates to Minkowski metric dτ2 = dt2dr2 / c2 for r<<RE.

The Einstein's universe metric seems to be the simplest metric with required by Einstein's property of non symmetry of metric [5].

## References

1. Explained in gravitation demystified.
2. Explanation of the above equation: It comes from the Newtonian force being F = GMm / r2 and M, the source of gravitational "repulsive force" being replaced by M = E / c2 according to Einstein's physics. E in this equation is difference of original mass of "dust" causing the "Newtonian repulsion" adjusted by in general tiny Ed(r) causing "attraction" (acting in opposite direction than the "repulsive" M). It is similar to a gravitational effect of a buble of air "repulsing" gravitationally their sorounding water in a tank of water plus tiny effect of attraction by Ed(r), the tiny gravitational energy of accumulated within the buble
3. Usually people neglect influence of a single photon on a Galaxy since it looks to them as "negligible". Though it is not exact but they think that no harm is done by small negligence of exactness of math. It is wrong approach as it is demonstrated below where exact calculations produce Hubble constant roughly as it is observed. And agreement of theory with observation is a good indicator that the theory may be right as Einsteinian gravitation explaining the phenomenon of redshift in Einstein's universe might be.
4. I think that Mileva Marić Einstein, might be considered the true discoverer of both teories of relativity, for the reason that Einstein was not able to manage "his blunder", while I could, and in rather straight forward manner, practically not knowing any tensor calculus being a sculptor rather than a cosmologist. Mileve Marić being an astronomer and a high class physicist, knowing everything about her husband's theories might have known the way of handling "his blunder" still in time for Albert to reveal to him the true nature of it. She might have a good reason not to do it and one may speculate that it might be her promiss to him not to reveal to anybody who was the true discoverer of the theory of relativity, or at least not to reveal that they both worked on it. A promiss that might be jeopardized while revealing the knowledge of how to remove the cosmological constant "the biggest blunder of Einstein's life", from Einstein field equation. Just a thought.
5. Einstein's 1950 paper "On the Generalized Theory of Gravitation", published in April 1950 issue of "Scientific American". This paper was not placed for some reason in detailed bibliography of Einstein's papers of 1973 "Gravitation", by Misner, Thorne, and Wheeler. The reason might be that this book assumes at the onset the symmetric metric of spacetime while Einstein excluded such possibility possibly for it potential of creation of energy from nothing utilized indeed later in "Gravitation" in the form of Big Bang hypothesis. For this reason "Gravitation" might be considered advocating a creationist cosmology, as contrasted by stationary cosmology of Einsteinian gravitation.

JimJast 19:18, 4 August 2011 (EDT)