Last modified on March 27, 2017, at 17:05

Essay:Demystified gravitation

  1. REDIRECT Gravitation demystified

This essay is an original work by W. Jim Jastrzebski.

JimJast 09:43, 22 August 2011 (EDT)

Demystifying gravitation


I'm writing this essay to explain to people interested in gravitation how it works according to Einsteinian physics.

Between 1905 and 1915 the theory of gravitation switched from Newtonian math to Einsteinian physics and many Newton's problems got solved, mostly due to the fact that the space is not flat ("flat space" is our "Euclidean space" [1]) but is curved having certain curvature of space and the time reflects faithfully this curvature of space through the gravitational time dilation in such way that the whole spacetime is flat.

This global flatness of spacetime is an important feature. Without it the movement of certain geometrical structures (tensors) that describe things in space and time (therefore in 4 dimensions: 3 in space and 1 in time) wouldn't have the property of keeping the strict conservation of energy and momentum between the now and the past or the future. There wouldn't be any conservation laws valid in the universe. The things would be spontaneously created from nothing and disappearing without trace from one moment to another just by the virtue of mathematical manipulation (as it happens in the Big Bang hypothesis in which spacetime is "curved") (luckily not in this universe of ours since then we would be forced to believe in "ghosts" while they would be created and disappear without no reason whatsoever). The creationist might be happy but the science wouldn't be as we know it today. Now in science neither energy nor momentum can be created from nothing as it is assumed in the Big Bang hypothesis with its curvature of spacetime. Conservation laws assure that energy and momentum are conserved separately (not as some relativists think, "together" as Wheeler's hypothetic "momenergy" in which energy is assumed to be able to be converted into momentum and v.v).

This essay is meant to explain this strange physics of gravitation, with which we deal every day but generally don't understand it, don't know how it operates and where the "gravitational force" is coming from, except for 3 physicists .[2] I hope it soon will be clear to the readers that gravitation contains no mystery and is probably the simplest physics around and could hardly work in a different manner.

There may be readers who might think that it should be left working as Newtonian math seemed to work when everybody and his brother thought that objects in the universe attract each other. Though it won't be true since gravitation doesn't work this way. Even according to Newton .[3] Newton was wise man and didn't believe in things that he thought are impossible. Then there were many things seemingly impossible. Though gravitation was obviously possible. How did it work?

The description of gravitation starts with the next section.

How gravitational force is generated

Qualitative aspects of gravitational force

Gravitation has been for long time treated as mysterious force attracting objects one to another, approximately according to Newton's law. Even Newton didn't believe in existence of such force, the same as Einstein, who called it "spooky action at a distance".

After Einstein discovered his equation for the energy of a particle

...... (1)

where is square of speed of light in vacuum adjusted by (ratio of proper time to coordinate time [4]) and is rest mass of a particle though the equation works also for relativistic mass with just the energy increased by kinetic energy of the particle, it turned out that this mysterious force of alleged attraction comes just as derivative of the above equation with respect to displacement of the particle .

It came from the fact that the time slows down in vicinity of any external mass just by right amount to cause force

...... (2)

where dx is displacement of particle in relation to the source of slowed time, that we call mass .

With slowing down of time in vicinity of mass , the energy of any particle in vicinity of mass diminishes according to equation (1) since all movements of the particle are also slower so all of them contain less kinetic energy expressed by . So there is certain apparent force, pushing mass towards mass taken before Einstein for "attractive force" of "gravitational attraction" while it had been just a force of apparent attraction: the tendency of particle to move towards the most most probable position of this particle, position of its lowest energy. Something that the particle is pushed towards, without any prompting force, which also shows the quantum nature of Einsteinian gravitation. We may say that gravitational force is a quantum effect of particle trying to assume its most probable position of its lowest enery.

It means that the force is only an inertial force with which particle presses towards its location of most probable position in spacetime. That's why introduction of curvatures of spacetime to physics the problem of gravitational force that Newton worried so much became moot.

This is the whole qualitatative mystery of how "gravitational force"' is generated in Einsteinian physics.

Now we show also its quantitative aspects to be sure that it is really the "gravitational force".

Quantative aspects of gravitational force

Since the energy of any particle is controlled by Einsteinian equation (1) then when time runs slower in vicinity of mass , the energy is diminished according to its derivative

...... (3)

The above equation implies that (if it is going to be the real "gravitational force" ) then

...... (4),

which may be derived from a rocket ship model moving with acceleration which according to the "principle of equivalence" of gravitation and acceleration is exactly the same as acceleration causing the same gravitational effect as the one given by equation (2). Remembering that dependence of bending of light ray in the rocket ship model is twice as big as the observed since the second half of bending comes from the curvature of space non visible in the flat space of our rocket ship model.

And so the "gravitational force" is the same as in (3) and substituting (4) at the right place we get as the result

...... (5).

And so the "gravitational force" comes out as , more accurately than the Newtonian force given by the Newtonian approximate equation

...... (6),

where G is Newtonian gravitational constant. The small difference between those two forces (Newtonian and Einsteinian ) comes from the difference between flat space of Newtonian math and the curved space of Einsteinian physics that is responsible for equation (4).

It is the whole mystery of gravitational force demystified by Einsteinian time dilation and the curvature of space. Turning out to be the inertial force, pushing the particle towards mass , that allegedly attracts it but in fact, its action only changes the rate of time around itself (slowing it down).

In the end, the apparent gravitational force of equation (2) is caused by the curvature of space and the time dilation that combined, cause the effects of diminishing energy of objects immobilized in curved space. When they aren't immobilized (they are left in free fall), they follow their natural positions of the greatest probability of being there in the spacetime without producing any forces. As e.g. the Moon following its natural path around the Earth moving at each point of its travel with appropriate acceleration compensating changes of its total energy as given by (1). In this way no additional energy can be created or destroyed and the law of conservation of energy is satisfied automatically for all the moving objects of the universe.

Demystifying universe

Introduction to Einsteinian universe

The Einsteinian universe theory (EUT), is a theory of universe from which the "expansion of univrese" was eliminated through the calculation of intrinsic redshift, that showed that the alleged expansion is a relativistic illusion due to the curvature of space.

The EUT is almost the same as Einstein's stationary model universe of 1917 that was Einstein's guess, very close to the present EUT model of universe. It has common with the present EUT model the proved only in 1985, and therefore unknown to Einstein, the lack of expansion, that in original Einstein's model has been achieved via introduction of cosmological constant, and therefore with purely mathematical means.[5]

To keep Einstein's field equation of this stationary model stable, and not knowing yet about the "Hubble time dilation", discovered by Edwin Hubble over a decade later, and suggesting to some astronomers that the universe might be expanding, Einstein added his "cosmological constant" to his field equation that stopped the alleged expansion. Later he called this cosmological constant "the biggest blunder of his life".

When the "cosmological constant" turned out to be unnecessary, in 1985, since it turned out that the Hubble redshift is not caused by expansion of the universe but by the time running slower at the deep space galaxies in curved space, which in EUT simulates accelerating expansion of space with acceleration

as observed in 1998 by the Supernova Cosmology Project team of astronomers and therefore predicted by EUT over 80 years earlier.

The agreement of value of alleged acceleration of space with the results of observations of Supernova Cosmology Project within one standard deviation showed that the expansion of universe is really an illusion but the news has been never published yet and is not likely to be published soon (except in conservapedia).

Publishing of such a news would destroy the credibility of estblishment cosmologists and their hypothesis of creation and evolution of universe called the Big Bang hypothesis, which assumes creation of energy in the Big Bang event.

Around 1929 Edwin Hubble discovered cosmological redshift that relativists started to consider a sign that the universe is expanding. Then Einstein gave up working on his stationary universe, and left his cosmological constant as it was. He also told his secretary not to let in anyone who wants to talk to him about the universe. [6]

The relativists were not able to remove cosmological constant since it might have ruined their Big Bang hypothesis and named it "dark energy" creating this way new mystery for themselves to solve (at the taxpayers expense) and so the "dark energy" remains ever since a mystery for them to solve.

The establishment scientific journals still resist the publication of the truth about the universe (that it operates according to Einstein's plan and that the Einsteinian gravitation, does not allow the expansion of universe, does not allow the Big Bang as a way of creation of the universe).

EUT suggests that the universe is stationary with Newtonian arguments based on relativity of time, which any intelligent high school student should easily comprehend and repeat the Newtonian physics contained in the following sections.

First easy to notice feature of EUT is that the universe has been always approximately the same. It has been never changing, beyond trivial movements of celestial bodies. The EUT is therefore described by the Perfect cosmological principle.

The theory is called "Einsteinian" since it's Einsteinian type theory, except a bit deeper since it includes elements that Einstein didn't know about or didn't care. For this reason there are also a few flaws which had to be dealt with and removed (as e.g. the mentioned "cosmological constant"). The new elements and recognition of flaws showed up after 1985 when it turned out that the universe is not expanding since its intrinsic redshift turned to be represented by Hubble constant equal


where is speed of light in vacuum and is Einstein's radius of universe, which means that Einstein must have been doing something right assuming the perfect cosmological principle already in 1917. And so the EUT may be also "right" as far as we know .[7]

The EUT is explaining how the relativity of time and space (notions of non existence of absolute time and absolute space) are working in stationary, neither expanding nor contracting, eternal universe that has neither the beginning nor the end.

The EUT which except explaining the mechanism of gravitational force as inertial force pushing masses towards their most probable positions of minimum energy 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 acceleration confirmed to one standard deviation by Supernova cosmology Project.

The EUT explains why the value of Hubble "constant" , which in reality is an exponential function, since the redshift is in reality an exponential function of distance (or time, since , distance is the same as time multiplied by speed of light), so the redshift

implies the density of space of universe as about


The EUT explains also why the above Hubble "constant", in which is apparent recession speed of a galaxy, is really


"Einstein's radius of universe" in the above equation equals roughly 13 billion light years is implied by the above value of Hubble "constant" and why it should be consistent with minimum of angular diameters of galaxies between redshifts Z=1 and Z=2 which is additional test of EUT.

Therefore, the EUT explains most of numerical results that the Big Bang cosmology is unable to explain. the EUT is in fact an extension of Einstein's general relativity but it may not be called by this name because the relativists highjacked this name for their pseudo scientific hypothesis of the Big Bang which, when used, might have caused a confusion.

Step by step derivation of cosmological redshift

To see the whole one page paper describing the derivation for physicists and/or astronomers look at Essay:Hubble redshift in Einstein's universe. To see step-by-step derivation for high school students look at following steps:

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 and this is rather important for further calculations. Additionally we assume that is what it usually is, is speed of light in vacuum, 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 around "point zero" is . 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 are radiated out from arbitrarily placed center of coordinate system called "point zero" (point with radial coordinate 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"
where 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 from "point zero" [9]

Step 2: Integrating the mass of galaxies

... of spherical shell of radius of galaxies around "point zero" and thickness gives number
where is the mass of whole thin (of thickness ) shell of galaxies of radius and

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

Substituting number from step 2 for the mass of galaxy resulting in
and so

Step 4. Integrating both sides of above equation

... over all spherical shells between "point zero" and to get total energy of "dust" absorbed from photons, and then differentiating both sides with respect to to get rid of integral on the right side of this equation, we get finally the starting where subscript now, as before, means "volume of universe of radius around point zero"

Step 5: Setting differential equation controlling the redshift

Substituting where is energy of photons at distance from their source and , number called in cosmology "curvature of space of Einstein's universe" (or "Einstein's radius of universe"), we get

Step 6: Solving the above equation

with initial conditions
and meanig selecting a solution that makes physicsl sense) one gets solution

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
where is proper time in deep space and 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 we get the relation between HTD in deep space and the curvature of space of Einstein's universe as
It suggests the existence of overlooked by cosmologists the antisymmetric part of Ricci tensor in time domain, named tentatively or Hubble Tensor (HT) such that , where is as before the (spatial) part of Ricci tensor, and 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 (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
where is "shifted" wavelength, 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 equal
It is observed as about
which implies that the radius of curvature of space of Einstein's model universe, is about 13Gly (13 billion light years). Very tough to measure but luckily very easy to calculate. And BTW, this value of implies the density of our universe about
which is within about one standard deviation from what is observed.

Step 11. Apparent accelerating expansion of space

After expanding Hubble "constant" into Taylor series around acceleration of this apparent expansion is approximately equal
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.

Metric of Einsteinian universe

Here is the metric of universe with radial coordinate only (angular coordinates dropped for isotropy of the model):


where is proper time, is radial coordinate, is coordinate time, is radius of curvature of space of Einstein's universe (Einstein's radius of universe). This metric approximates to Minkowski metric

for <<.

The EUT metric seems to be the simplest metric with required by Einsteinian property of non symmetry of metric.[10]

Critique of the Big Bang hypothesis (BBH)

First of all there is no expansion of universe as it was proven by application of high school calculus to simple task of predicting the amount of intrinsic redshift in the universe in BBH. Lack of expansion disqualifies the BBH.

Since humans are not easily convinced by one proof of invalidity of certain assumption and easier by multiple proofs, there are more flaws of BBH:

  • curvature of spacetime of the universe, allegedly causing gravitation, while gravitation is most likely caused by particles assuming the most probable position in flat spacetime formed by 10 curvatures of spacetime, while "curved spacetime" might produce disappearing of physical objects not observed in nature (unless they are socks in a washing machine), and creation of physical objects "out of blue" (neither observed in nature),
  • symmetric metric tensor of spacetime, allegedly causing a need for assuming that spacetime is "expanding" while Einstein specifically presented the demand that metric tensor must be non symmetric[11] and possibly non Riemannian,
  • quasars at cosmological distances (forced by the lack of alternative mechanism of the redshift except the expansion),
  • "dark energy" (forced by the lack of alternative mechanism of the redshift except the expansion),
  • to be continued

History of Einsteinian gravitation (personal experience)

Atempt to get the rate of expansion of space

I was all my life interested in physics though not in gravitation. I thought that gravitation works approximately as Newtonian equation say and there is nothing special about Newtonian force being

Of course I "knew" that the universe "is expanding" and speed of expansion (called Hubble constant) is about 70 km/s/Mpc. I also knew that photons interact with masses in the universe that have to influence those photons slightly diminishing their momentum. All this was "known". What was not known was how much of their momentum photons lose flying through the universe, due to the presence of matter in the universe. I tried to calculate it approximately but got the results always between zero and infinity depending on my approximations, which obviously meant that my approximations were wrong.

In February 1985 I decided that it is crazy not to be able to calculate such a simple thing. I decided not to use any approximations this time and I calculated, with high school calculus, Newtonian math, and a little of Einsteinian physics (limited to equation ) how much of cosmological redshift in the universe is due to the expansion of space and how much is due to the Newtonian interaction between photons and the masses of universe that as I learned from Ned Wright [12] was called dynamical friction.

I wanted to find out what is the real rate of expansion of space after subtracting the result of my calculations from the Hubble constant that, as I thought then, contained the total redshift: coming from the expansion of space and from dynamical friction.

I thought that I can adjust the Hubble constant by subtracting from it to get the real rate of expansion of space. I thought that the rate of expansion was too high since nobody bothered with making those calculations of dynamical friction. When I asked astrophysicists they all kept saying that the dynamical friction must be "negligible" for photons since if it were not then someone would have calculated it already, and no one heard that it was ever done. Consequently, no one knew how "negligible" it is.

Surprise! Surprise! No expansion?!

The result turned out to be , where is Newtonian gravitational constant, and is the mass density of universe [13] but also that it is the whole Hubble constant of our universe. There was nothing left for the expansion of space.

It means that the cosmological redshift might be the result of interaction of photons with the masses of universe due to dynamical friction and none due to the expansion of space. Necessary conclusion was that the space of our universe is not expanding and the redshift is due dynamical friction of photons only (as Fritz Zwicky once suspected, just couldn't prove it with his math).

Could I with mine (or rather Newton's and Einstein's)?

Checking the results and the first rejection

To be sure that I didn't make any mathematical error I gave those results to a math professor who was teaching general relativity a few years earlier to check them,[14] and when he assured me that there was no math error I sent the results to "Nature", which I subscribed to at that time, on February 12, 1985, for final verification and possible publication. Then I got a response from "Nature": ... 20 February 1985, "I am afraid that we cannot publish your paper which is too speculative to be appropriate for Nature." Dr Philip Campbell, Physical Science Editor.

Studying general relativity

After the rejection letter from Dr Campbell, I asked him which part of my paper he considered "speculative" and sent the paper to "Physical Review Letters". It rejected it as well but after the referee said he does not understand it recommending rejection. Then I asked for a referee who understands gravitation and got one with whom I discussed the subject for many weeks learning in the meantime the general relativity mostly from his questions. Then I enrolled in the general relativity two semester course at Harvard University, Cambridge, MA to find out why such pretty paper can't be published. On March 28, 1985 came response from Dr Campbell: "Maybe our judgement is at fault but, based on that judgement, our decision must remain against publication."

Reformatted value of Hubble constant

The first thing that I learned from my general relativity course in Harvard was that my result

can be written as


where is as before Hubble constant, the same as , is speed of light in vacuum, and is Einstein's radius of universe.

To make a long story short I wrote another paper based on my newly acquired knowledge of general relativity, using in it already my reformatted Hubble constant being now equal and arguing with a referee for several weeks explaining also except the reasons for the necessity of the universe being stable the temperature of MCBR (Microwave Cosmic Background Radiation) and how it allows the calculation of average size of pieces of non luminous matter.

The referee seemed to be impressed but not knowing what else might be my weak points and being not able to refute my results for any mathematical or physical problems decided that he will reassess their rejection on the basis of their little interest to the readers of "Physical Review Letters". Which the referee did and so my next rejection was for the presumed little interest of the readers in cosmology.

Ratio of radius of Earth orbit to itself

In the meantime I became the nuisance to the professor teaching my general relativity course. After he told us that we don't need just listen but are supposed to ask questions if we don't understand something. The first question he heard from the audience was "how come the ratio of radius of planet orbit to radius of Earth orbit is exectly one for the Earth?" After everybody started laughing at the silliness of the question I got less intimidated by the presence so many wise physicists around me and asked "if space expands with speed greater than the speed of light, what will be the speed that galaxies move with in respect to a guy connected by long pole to one of galaxies? Wouldn't it be greater than speed of light?"

The question was followed by long silence during which our teach was thinking intesively. After the silence became too long, he said "I teach this course for four years already and nobody ever asked me such a question so I need more time to think about it, just come to me tomorrow before the lecture so I have more time to think. I'm not really a relativist but a particle physicists, I just teach this course".

The question wasn't answered neither next day ("no time") nor ever. I just learned that general relativity is not a simple subject known to any physicist. So I started learning it and think about it on my own.

Getting rid of cosmological constant

First idea that came to my mind was that if the Einstein's value of cosmological constant stabilizes the universe then one should come up with such a constant that changes together with (contraction of Ricci tensor) in such way that their sum vanishes. I called this hypothetical constant "contraction of Hubble tensor" in honor of discoverer of Hubble redshift. To have a contraction one needs also a tensor itself so I needed a Hubble tensor as well. It would be a tensor having such properties that its contraction multiplied by half of metric would be equal exactly (Einstein's value of cosmological constant). I just learned lately from a mathematician in my school, that such arrangement is possible if the metric is non symmetric. Which it is, since it was metric required by Einstein for the spacetime.[15]

Hubble tensor

Furthermore, to make the spacetime flat to comply with the principle of conservation of energy and momentum (that followed from Noether's Theorem) I needed the missing antisymmetric part of Ricci tensor that would represent the "curvature of time" (the term coined by Richard Feynman) that would compensate the curvature of space represented by Ricci tensor to make the spacetime flat (Minkowski). Hubble tensor seemed an ideal candidate to fill up this place.

The metric that I assumed for Einsteinian universe is non symmetric and degenerate (with vanishing determinant) and yet is approximates to very neet flat Minkowski metric


I sent this proposal of new Einstein field equation in January 1986 to "Rapid communications of Physical Review Letters". They received it on January 27, 1986, and rejected on March 14, 1986 by the same referee who rejected my previous stuff about Newtonian calculation of Hubble constant in Einstein's universe.

Future of Hubble tensor

The assumption of existence of Hubble tensor leads to very neat Einsteinian equation

that is the same as the present Einstein field equation but without his cosmological constant replaced by

, where is contraction of Hubble tensor.

To be continued... JimJast 12:08, 24 August 2011 (EDT)

See also


  1. "Flat" or "Euclidean" means "3-space" (x, y, z space) in which any "2-sphere" (since its surface has only 2 dimensions) of surface area contains volume of space .
  2. According to the legend only two physicists, Albert Einstein and Arthur Eddington, understood relativity since when a journalist told Eddington that he heard that only three people in the whole world understood relativity, Eddington asked him: "Yeah? Who is the third?"
  3. Quote from Newton [according to wikipedia]: "That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one another, is to me so great an absurdity that, I believe, no man who has in philosophic matters a competent faculty of thinking could ever fall into it."
  4. This ratio is expressed usually as
    in sign convetion used in mathematics, where distances in spacetime are considered to be positive numbers, or
    in sign convention used in physics, where times are considered to be positive and distances negative.
    in both conventions is time-time component of metric of spacetime, which means that
    is not a constant but a function of displacement in relation to the particle and we may take its non vanishing derivative with respect to displacement to calculate "gravitational force".
  5. And one has to remember that math is not science.
  6. Story told to the author by Roy J. Glauber who was at the time Einstein's assistant in Princeton University in N.J. and later, for one year, author's physics teacher at Harvard, Cambridge, MA.
  7. Theories are never "right" unless they pass all the possible tests of their validity, which the Big Bang hypohtesis was not able to pass and that's why it has to be replaced with EUT (also unsure but at least better theory than the Big Bang that contains errors, which we don't see in EUT, hoping that in EUT there are none).
  8. Explanation of the above equation: It comes from the Newtonian force being and , the source of gravitational "repulsive force" being replaced by according to Einstein's physics. in this equation is difference of original mass of "dust" causing the "Newtonian repulsion" adjusted by in general tiny causing "attraction" (acting in opposite direction than the "repulsive" ). It is similar to a gravitational effect of a bubble of air "repulsing" gravitationally their sorounding water in a tank of water plus tiny effect of attraction by , the tiny gravitational energy of accumulated within the bubble
  9. 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.
  10. 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.
  11. Einstein, A., "On the Generalized Theory of Gravitation", April 1950, "Scientific American", dropped from bibliography of Einstein's papers in Misner, Thorne, and Wheeler's 1973 "Gravitation".
  12. Edward L. (Ned) Wright, Astronomy professor from UCLA, who advises through e-mail on problems within astronomy to anybody who has such problems.
  13. See step-by-step calculations in
  14. Prof. Tadeusz Bałaban, presently tenured professor at Rutgers University in New Jersey, who checked my calculations in February 1985 before saying that they don't contain any mathematical error, after which I sent them to "Nature", where the editor rejected them for his disbelief of them without any peer review.
  15. Einstein, A., "On the Generalized Theory of Gravitation", April 1950 issue of "Scientific American". Einstein's paper has been dropped from the bibliography of Einstein's papers in 1973 "Gravitation" by Misner, Thorne, and Wheeler.