(C) 2009 Jim Jastrzebski
GRAVITATION FOR EVERYONE   April 6, 2009
Why things fall? The explanation for high school students and physicists.
"You do not really understand something unless you can explain it to your grandmother." [Einstein]
Demystifying Einstein's gravitation and recovering Einstein's Universe merely through the principle of conservation of energy.
History of Einstein's Universe: In 1687 Izaak Newton described the math of gravitation. He didn't know its physics but not believing in action at a distance he refused to believe in gravitational attraction that his theory suggested. Another such non believer was Albert Einstein who called action at a distnce "spooky". Both were right. In 1915 Einstein explained the gravitation through the curvature of spacetime. It was the first physical theory of gravitation, imperfect yet since the original field equation turned out to be unstable. In 1917 Einstein discovered his cosmological constant Λ that stabilized the field equation and then the only flaw left was the symmetric metric tensor of spacetime inherited from the Riemannian geometry that didn't allow the conservation of energy. Symmetric metric couldn't be the physical reality since there were no indication that energy is not conserved. In 1950 Einstein fixed this flaw by recanting his old assumption of symmetric metric tensor and declaring that the metric tensor of spacetime must be non symmetric leaving to astronomers to determine the value of dynamical frictions of photons. Misner, Thorne, and Wheeler ignored Einstein's gravitation in their 1973, over 1200 page cosmologists' bible "Gravitation" and assumed creation of energy by assumption that the observed redshift is due only to the expansion of universe. It was the biggest blunder of their lives. Wheeler's version of gravitation became a pseudoscience based on an assumption that energy can be created from nothing in amounts needed to support his theory. Without ever calculating the amount of dynamical friction of photons. Despite that it was obviously false, at least according to the Patent Office clerks, it became for the next 25 years a mainstream version of gravitation opposing Einstein's gravitation since Einstein's universe was stationary. In Einstein's gravitation still remained a question how much redshift there is due to the dynamical friction of photons i.e. what is the Hubble constant of Einstein's Universe. In 1985 I got its value (the derivation is here). It turned out that the value is equal the speed of light divided by Einstein's radius of universe which meant that Hubble redshift tensor is the mirror image of the tensor of curvature of space or that spaceime has intrinsically flat geometry. It has been already suspected by astronomers. Wheeler's myth of curved spacetime fell. Not many people noticed though since scientific journals declined to publish the news as not interesting enough.
     To be continued in Introduction at http://geocities.com/jim_jastrzebski/sci/gravitation.htm
Table of Contents

Introduction

 Continued from preamble

Despite the beauty of the result it has been never published. The editors of scientific journals claimed that their reders aren't interested in a news that according to Einstein's theory, and astronomical observations, our universe is stationary. So I show it below for those few who are interested how Einstein's curvatures of spacetime cause the illusion of gravitational attraction and the illusion of accelerating expansion of space. Maybe at least it'll have an effect of stopping imbuing the physics students and high school students with the medival idea of "gravitational attraction" that Newton was so much against, and rightfully so as it turned out.

Since the news about Einstein's Universe and about the following from Einstein's theory the 1985 general time dilation was never published in any scientific journal the astronomers still don't know it even today and they think that the redshift they see as the Hubble redshift results form recession velocity of galaxies in expanding universe. The necessary in such scenario creation of energy they treat is a "natural phenomenon" resulting from curvature of spacetime as cosmologists, who have little appreciation for physics and the principle of conservation of energy, keep suggesting to them.

Richard Feynman called cosmologists idiots for assuming that "some obvious and correct fact, accepted and checked for years, is, in fact, false". Wheeler, the proponent of expanding universe hypothesis, the same who transgressed against good science by not mentioning Einstein's statement about the metric tensor of spacetime, maintained for many years that Einstein's cosmological constant Λ is a blunder, and it should be set to zero. In 1998 when the techology reached the level that Wheeler's assertion could be tested the "Super Nova Project" proved Wheeler painfully wrong. The alleged expansion of universe looked accelerating, just opposite to Wheeler's prediction but in precise agreement with the conclusions of Einstein's theory.

After 1998 disaster of predicting an opposite situation to the acctual the cosmologists patched their hypothesis of expanding space with assumed ad hoc "repulsive gravitation" (that they apparently just discovered) and with the existence of exotic "dark energy" that allegedly were using this "repulsive gravitation" to expand the universe faster and faster.

That the universe is not expanding at all was never mentioned by editors of sceintific journals who knew about it (e.g. editors of "Nature", "Phys. Rev. Lett.", "Science" and others). The editors might have known for a fact that cosmology is a pseudoscience created to employ scientists who couldn't earn living in any egitimate branch of science and yet needed employment too. The cosmologists, who apparently aren't smart enough to keep low profile and cautiously admit that "they actually don't know" and limit their activity to picking up their salaries, insist that space is expanding, spacetime curved (of unknown yet intrinsic curvature), the metric tensor of spacetime symmetric, and energy constantly created, most likely through divine intervention, an idea supporeted by most theists and even by the Pope. They consider investigating the nature of "dark energy" the most important problem of physics of 21st century.

Einstein's theory doesn't allow neither "repulsive gravitation" nor creation of energy and because of it it requires intrinsically flat spacetime (as long as the space is curved and time dilated), it doesn't require "dark energy" and it is so simple that high school education is sufficient to understand it and so even a sculptor can explain it. What I'm just doing below.

Einstein's gravitation can be really explained to anyone's grandmother especially when the granny attended a high school, liked physics and math, and is not prejudiced agains Einstein, which almost never happens to grandmothers but often to physics professors.

Because of the phenomenon of redshift of galaxies observed since 1912, Einstein's stationary universe of 1917 hasn't been accepted by gravity physicists who even today, after nearly a century of success of Einstein's gravitation still count on "quantum gravity", which they hope is going to disprove Einstein's theory and prove their own hypothesis that the universe is expanding, necessarily in violation of the principle of conservation of energy. Unfortunately for gravity physicists who dream about abolition of conservation of energy under pretext of reconciling Einstein's gravitation with quantum mechanics Einstein's gravitation is already a quantum theory, as all physical theories should be ("physical theories are often smarter than their creators" [Hertz]).

The explanation of gravitation is done here in such simple way that it reminds a little poem by Einstein: "A question that sometimes drives me hazy: am I or are the others crazy?" It is since physics of Einstein's gravitation is considered by most people, even by physics professors, tough. And yet I didn't see anything tough in it yet. So the reader may see that I'm justified thinking that I might be crazy. But on the other hand, Niels Bohr said that in many cases it might be said: "Your theory is crazy, but it's not crazy enough to be true." So what I say might just be crazy enough to be true and besides it agrees with what Einstein's theory says about gravitation, so at least if I'm crazy I'm in a good company.

Since February 1985 I've been checking my result based on Einstein's 1950 abandonment of symmetric metric tensor for the agreement with Einstein's physics and Newton's math, learning in the meantime more math and physics, and my result, that Einstein's gravitation explains the illusion of accelerating expansion of space, has been always the same. Recalling another Einstein's opinion: "Insanity: doing the same thing over and over again and expecting different results" and being sane (in my opinion) I finally decided to give up checking the result and starting to write about them.

For the attempts to inform astronomers about Einstein's results I was banned for life from a couple of moderated astronomy and physics fora on the internet, so I decided to write this article for everybody who wants to understand gravitation but knows only high school math and physics. Which is true for most physicists and astronomers so at least they should be happy being able to get a document from which they may finally learn why things fall.

Then there is a point why we should understand Einstein gravitation being already physicists or astrophysicist and never having the need of understanding Einstein's gravitation before. The reason is that since then we won't believe in creationist propaganda that the universe was created some 14 billion years ago, and possibly save ourselves engaging in working on the Big Bang cosmology (a.k.a. Big Bang general relativity, BBGR for short) that becomes lately more and more popular despite being false (its falsehood assured by the violation of the principle of conservation of energy). The idea of creation of the whole universe from nothing is popular especially among theists who may like the idea of action at a distance and matter being freely created, possibly by creatures populating the spiritual world. Support of theists for the BBGR has been documented by March 2008 Templeton Foundation's award of $1,600,000 to an astronomer, cosmologist, mathematician, and Catholic priest, Michael Heller from the Papal Accademy of Krakow, Poland, "in recognition of scholarship and research that has pushed at the metaphysical boundaries of science". So we might be in a danger of establishing laws against engaging in science without a license which may end all the progress and would surely force me (rather a sculptor than a scientist) to shut up. Even now people are banned for life from establishment's scientific fora for propagating Einstein's ideas which gravity physicists consder contradicting the idea of creation of universe. So I'd like to explain it all to the "physicists who don't understand gravitation", who have been the only ones who've never believed in theistic ideas about the Big Bang since they belived in the principle of conservation of energy that contradicts the inventors of the Big Bang and their constant creation of energy from nothing "that has pushed at the metaphysical boundaries of science".

And how can we be sure that Einstein's gravitation really works the way that I desribe it? Well, it is like with everything in science: while a theory can't be proven, we have observational evidence that confirms the theory in a sense that "it was not yet proven wrong by any observations", while a hypothesis that the universe is expanding is proven wrong by the necessity of contineous violation of the principle of conservation of energy through an assumption in the Big Bang hypothesis of the vanishing value of dynamical friction of photons while allowing the conservation of energy produces Hubble constant of about 70 km/s/Mpc (observed). Besides, Einstein's theory predicts not only the illusion of expansion but also that it is the illusion of accelerating expansion with the value of dH/dt = -Ho2/2 (observed in the universe by "Super Nova Project" astronomers), predicted high redshift of quasars (observed by Halton Arp), predicted (and observed) "anomalous" acceleration of space probes Pioneer 10 and 11, predicted (and observed) minimum of angular diameters of galaxies around Z = 1.6, predicted (and observed approximately) density of space 6x10-27 kg/m3, predicted size of non luminous matter pieces of 2 m across, none of them predicted by the BBGR. Since there is no rational reason for supporting BBGR against Einstein's GR, "pushing at the metaphysical boundaries of science" seems to be the only reason for the BBGR still being around in 21st century. But it doesn't have to be around since a stationary, complying with the principle of conservation of energy Einstein's universe is described by simple physics able to be understood by a high school student. And even his granny. Despite this the majority of physicists declare that they "don't understand gravitation". This article is meant to be a cure for the poor understanding of Einstein's gravitation by physicists.

We have here a collision of physics (redshift of photons interacting gravitationally with the rest of universe) with assumed math (the symmetric metric tensor that prevents photons from having readshift in static universe). That's why Einstein solved this contradiction in 1950 by assuming non symmetric metric tensor for the spacetime which allows the Hubble redshift in stationary universe. Simple calculation reveals that the Hubble resdhift observed in our universe is exactly equal to the redshift resulting from dispersion of kinetic energy of photons in a stationary universe. See Jastrzebski, W. J. The general time dilation: relativistic redshift in stationary clouds of dust The only sane conclusion is that our universe is stationary. Which Einstein was sure of already since 1950. Yet, the matematicians, to make their math easier, assumed the symmetric metric tensor of spacetime and got an artifact of illusion of expanding space. And Feynman warned cosmologists: "Let me also say something that people who worry about mathematical proofs and inconsistencies seem not to know. There is no way of showing mathematically that a physical conclusion is wrong or inconsistent. All that can be shown is that the mathematical assumptions are wrong. If we find that certain mathematical assumptions lead to a logically inconsistent description of Nature, we change the assumptions, not nature." Yet in the case of Big Bang the view of nature has been changed to accomodate for mathematical assumptions.

This creation of energy (or matter, from nothing) invalidates BBGR as a scientific hypothesis (or, which is equvalent of the above, it invalidates the assumption that the metric tensor of spacetime is symmetric) and so we don't need to dwell on it and we may move to the description of Einstein's gravitation and later to the events that brought such silly hypothesis as BBGR around. We may only mention that Einstein's 1950 discovery was just that the metric tensor is non symmetric, which was not even mentioned in Misner, Thorne, and Wheeler's 1973 monography "Gravitation" since those gentlemen assumed at the onset of their monography a symmetric metric tensor, as also Einstein did but for Einstein it was in 1915 when no one yet heard about the Hubble redshift. Misner, Thorne, and Wheeler did it in 1973, 23 years after it was known that Einstein maintains that non symmeteric metric tensor has to be assumed. Yet the gravity physicists didn't try then to examine the proposition allegedly because it would be not as elegant a metric tensor as the symmetric one. To which Einstein said: "If you are out to describe the truth, leave elegance to the tailor."

Glossary of terms pertaining to gravitation

Abandonment of symmetric metric tensor. Einstein might have realized in 1950 that symmetric metric tensor implies violation of the principle of conservation of energy and said that symmetric metric tenso must be changed to a non symmetric one. Einstein's move has been ignored by astronomers and cosmologists who apparently didn't mind the violation of the principle of conservation of energy or even liked it possibly being theists as e.g. a cosmologist and Catholic priest, Michael Heller, who got $1.6 million from The Templeton Foundation for "pushing on metaphysical boundry of science".

Action at a distance is an action through empty space without any carriers of this action that, as it is now known, have to carry energy from one object to another since energy has to be carried on some carrier of energy, e.g. photons, gluons, or whatever, but can't get through "empty space" which would be analog of energy disappearing in one place and appearing without any good reason in anohther, which would contradict the iron principle of physics, the principle of conservaton of energy. Einstein called action at a distance "spooky".

Anaxagoras a Greek who said "nothing comes from nothing".

Angular diameter of galaxies as function of distance of those galaxies from us shows a minimum at about redshift Z = 1.6 and it seems to confirm Einstein's idea that the universe is a 3-sphere of "Einstein's radius".

Applied mathematicians are guys who can handle equations better than computers though not the same fast so they may never be able to replace computers, however one of their subspecies, gravity physicists, hope to replace physicists one day.

Atheism is a symptom of critical mind and a belief, that the universe is not controlled by supernatural beings (like "ghosts" or even Santa Claus) but rather by a string of events each possible to be explained without an action at a distance of some "ghosts". It is a general disbelief in existence of supernatural as e.g. the creation of matter from nothing would be. That's why it may be important for theists to "prove" that creation of matter (or energy) from nothing, as it is assumed in BBGR, is possible. Atheists didn't believe neither in creation of something from nothing, nor in ghosts, nor in Santa Claus. They might believe though that they "may have another beer..."

Big Bang hypothesis "Big Bang" is a term coined by Hoyle to ridicule the hypothesis that the whole matter of the universe has been created about 14 billion years ago as a small dot, smaller than any pixel on this screen, and it has been expanding ever since (Wheeler, 1973). The hypothesis has been accepted by the majority of gravity physicists (at least by those devoid of sense of humor) and included into their version of general relativity in which energy in "negligible amounts" (how "negligible"?) is created. Since about the half of the 20th century the consensus of gravity physicists considered BBGR to be true, and papers falsifying this cosmology for violaton of the principle of conservation of energy were recommended for rejection by referees of scientific journals (gravity physicists themselves) with editors complying with the recommendations. There was even a proposal directed to Alan Guth (one of believers in this cosmology), to create another universe in a lab by creating proper conditions for such an event (the proposal was witnessed by this author while driving Alan, and his friend from Columbia U, to some meeting downtown Boston, MA since both guys were too drunk to drive themselves after the weekly "Early Universe Seminar" at Harvard U in Cambridge, MA, for which Harvard U provided free beer to promote science).

BBGR explains the Hubble redshift as caused by the expansion of universe and because of this it postulates constant creation of energy in "negligible" amounts, just sufficient to compensate for the dynamical friction of photons that would exist in the world in which energy is conserved but can't exist in spacetime with symmetric metric tensor postulated by Wheeler's physics in which the conservation of energy is dropped, presumably to allow God to create matrial things from nothing and to allow to reconcile science with religion. This creaton of energy must happen through divine intrvention to keep the metric tensor symmetric since there is no other mechanism available in physics to create energy from nothing. The value of this dynamical friction has been never calculated by BBGR theorists just assumed as negligible as it should be in a spacetime with assumed symmetric metric tensor. I calculated the real dynamical friction of photons in 1985 and it turned out to be as it should be in Einstein's universe. I draw from it a very unpopular among BBGR gravity physicists and astronomers conclusion that our universe is Einstein's universe (Jastrzebski, 1985) (the Hubble constant coming out as 70 km/s/Mpc). For which I was banned for life from several moderated Internet fora losing connection to scientists who still believe in a scientific explanation rather then religious.

Richard P. Feynman in his comments on gravity physicists (point 4 of his critique) called gravity physicists idiots for assuming that an "obvious and correct fact" [like the principle of conservation of energy] "accepted and checked for years, is, in fact, false (these are the worst: no argument will convince the idiot)". In this controversy I support Einstein and Feynman, while theists, at least those from Kentucky, consider it "an atheist plot".

Conservation of energy means that in an isolated system (one with no connection to the outside of it) there is always the same fixed amount of energy. No energy in this isolated system can be created or destroyed. This assumption is dropped in Big Bang hypothesis to accomodate for a possibility that the universe is expanding. The Big Bang hypothesis is called rather illegally general relativity but it is only a magical hypothesis based on Riemannian geometry describing quite inaccurately the Einstein's physics of gravitation trying to make a mathematical theory out of it in which energy is not conserved (which is clearly not happening in the real world since the whole universe is necessarily an isolated system since there is nothing outside of it). Here we are considered only with accurate description of Einstein's physics, and so not with description of the Big Bang theory. We are showing in this article that (i) conservation of energy is a basic part of Einstein's theory of gravitation without which it isn't working at all and (ii) that with the conservation of energy being a valid assumption it is not necessary to assume that the universe is expanding.

Copernican Principle says that space is (roughly) homogeneous. The "Perfect Copernican Principle" states that the Copernican Principle is time independent. Einstein's general relativity states that our universe behaves as required by the Perfect Copernican Principle.

Cosmic Microwave Background Radiation before 1963 was thought to be the thermal radiation, of temperature about 3 K, of the non luminous matter of the universe being in thermal equilibrium with redshifted starlight. It was measured by the radio engineers from the density of noise that the radio engineers noticed as coming from the sky and it couldn't be assigned to any known source of noise. They assigned it to the thermal energy of non luminous matter of the universe assuming that the temperature of the uiverse was about 3 K (accodring to the author's recollections of his pre 1955 high school lecture of a radio technology textbook and that's why the author knew "since always" that the temperature of the universe is about 3 K and he was surprised that it has been "discovered" only in 1963 by Wilson and Penzias).

Cosmic Microwave Background Radiation after 1963 (a.k.a. CBR or CMBR) is radiation that comes from the sky as black body radiation of temperature 2.716 K being most likely the thermal radiation of non luminous matter of the universe being in thermal equilibrium with redshifted starlight as it was assumed before Wilson and Penzias "discovered" that it might be the radiation left in the universe as the result of the Big Bang event for which "discovery" (see "Cosmic Background Radiation before 1963") they got Nobel Prize in 1978.

Cosmological constant, a.k.a. Einstein's cosmological constant or Λ is a constant of Einstein's field equation and its nature is similar to constant of integration that needs to be determined while calculating integrals. The value of this constant has been a subject of Big Bang controversy since the value calculated by Einstein in 1917, ΛE = 4 π G ρ / c2, implies stationary spacetime (idea supported by about 5% of cosmologists, astronomers, and this author) while most astronomers and gravity physicists assume that the Hubble redshift is the Doppler shift caused by recession of galaxies which implies that the universe is expanding. Therefore the value calculated by Einstein is the very value not accepted by most astronomers and gravity physicists. They assume various values of Λ consistent with expanding space. After discovering cosmological constant Einstein was constantly bothered by visits from cosmologists who kept proposing their pet values of cosmological constant. Finally Einstein told his secretary not to let in anyone who wants to talk to him about the universe (source: prof. Roy Glauber, Einstein's co-worker, telling the story to the author, his student at Harvard). Then Einstein resolved the Big Bang controversy with a joke, telling George Gamov that the discovery of the cosmological constant was "the biggest blunder of my life". The cosmologists (at least those deprived of sense of humor) treated Einstein's joke as his admitence that he was wrong discovering the cosmological constant and they assumed its value as zero (see 1973, Misner, Thorne, and Wheeler, "Gravitation", p. 411). Before 1998 the gravity physicists supported by Stephen Hawking who advised to try to make finally some observations confirming the Big Bang assumptions (namely that the cosmological constant doesn't exist, that it was just "Einstein's blunder") and so the expansion of the universe is decelerating, as required by the zero value of cosmological constant. Around 1998 it turned out that the value of cosmological constant can't be zero since the alleged expansion of universe, instead of slowing down looks like speeding up. Then the gravity physicists proposed the existence of "dark energy" of unknown yet properties as the way out of discrepancy between the Big Bang theory and observation of accelerting expansion. The nature of "dark energy" is investigated ever since.

Critical mind is a mind that does not believe in anything that is not confirmed by a reliable experiment or a reliable observation. Taking also under consideration that "reliable" depends on interpretation which may be source of errors in otherwise neat theory. Newton's theory of gravitation once was such a neat theory (not to Newton himself though, since he had a critical mind himself and he didn't believe in "action at a distance") that his theory indicated. He turned to be right (there is no "action at a distance" in gravitation). The list of things that critical minds don't believe in would be too long for this article so it is skipped here.

Curvature of space means that at some places of the universe there is "more space than at a point far enough from any material objects". It means that within the same 2-sphere of the same surface area, there is a greater amount of space (geater volume of space) than at a point far enough from any material objects (e.g. the same 2-sphere can hold greater amount of water). Furthermore it turns out that whenever there is "more space" also the time "slows down" in such way that product of the amount of space and the amount of time ("volume of spacetime") is constant. It means that the spacetime of the universe is intrinsically flat. This is a necessary result of the principle of conservation of energy. So one may say that curvature of space, or concurently, the time dilation, is the result of one of these features and the other resulting from the inability of nature to make energy from nothing.

Curvature of spacetime means that space is curved and that the time is dilated. In Einstein's spacetime the dilated time implies increased amount of space in such way that their product (volume of spacetime) is the same in the whole universe meaning that the spacetime of the universe is intrisically flat. Einstein's gravitation is considered having flat geometry.

Dynamical friction of photons is a relativistic effect of photons reaching the observer with a redshift depending exponentially on the distance that they travel, dτ/dt = exp(- r / RE), where τ is proper time (at source of light), t is coordinate time (at observer), r is distance from observer to source of light, RE is Einstein's radius. The reason for this effect is slowing of the time rate at the source of this light called here the general time dilation as opposed to gravitational time dilation. It is named dynamical friction (of photons) through analogy to the dynamical friction, a Newtonian effect of the disperssion of kinetic energy of things that move through the universe and interacting gravitationally with its matter losing in the process the kinetic energy to the matter of the universe (the effect opposite to slingshot effect used to accelerate cosmic probes while rejecting them from the Solar System). Within the Newtonian magic (phenomanological description of gravitation by Newton) the dynamical friction of photons is represented by the tired light effect in which the photons move against dynamical gravitational field c2/ RE, where c is speed of light and RE is Einstein's radius, as it might be observed in "anomalous" accleration of space probes Pioneer 10 and Pioneer 11.

Einstein's cosmological constant see Cosmological constant.

Einstein's radius a.k.a. Einstein's radius of the universe, RE = c / sqrt(4 π G ρ), where c is speed of light, G is Newtonian gravitational constant, and ρ is energy (or mass) density of space. It's the radius of 3-sphere being the 3-space of Einstein's universe.

Einstein's theory of gravitation a.k.a. Einstein's general relativity, a theory of gravitation that explains the gravitational force not as some magical force of "gravitational attraction", that even Newton who didn't believe in miracles refused to accept (for its implied action at a distance) but as a force resulting from the internal energy of the particle itself, diminishing along certain direction because of curvatures of spacetime. Gravitational force in Einstein's theory is therefore a force pushing a particle in a direction of the diminishing internal energy of particle. The excess energy (excess energy resulting from diminishing internal energy) is changing into the kinetic energy of movement of particle with the total energy of particle being always the same. In which we may see the working of the iron principle of physics, the principle of conservation of energy that can't be violated in a scientific theory and so it can be use to tell a scientific theory from a fiction. The curvatures of spacetime is all that is needed to explain the shape of diminishing energy of particle in space and in turn this shape is desribed by the math (or rather geometry) of Einstein's theory. That's why Einstein's theory is called a "geometric theory of gravitation". There are no external ("fundamental") gravitational forces acting on partiles. All is just decided by the curvatures of spacetime.

Einstein's universe is a stationary (neither expanding nor contracting) universe described by Einstein's field equation with Λ = ΛE, which is called "Einstein's value of cosmological constant". Einstein discovered Einstein's value of cosmological constant in 1917, but it was rejected by the theorists of BBGR since such value stabilizes Einstein's field equation and tells that the universe is not expanding contrary to the opinion of BBGR theorists.

E = m(v)c2 is an identity discovered by Einstein in which E is total energy of a particle, m(v) is its inertial mass and c is invariant speed of light. Some of this energy is called "kinetic energy" if it is due to linear motion of the particle with velocity v, but if it happens to be due to some rotations within the particle that aren't visible outside of it then it is called "internal energy" or the "invariant energy" of the particle since it is the same for all observers of the particle of this type regardles of their total energy (internal energy + kinetic energy). Since around half of 20-th century m started to mean the rest mass (invariant mass, formerly called mo), the equation E = mc2 became only equation for internal energy of particle and total energy had to be changed to E = m(v)c2(x) to distinguish m(v) from m while m(v) is related to m by equation m2(v) = m2 / (1 - v2 / c2), and to distinguish c(x) from c, which now means speed of light that a particle sees itself, at its actual position in space and not necessarily what observer sees which is his "coordinate speed of light". The value c(x) depends on the curvature of spacetime and is related to c by equation c2(x) = c2goo1/2 where goo is the time-time component of the metric of spacetime). Total energy of a particle is still an invariat number which means that it does not change while the particle changes, in a free fall, its position in relation to a coordinate system.

Evolution is something that is responsisble for creation of all spiecies, also those extinct, but many of them being just auxilliary structures for developing other more sphisticated structures. Evolution acting randomly not always succeeds as it happened e.g. with H. sapiens who has been evolved, with original purpose of converting bananas into proteins to feed with them the saber-toothed tiger, but something went wrong with that scheme and the saber-toothed tiger got extinct by H. sapiens whose brain evolved faster than that of the saber-toothed tiger and survived not as evolution intended, as the saber-toothed tiger's food, but as a pest destroying the other species which evolution planned to populate the planet Earth with, creating a paradise on Earth. Luckily H. sapiens managed to invent also H. bomb that soon will wipe out the H. sapiens from the face of the Earth and so the evolution will be able to start again from scratch, with its purpose being easier to achieve because of the anhanced radiation that H. sapiens will leave on the Earth as its main achievement. So maybe the real purpose of creation of H. sapiens was not to provide proteins for the saber-toothed tiger but to accelerate the mutations by releasing more radiation? Then the evolution might not be as stupid as some of us think it is but it is doing the best it can to bring the paradise on Earth around? And so we shouldn't call it "it" but more respecful name like "SHE" and start believing in "HER"? Another bad news for theists who don't believe in HER but aren't listening to Pascal's advice that believing is safer than not believing.

Feynman's opinion about gravity physicists provoked by his attendence at the World Conference on Gravity. It illustrates Feynman's disappointment with qualification of gravity physicists as scientists.

Field equation, a.k.a. Einstein's field equation, is a set of 10 differential equations describing the geometry of the spacetime. The equations may be combined together in tensoral form as follows:
Final version (of 1917, with Λ): Rαβ - (R / 2 - Λ) gαβ = 8πTαβ
where Rαβ is Ricci tensor, gαβ is metric tensor of spacetime, Λ is cosmological constant, and Tαβ is stress-energy tensor
Original version (of 1915, "elegant"): Rαβ - (R / 2) gαβ = 8πTαβ (all variables as above)
Wheeler considered Einstein's original version of field equation "elegant" and Λ "the biggest blunder of Einstein's life" since, as he has written in his "Gravitation" (by Misner, Thorne, and Wheeler, p. 411): "had Einstein stuck by his original equation, he could have claimed the expansion of the universe as the most triumphant prediction of his theory of gravity". Einstein said "If you are out to describe the truth, leave elegance to the tailor".
Comment: Had Einstein stuck by his original equation as Wheeler advised, he would make the same fool of himself as Wheeler and his team of Big Bangers did when it turned out around 1998 that the alleged expansion of the universe looks accelerating instead of decelerating as the original equation od 1915 "predicted".

It would be even worse when it turned out that the universe is not expanding at all since its expansion would violate the principle of conservation of energy. Einstein, being a patent office clerk and an atheist might have not supported the violation of conservation energy since the Patent Office had a policy of rejecting applications for perpetual motion machines without even testing their viability, and atheists didn't believe in supernatural, while Wheeler, being one of 126 dopes attending the World Conference on Gavity, as his student Richard P. Feynman disclosed in the book "What Do You Care What Other People Think?", might consider the ability of creation of energy from nothing, a necessary atribute of God and even of his angel whose picture blowing a horn adorns page 1218 of his "Gravitation".

It might be good to put more light on Big Bang controversy by mentioning here that according to the University of Kentucky gravity physicists, "Einstein's universe" is an atheist plot and "The Templeton Foundation" gave over million dollar award to a cosmologist Michael Heller "in recognition of scholarship and research that has pushed at the metaphysical boundaries of science.".

Flat geometry means that a certain aspect of the space under consideration is the same everywhere, as in Euclidean flat space. A convinient thing for measuring flatness of 2-dimensional (2D) surface may be the ratio of a circumference of a circle on this 2D surface to radius of this circle. If it is always 2π the surface is flat. It may be also the ratio of area of circle to radius squared. If it is always π the surface is flat. For 3D space it might be more things: as above the ratio of circumference to radius and the ratio of area of surface to radius squared (2π and π respectfully for all circles imply flatness) and also the ratio of area of surface of 2-sphere (our regular 3D sphere) in this 3D space to the radius of this 2-sphere squared (if always 4π the space is flat) or the ratio of 3-volume of 2-sphere to the cube of radius (if always 4π/3 the space is flat). For 4D spacetime it may be even more things.
It seems that spacetime of our universe is flat but the space is surely curved and its radius of curvature is about 4.3 Mpc since speed of light divided by this radius turns out to be 70 km/s/Mpc. For space being curved and spacetime being flat the 4th dimension (our "time") must fit the curved 3D space so that the 4-volume remains the same as in Euclidean 4-space (our "spacetime"). So the flatness of spacetime by adjusting the time rate to the curvature of space seems to be the sufficient reason for the illusion of accelerating expansion of space. Especially when parameters of this apparent expansion predicted with Einstein's theory fit the observations.

General time dilation is an effect valid for any space containing energy in any form. It is an effect of proper time running outside an observer slower than observer's coordinate time. In homogeneous space it is inversely proportional to the exponent of distance from the observer to the observed point in space, and proportional to the square root of density of energy in that space. The effect, in agreement with Copernican principle, shows up for any observer in the universe supplying the reason for the Hubble redshift since the density of space 6x10-27kg/m3 (approximately what is observed) implies the observed Hubble constant of 70 km/s/Mpc. Calculations based on conservation of energy while calculating the dynamical friction of photons indicate that this effect reflects the curvature of space in such way that the tensoral sum of three dimensional Ricci tensor of the curvature of space and the tensor of general time dilation (known as K tensor) vanish.

Gravitational energy is something that used to be called internal energy of a particle E = mc2(x), where m is mass of a paticle and c(x) is coordinate speed of light. The derivative of this energy with respect to displacement turned out to be gravitational force with negative sign (dE/dx = -F) then E = mc2(x) satisfies the definition of gravitational energy.

Gravitational force in Einstein's gravitation is the same as in Newtonian gravitation, F = mg, where m is mass of particle and g is acceleration of the particle in free fall. Also it is the derivative of gravitational energy with respect to displacement with minus sign F = -dE/dx . However unlike in Newtonian gravitation where gravitational energy is not a well defined quantity in Einstein's gravitation it turns out to be a well known and precisely localized quantity E = mc2(x).

Gravitational time dilation is an effect of time running slower at the source of light located at an object containing energy in any form (e.g. planet, star, or galaxy). The effect is approximately (for masses M << c2 r / G) proportional to the amount of the energy located in the object (contained in its mass M) and inversly proportional to the distance form the object: dτ/dt = 1 - G M / c2 / r, where τ is proper time (at source of light), t is coordinate time (at observer), G is Newtonian gravitational constant, c is speed of light, r is distance from observer to the source of light.

Gravity physicisists a subspecies of applied mathematicians who due to their ideas about physics believe that physics can be replaced by math. See Richard Feynman's impressions from his participation in a World Gravity Conference that he applied to gravity physicists who propagated BBGR. Feynman also warned gravity physicists saying: "Let me also say something that people who worry about mathematical proofs and inconsistencies seem not to know. There is no way of showing mathematically that a physical conclusion is wrong or inconsistent. All that can be shown is that the mathematical assumptions are wrong. If we find that certain mathematical assumptions lead to a logically inconsistent description of Nature, we change the assumptions, not nature."

Homogeneous means that the thing under consideration is the same everywhere. Things with some feature of their geometry being everywhere the same as in Euclidean geometry are called flat. E.g. a surface is "flat" if the ratio of area of a circle to its radius squared is the same everywhere on this surface and equal π. So by the above criterium the survace of the Earth isn't flat as some people ("flatearthers") think that it is. They think so because the "curvature" of the Earth is too small to be noticed. They can't notice it, that's why they think it's none. Similarly as with gravitational force that depends on tiny changes of speed of light outside of particle, but so small that people thought that there are none and so this force must be caused by "gravitational attraction" which as it was discovered by Einstein doesn't exist in the nature but only in what some humans think about this nature (hopefully no one who's read this article).

Hubble constant a.k.a. Hubble parameter, is the velocity of (apparent) expansion of the universe. It is equal to the ratio of (apparent) recessional velocity of galaxies to the distance to them. Hubble constant of Einstein's universe is equal H0 = c / RE, where c is speed of light and RE is Einstein's Radius. The time derivative of this velocity, dH/dt, is an (apparent) acceleration of the expansion of space. For Einstein's universe it is dH/dt = - H02 / 2.

Hubble law is the relation between distance to an astronomical object and its redshift, approximately exponential with distance. Quasars redshifts don't seem to fit this law and so their redshifts seem to be produced by a different mechanism than redshifts of galaxies (or the same mechanism if we allow the Einstein's version of general relativity, which then would be the general time dilation). However the Big Bang hypothesis can't tolerate a different mechanism of redshift than Doppler shift.

Hubble redshift is a phenomenon of light coming from distant galaxies with redshift increasing approximately exponentially with distance from the observer. The Big Bang hypothesis assumes that this redshift is Doppler shift due to the expansion of the universe (Misner, Thorne, and Wheeler, 1973, plus correction for the exponential change of this redshift with distance observed only in 1998). Einstein's gravitation explains it by the rate of time slowing down in curved space proportionally to the curvature of space. As it follows from the principle of conservation of energy, the sum of time dilation and curvature of space vanishes in curved space since d2τ/dtdr + 1 / R = 0, where τ is proper time at galaxy, t time coordinate of observer, r is radial coordinate of observer, R is radius of curvature of space (Jastrzebski, 1985, that predicted redshift with observed parameters but it is still not accepted due to belief of gravity physicists in the expansion of universe).

Identity is an "equation" that is valid no matter what, while a "real" equation, that is not necessarily an "identity" is valid only for some particular case, a particular combination of variables in that equation. E.g. y = x2 is valid only for a particular sets of numbers, e.g. for (x,y) = (3,9) but not for (x,y) = (3,7). Finding what particular sets of numbers are valid for a given "equation" is called "solving the equation". One can't "solve" an "identity" since by definition it is "valid no matter what". So "identities" are just ways of stating that something is "the same as something else". E.g. the identity E = mc2 means that energy "E" and inertial mass "m" are the same things at certain coordinate speed of light "c". May be in different units, and then "c" takes role of a unit, but they are physically indistinguishable. There are no physical features that can be used to tell one from another.

Illusion of accelerating expansion of space is an illusion that the space of the universe is expanding and that this apparent expansion is acceerating. This is an artifact caused by the interpretation of the Hubble redshift as Doppler shift, allegedly caused by the recession of galaxies.

Illusion of gravitational attraction is an illusion that all particles of the universe attract each other. This illusion is caused by the curvature of spacetime that changes the coordinate speed of light around each particle in such way that there shows up a gradient of particle's internal energy (dE/dx, where E is internal energy of the particle and x is displacement vector in the frame of this particle) that when the particle is immobilized it produces an inertial force equal -dE/dx. This force has been called gravitational force and it is a force pushing the particle in the same direction towards which it starts falling when it is in free fall (in direction g/g). Then it makes an illusion that this force is due to external "Newtonian attraction" as indicated by the known Newtonian equation (F = G M m r / r3 = mg), while it is an internal push from the inside of particle itself, due to the Einsteinian push, as the value of this push (F = -dE/dx = mg) indicates. Both forces are the same mg and that's why it was so difficult to discover the thruth about gravitation and which part is accidental math (which turned out to be "Newtonian attraction") and which part is real physics (which turned out to be Einsteinian push, see Basics of Einstein's gravitation for derivations).

Isotropic is a property of being the same in any direction. The Copernican Principle states that the space of our universe is roughly homogeneous and therefore also "isotropic".

K tensor (a.k.a. Konieczny tensor) is a 3 dimensional tensor describing the physical properties of general time dilation.

Magic is a machanism through which a mathematical (phenomenological) model that uses non existing entities works. The same as the ordinary magic it works through accidental similarity of the model to the physical phenomenon. A good example is Newtonian gravitation with its gravitational forces that act at a distance. Newton didn't believe in such forces considering them mathematical entities that are only imagined and therefore a kind of magical things, as unicorns. Yet knowing their magical nature through the Newtonian equations that describe their behavior we may use these magical things in calculations and get almost true results due to the similarities between the equations that control their magical behavior to the real equations that control the behavior of the real gravitation (unknown in times of Newton). The magical things are usefull as long as one does not consider them real and does not conclude about the real world as if those magical things were real. A newer example of magical thing is the expansion of the universe that has been considered real by many 20th and 21st century gravity physicists despite that it requires dropping the principle of conservation of energy that most of them consider accepted and checked for years . Dropping a well tested principle for some, at best highly hypothetical, and at worst totally impossible, thing changes the phenomenological theory into a magical theory usually favored by theists though not by all of them since many of them believe that facts shouldn't be explained by magic.

Mathematical theory, a.k.a. "phenomenological theory" or a magical theory. A theory that predicts everything that happens within its domain of application but doesn't explain why it happens just this way. Examples: Newton's gravitation after it was known that attractive gravitational forces don't exist, the quantum mechanics. The "Copenhagen school" maintains that theories can be mathematical only and so the question "why"? is empty in epistemological sense. Some proponents of BBGR maintain that questions to which scientists don't know answers shouldn't be asked at all since their may be asked on wrong assumptions, which I learned after asking prof. Baez, one of gravity physicists, about conservation of energy in gravitation, and was told by him that "in gravitation the energy is not conserved", which prompter me to figure it out on my own (and place it in section titled "Basics of gravitation" for all curious people to learn how energy is conserved in Einstein's gravitation).

Paradise on Earth is a projected purpose of evolution and situation on the Earth when the Sun has already used its all available H and is powered by burning He but it makes no diff to species populating the Earth, since the life on the Earth is now 100% inorganic (see wiping out the organic life by H. sapiens in 20-something century and leaving it with industrial robots that evolved into intelligent species) and all Earth organisms are eternal since made of 100% repleceable parts so they don't need even to eat one another just enjoy life and art since they are even more then their ancestors sensitive to beauty.

Physical theory a theory which its all predictions are true: really happen in the real world. Examples: corpuscular theory of gases, Einstein's special relativity, Einstein's gravitation, a.k.a. Einstein's general relativity (EGR) as opposed to the Big Bang general relativity (BBGR) which is a creationist hypothesis propagated between others by The Templeton Foundation with awards bigger than Nobel Prize to reconcile religion with science.

"Physicists don't understand gravitation" is a quote from my first semester ("Physics 1") physics teacher prof. Zharnecki who expressed the opinion of majority of physicists. I myself heard the opinion many times before. Most likely the opinion is caused by way the physics of gravitation is thought. This poor way of teaching leavs over 99% of physicists without a slightest idea what is the reason for gravitational force which is the most common force in nature that every living thing feels whole its life. The physicists just know it is not "gravitational attraction" since they heard rumors that in Einstein's general relativity a.k.a. "Einstein's theory of gravitation" there is no such force acting through vacuum (also Newton was squarly against existence of such force) yet teachers of physics assume that (screw Newton and Einstein) the best way of teaching physics is to imbue high school students, who later become physicists and astronomers, with the idea of "gravitational attraction being a real force of nature" (even one of "four fundamental forces" of nature). They know that later it is going to prevent physicists from understanding gravitation and not to protest (except those with critical minds which is only about 5% of any population) when the gravity physicists tell them that "the universe is expanding". And it may hold true even if they learn that the expansion of the universe contradicts the principle of conservation of energy (remember, we are talking about ones without critical minds). This lack of opposition will allow theists to maintain that it "might indicate" that there are more forces in the universe than the physicists know about. The physicists just "won't understand gravity" and so they won't suspect a large scale theistic fraud, a.k.a. the Big Bang.

Pseudo Riemannian geometry differs from the four dimensional Riemannian geometry by signature which in Riemannian geometry is + + + + while in psedo Riemannian geometries it is either + - - - (so called modern signature, with positive timelike intervals, used mostly in physics) or - + + + (so called traditional signature, with negative timelike intervals, used mostly in math).

Quantum features of Einstein's gravitation follow from the fact that an atom exchanges energy with another atom through exchange of a photon or a nucleus with another nucleus by the exchange of a meson. It means that whenever one object loses part of its gravitational energy (mc2) it happens in quanta. Calculations of probabilities of finding a particle somewhere in its direction of diminishing internal energy is more probable than it is at its old position or any ohter. So Einstein's theory turns out to be also a quantum theory that explains things with probabilities of certain events being more probable than others.

Quantum gravity is something the gravity physicists hope to release them from necessity to understand Einstein's gravitation. They hope that it will be a "real" theory of gravitation since they believe that the theory of gravitation should be a quantum theory and due to their way of understanding of physics of gravitation they don't realize that Einstein's theory is a quantum theory.

Quasars, a.k.a. quasi stellar sources of light are astronomical objects that look like stars (light points) but have redshifts much greater than the ones reasonably corresponding to their distance as calculated ftom their luminosity. Their existence indicates that they may be much closer to us than it would be indicated by the Hubble law and therefore their redshift may be due to something else than Doppler redshift assumed by the Big Bang hypothesis.

Real World is since about half of 17th century a subject of what physicists like Einstein investigate. It is thought by those physicists to exist objectively i.e. regardless of what H. sapiens thinks about it.

Ricci tensor is a tensor which spacial part is describing the curvature of our common 3-D space.

Riemannian geometry geometry proposed by Einstein as the geometry of spacetime when he still didn't understand fully the implicaitions of application of this geomtry to solutions of gravitational problems. In 1950 Einstein started to realize that to unify gavitation with electromagnetism (as it is done today) the spacetime can be only approximately described by symmetric metric tensor and he proposed that the metric must be non symmetric (Einstein, "On the Generalized Theory of Gravitation", Scientific American, April 1950). In reality it must be also a degenerate, which makes Riemannian geometry non applicalble in gravitation sice Riemannian metric is non degenerate.

σ (sigma) is standard deviation of the measured value. It is a measure of the accuracy of measurments. Physically it is the effective amplitude of the noise that is added to the measured variable by the measurments regardless of their number. It is also known as the rms (root mean square) value of the noise of measured value.

Spacetime is space and time considered together as one four dimensional object. The spacetime has a property that the sum of general time dilation in any direction plus the curvature of space in this direction, vanishes which means that the spacetime is flat (proper volumes of spacetime are isotropic).

Structure of time dilation means how amount of time dilation changes along any particular direction. We know how much because it is relatively easy to figure out what mass causes what time dilation since we know the relation between gravitational field and gravitational time dilation (see Appendix 1 for the relevant equations) and the relation between gravitational field and mass that generates it we know from Newton's equation. Time dilation reflects exactly the Newtonian gravitational potential except that it does not contain its ambiguity about its absolute value since it has natural zero value at infinity.

Support of theists for the BBGR has been documented by March 2007 Templeton Foundation's award of $1,600,000 to a cosmologist, mathematician, astronomer, and Catholic priest, Michael Heller from the Papal Accademy of Krakow, Poland, "in recognition of scholarship and research that has pushed at the metaphysical boundaries of science".

Symmetry is estetics of idiots. Discsovery of counter position (a.k.a. "contraposto") in Greek sculpture (around 600 BC) in wich pelvis is tilted in opposite direction to shoulders, as estetically superior to symmetrical position of figure cultivated in previous sculptures, has been a breakthrough in art that has been cotinued in sculpture ever since.

Theism. Philosophy, or rather a belief, that the universe is controlled by supernatural beings (a.k.a. "ghosts", also "holy") or a supernatural being (a.k.a. "God") instead of evolution. The belief is supported by the belief, also of gravity physicists, most astronomers, and applied mathematicians (though neither "pure mathematicians" who consider BBGR being pseudoscience, nor the majority of physicists who, as they claim, don't understand gravitation) that energy (and therefore the matter being the same as energy as shown by Einstein's equation E = mc2) can be created, in "negligible amounts" though, from nothing, since otherwise the supernatural beings wouldn't have any possibility to inervene in the affairs of real world. The mathematical theory going with this belief is a belief that the metric of the spacetime is symmetric (despite that "symmetry is the estetics of idiots" and Einstein, who believed rather in evolution than ghosts, has maintained that the metric of spacetime must be non symmetric. "Atheism" is opposite to "theism".

Time dilation means roughly that the time in any particular point in space runs slower than at a point far enough from any material objects. It means that presence of material objects slows down the rate of time in their vicinity.

Tired light effect is an apparent effect of photons reaching the observer with seemingly smaller frequency than they were emitted with behaving as if they lost part of their energy on the way. It turns out that photons don't do this and they just start from their sources with smaller energy since the time runs slower at the source due to general time dilation.

University of Kentucky, which introduced creationism as scientific discipline, declared that opposition to the Big Bang Cosmology is an atheist plot.

Basics of Einstein's gravitation

What's "gravitational force"?

 Before Einstein's discovery of the reason for gravitational force in 1915 the gravitational force was thought to be a mysterious attractive force, acting, against common sense, at a distance, between all material particles of universe attracting all of them to all other particles of the universe in agreement with Newton's math (see equation 1.4 in Apendix 1 for Newton's equation).

The main problem was where was this mysterious force coming from and how it is reaching the particle through the empty space. It turned out that this force is coming from diminishing energy of particles in certain direction in space that showed up to be the direction of "increasing amount of space". It turned out that in this direction of increasing amount of space the time is running proportionally slower and those two combined effects of curvatue of space and time diation that are coupled to each other make the spacetime such that the coordinate speed of light slows down in vicinity of masses by amount that produces gravitational force.

Slowing of coordinate speed of light causes the internal energy of the particle that is equal E = mc2(x), where m is mass of particle and c(x) is coordinate speed of light that depends on the position of particle x. It diminishes the internal energy of a particle in this direction. Diminishing energy pushes the particle into the direction of diminishing energy and as always in such cases (of diminishing energy) the force shows up and it's equal F = -dE/dx (where dE is a change of the internal energy of particle along distance dx). This force is a "gravitational force" since it is also equal F = mg (where g is the acceleration that the particle moves with while it is in a free fall).

The acceleraion must be such since the excess internal energy that the particle loses while speed of light gets diminished, changes into a kinetic energy of the movement of the particle and so the energy is conserved automatically. And of course in a free fall the total change of energy of particle must be automatically zero. This is the reason for conservation of energy in gravitation. No mysterious "gravitational potential energy" that changes into kinetic energy of movement causing as the effect the conservation of energy: it is other way around: since nature can't make energy from nothing (remember Anaxagoras) there is acceleration g in a free fall that follows form this inability of nature to make energy from nothing.

So finally we know what gravitational force is. It is an inertial force with which any object restricted from following its path of free fall pushes at whatever is restricting it from following the path of free fall.

It is a force coming from within the gravitating object itself. This push is called gravitational force. Integral of this gravitational force along displacement is gravitational energy the same as in Newtonian physics. What is different than in Newtonian physics is that this energy has a well defined location in space, namely the gravitating particle itself. And its value, as we show in section "How gravitational force is generated", is E = mc2.

The push varies according to structure of time dilation in space and the curvature of space (or according to structure of spacetime) which in turn varies according to distribution of energy (E = mc2) in this space. So the gravitatinal force is an inertial force generated by certain, in general varying (and the results of changes propagate with speed of light), distribution of energy in this space and nothing else.

The gravitational force will be derived below and then it'll be seen why it is the same as in the Newtonian gravitation despite having completely different nature. Why it must have a quantum nature automatically, why there is neither special "gravitational energy" (beyond mc2).

The whole Einsteinian gravitation, except mentioned above time dilation, contains in it only curvature of space and conservation of energy as basic physical entities. There is nothing more needed to explain all the gravitational phenomena in the universe including its apparent accelerating expansion.

Let's start with the most interesting part (that "no physicist understands") namely how gravitational force is generated in Einstein's gravitation.

How gravitational force is generated

In our frame of reference the total energy of any particle is (see Landau and Lifshitz, Theory of fields)
  E = m(v) c2(x) (1)
where m(v) is inertial mass of the particle or
  m(v) = m / [ 1 - v2(x) / c2 ]1/2 (2a)
where m is the rest mass of particle, an expression familiar from special relativity, where v is velocity of the particle in our frame of reference, x is displacement within this frame, and c(x) is coordinate speed of light within this frame, or
  c(x) = c [goo(x)]1/4 (2b)
where goo is time-time term of metric tensor of spacetime, or (dτ/dt)2, where τ is the proper time of point in space and t is the coordinate time of observer observing this point in space, usually at some distance from him/her. So the time at point in space (the proper time of that point in space) is different than the time shown by observer's watch (the coordinate time of observer).

The derivative of energy (1) with respect to displacement x (a derivative that when with opposite sign is called "force that pushes the particle" since the particle always tries to achieve a lower energy level), putting c(x=0) = c and m(v2=0) = m, is
  dE/dx = c2 [dm(v)/dx] + 2 c m [dc(x)/dx](3)

Since for a particle at rest v = 0 then from (2a) dm(v)/dx = dm/dx = 0. What is left is dc(x)/dx and in the real world the light ray bends in the vicinity of a material object twice as much as it is required by curvature of space and so it means that around materal objects speed of light changes and it better be dc(x)/dx = - g / 2c because otherwise energy (1) couldn't be potential energy. But if it is then in a static situations we should get some force F(x) = - dE/dx acting on a particle pushing it in the direction of its diminishing energy. Substituting dm(v)/dx = 0 and dc(x)/dx = - g / 2c into (3) we have
  F(x) = - dE/dx = g m (4)
and therefore exactly the same as Newtonian inertial force. And it is our static gravitational force the same as Newtonian gravitational force. Except that we figured out this force directly from the diminishing speed of light in direction of gravitational field g (which we intend to show that it is a necesary consequence of time dilation and curvature of space). And then our potential energy is the internal energy of a particle, given by (1), and located not in the field but within that particle. Now we have to prove that our working assumption that dc(x)/dx = - g / 2c is actually true.

The proof

We need to find out how the coordinate speed of light c(x) is related to gravitational field g. To figure out this relation we need to remember the following facts:

The angle of deflection of light ray in vicinity of material objects is twice as large as it would be predicted by existence of Newtonian gravitational field due to time dilation. Einstein's guess was that half of this angle of deflection is due to time dilation that simulates the Newtonian gravitation and the other half due to the curvature of space that has no counterpart in Newtonian gravitation.

Next fact is that when the time slows down everything is running more slowly in the same space. When one side of a light ray runs more slowly than the other the light ray bends in direction of smaller c(x) and the angle of deflection is
  φ = - [dc(x)/dx] t (5)
where dc(x)/dx is change in speed of light per unit of distance across the ray and t is the time of light passing the area of changed speed of light.

In a flat space the angle of deflection of light ray would be due only to the change in speed of light across the ray. In a situation when space is curved the curvature of space bends the light ray without any change in the speed of light since then both sides of the light ray move in the (curved) space straight. The light gets bent due to the space curvature without a difference betwee speeds of light across the ray. So to find observationally dc/dx we need to take a half of the observed angle of deflection of light in gravitational field g and apply equation (5) to it.

Angle of deflection of light ray may be derived from an example with a rocket ship in space, sufficiently far from all material objects not to feel any influence of those objects, accelerating let’s say as much as the particles that fall on the earth. If there is a light ray that enters the rocket ship perpendicularly to the direction of acceleration of the rocket ship the observer in the rocket ship will feel the gravitational field but the light ray won't and so it will move along a straight path in relation to fixed points outside the accelerating rocket ship. The accelerating with the rocket ship observer however will see the light ray bent towards the rear end of the rocket ship (assuming that the rocket ship accelerates forwards). In the relation to the rocket ship that is accelerating "up" with acceleration g the ray is dropping "down" with the same acceleration g. The height of the drop is (integrating the acceleration g twice with respect to time)
  h = g t2 / 2 (6)
where t is the time that takes the light to cross the rocket ship. In relation to the accelerating with the rocket ship observer the light is moving along a parabola, which for our purposes may be approximated very well by an arc of a circle. The tangent to this circle at the point where the ray enters the rocket ship crosses halfway through the rocket ship the tangent to the circle at the point of leaving the rocket ship. It makes the angle between these tangents (the angle of the deflection of the ray), substituting h from (6):
  Θ = h / (t c / 2) = g t / c (7)

According to Einstein's principle of equivalence of acceleration and gravitational field this case is identical to the case when the light ray moves across a rocket ship that is standing on the earth, and so the ray bends in the gravitational field g, the same as the ray seen by the observer in the accelerating rocket ship. Since half of this angle comes from the curvature of space and the other half from the change in speed of light across the light ray we take φ = Θ / 2 and get change in speed of light from (5) and (7) as
  dc(x)/dx = - g / 2c (8)

Quad Erat Demonstrandum

Vanishing gravitational force in free fall

So we've proved that energy (1) is potential energy Ep that produces a required gravitational force by changing itself along a distance due to the changing speed of light in curved space where the time is dilated. The value of gravitational force might be obtained by reversing the direction of the derivative of potential energy with respect to distance as in F = - dEp/dx.

It turns out that in the real world it is not a gravitational "pull" by "attraction" of some external body but inertial "push" by inrtia of the particle in space where there is a change of internal energy of the particle as a function of displacement. So it is not a body attracting other bodies but other bodies are pushed by themselves towards an "attracting" body with this body not attracing them but just modifying the spacetime around herself by her presence in such way that those other bodies get themselves pushed towards the "attracting" body. "Attraction" is a figure of speech here and what is reall is the "push" towards this "attracting" center.

Now we need to do the test with free fall to see if energy of a particle in free fall doesn't change.

Since in a free fall in gravitational field with acceleration g, velocity v2 = 2 g x, where x is the distance by which the particle has fallen (in direction of its acceleration g) then substituting v2 into (2)
  m(v) = m / (1 - 2 g x / c2 )1/2 (9)

Differentiating with respect to x and ignoring small higher order terms
  dm(v)/dx = m g / c2 (10)

After substituting (8) and (10) to (3) we have a change of total (potential) energy of a free falling particle as
  dE/dx = 0 (11)
which shows that there is no change in the total energy of a free falling particle and so the whole kinetic energy of a free falling particle is a part of its diminishing internal energy (1).

This concluds the explanation of basics of Einsteinian gravitaton. The rest of this page shows how conservation of energy is responsible for an effect called here the general time dilation that in turn is responsible for the Hubble redshift (illusion of accelerating expansion of space) and possibly also for the high redshift of quasars.

Illusion of accelerating expansion of space

If one assumes that the reshift of galaxies is due to their recession velocity then the universe looks as if it were expanding and its expansion were accelerating. This is so since the light coming from distant galaxies has on average a smaller frequency than the light generated by the same sources close to the observer. The reason for this smaller frequency of photons was assumed in BBGR to be a recessional velocity of galxies but it turned out that the time at those galaxies runs slower than at the observer and so the effect simulates the expansion of space. Furtheremore the simulated expansion looked as if the space were expanding with acceleating expansion. This effect of the time running slower at the greater distance turned out to be necessity if energy couldn't be made out of nothing and the simple derivation of this effect, from the principle of conservation of energy, is presented in this document.

Hubble constant of this apparent expansion comes out as H0 = c / RE, where c is speed of light and RE is Einstein's radius of the universe, and the acceleration of this apparent expansion comes out as dH/dt = H02 / 2, which is observed within a fraction of standard deviation.

Since the theory can't be falsified for the time being by observations since it predicts correctly the observational results within one σ (which in astronomy means a perfect agreement), then we may look at the cosmic background radiation (CBR) to see how it is doing over there.

This radiation cannot be just the redshifted starlight since then it could not have the black body spectrum that it has. It seems therefore that it has to be the radiation from non-luminous matter that is in thermal equilibrium with the redshifted starlight. If it is so then we can calculate the average size of the pieces of non luminous matter of the universe. This is because the probability P of a photon hitting an obstacle of diameter D on it's way, and transferring to it its energy, which then becomes thermal energy, is approximately proportional to the area of the obstacle D2 and to the number of obstacles along the photon's way (inversely proportional to the cube of the distance L between obstacles P ~ D2 / L3. Since for a fixed mass density of the whole space (already determined from Hubble parameter) the distance between obstacles is proportional to their linear size L ~ D, the total probability of the photon hitting an obstacle becomes inversely proportional to the linear size of the obstacle: P ~ D2 / D3 ~ 1 / D.

So, knowing the temperature of the redshifted starlight, presumably re-emitted as a thermal radiation from the non luminous matter, and assuming specific density of the matter that the non luminous matter is made of, one may determine the average size of the pieces of non luminous matter of the universe (see Appendix 3 for details).

Origin of Big Bang GR

The "expansion of space" has been suggested by astronomers because of galactic redshifts discovered by an American astronomer Vesto Slipher in the years following his discovery of blueshift of Andromeda in 1912. Most of these shifts turned out to be redshifts, with ratio of red to blue 4:1. Then these redshift, because of absence of better explanations at the time, were interpreted by astronomers as Doppler redshifts that were taken by theorists for a proof that the universe is expanding. In 1931 Georges Lemaitre proposed an explanation of this expansion, known later as the Big Bang.

In the Big Bang "theory" it has been assumed (after Einstein, who changed his opinion only in 1950 when he proposed a non symmetric metric tensor) that the geometry of spacetime is pseudo Riemannian and that the metric tensor of spacetime is symmetric. At such conditions it is impossible to have redshift of photons that move along closed loops in stationary space (Hubble type redshift). Therefore it has been considered an established fact that the Hubble redshift is a result of the expansion of the universe and that in a stationary universe there wouldn't be any Hubble type redshift.

But it has been overlooked that the principle of conservation of energy implies the existence of dynamical friction of photons which would cause Hubble redshift anyway. Therefore the metric tensor of spacetime couldn't be symmetric. Einstein realized this only in 1950 when he proposed a non symmetric metric tensor for the spacetime.

In the meantime it was the 1929 line of reasoning of Fritz Zwicky (an astronomer) who maintained that because of asymmetry of gravitational interaction between photons and the universe there must be a Hubble type redshift in any light. It was called tired light effect but ignored in favor of the expansion of the universe by the gravity physicists for whom it was too exotic an effect not fitting in any way the general realtivity since they strongly believed, against reason, in Riemannian geometry of spacetime with its symmetric metric tensor.

Zwicky didn't know how to calculate the redshift properly and so he didn't get results that he could use to convince the opponents of tired light effect. It was the common problem of many astronomers and astrophysicists who apparently tried to do calculations in Newtonian, approximate (and illegal) way to get the tired light effect. The first one who calculated the redshift of photons rigorously, not using any Newtonian approximations and so he got in 1985 the right result (after many years of trying to do the same approximate Newtonian calculations) seems to be this author. But he was a sculptor, therefore with zero credibility, so no one wanted even to see his result (except referees).

The referees didn't find any formal problems with the result but none of them had any advanced knowledge on gravitation and so all recommended the rejection of the solution for the reason of not proposing any new physics (which they thought is necessary to solve such a profound problem that even they didn't understand). So the first results solving the problem exactly along Einstein's lines of reasoning were rejected by Nature, Physical Review Letters, Science, The Astronomical Journal, and even Nuovo Cimento, not to mention many popular science journals and Physics Today.

Zwicky's idea was not even mentioned in over 1200 page "Gravitation" by Misner, Thorne, and Wheeler, supporters of the expanding space hypothesis, and of the non existing cosmological constant, who being gravity physicists might have not even known about dynamical friction, and maybe that's why they had never calculated its value for photons assuming zero value as best fiting their purpose. If they had calulated the value of dynamical friction of photos there wouldn't have ever been a need to assume that the universe is expanding and that energy can be created from nothing, not even to mention that the true nature of quasars could be discovered decades earlier to satisfaction of H. Arp (an astronomer) and J. Narlikar (a pure mathematician) who seem to be the most informed opponents of the expanding space hypothesis.

The plot thickened when Arthur Eddington suggested in 1929 that according to the general relativity (as he understood it) Einstein's universe is unstable with respect to the small fluctuations of radius of curvature of space and so the universe has to either expand or contract. It is an analog of suggestion by some gravity physicists that the orbits of planets are unstable with respect to small fluctuations of their radii since centrifugal force increases with radius and therefore leads to even greater increase of the radius (which would be true if other factors, like conservation of angular momentum, didn't take part in this phenomenon).

So Eddington's suggestion may be ignored as long as all the factors taking part in the stability of the universe are not taken under consideration. Besides, it is risky to tell how a system of 1011 bodies is going to behave if we don't know yet how to predict analyticaly the bahavior of three bodies.

However, most gravity physicists lead by Gamov and later by Wheeler embraced the idea (baptized "Big Bang" by Hoyle) that was proposed in 1931 by George LeMaitre, a priest of Jesuit background, as the only possible way of creation of the universe by God.

Since the Hubble redshift was already included in the so called Big Bang general relativity as a result of expansion of space, and the expansion of space became the basis of the Big Bang GR, the conservation of energy had to be dropped to avoid the contradiction within the theory and the dynamical friction of photons had been assumed exactly zero to the bewilderment of those astronomers who still believed that energy is conserved.

In the Big Bang GR the contradiction between expansion of the universe and the conservation of energy had been decided by the theorists against the conservation of energy. The dynamical friction has been assumed to be limited only to Newtonian physics despite that Einsteinian physics as more general should explain all observed Newtonian effects. This limitation of dynamical friction to all particles except photons in the Big Bang GR is an equivalent to an assumption that while all other particles are subject to the principle of conservation of energy and so to the dynamical friction the photons aren't (they are supposed to have zero redshift in a stationary universe), and so, while photons are moving through the universe, carrying energy and modifying the gravitational field, the energy needed to compensate for the dynamical friction of photons is assumed to be created from nothing by the theorists like Wheeler and others. It is a point where divine intervention into the affairs of the universe is to be assumed by the gravity physicists.

It has been tacitly assumed by astrophysicists (in order to understand the gravity physicists) that this energy is so small that assuming that it is created from nothing won't change any observational results [source: Dr. Bohdan Paczynski, astrophysicists].

Actually there exists even a back-of-envelope Newtonian calcultion that convinces astrophysicists that this is really the case. Consequently the amount of this energy has been never calculated, just assumed on the basis of this back-of-envelope calculations to be negligible. Unfortunately for the Big Bang GR it isn't negligible and consequently it is a fatal flaw of this hypothesis. Obviously one has to retun to Einstein's GR with global conservation of energy and consequently with non symmetric metric tensor and possibly assume the Finsler (non Riemanian) geometry of spacetime.

Such situation with the mysterious redshift of galaxies lasted until 1985 when it turned out that if the principle of conservation of energy holds for the universe then its alleged expansion is only an illusion (Jastrzebski, 1985). It turned out that this illusion must be caused by a relativistic effect of general time dilation officially unknown to most gravity physicists due to their lack of interest in such an effect, according to "Phys. Rev. Lett." editors.

Because of this projected lack of interest, the news about general time dilaton were never published neither by "Phys. Rev. Lett." nor by "Nature", "Science", "Nuovo Cimento" (disfunct since then), "The Astrophysical Journal", nor any other scientific journal despite that no formal errors were ever specified by the referees of those scientific journals. The referees objected only to the lack of any new physics in Jastrzebski's paper and him explaining everything through Einstein's general relativity.

Jastrzebski has been explaining many mysterious features of cosmology as simple relativistic effects of Einstein's gravitation, explaining physics of the illusion of accelerating expansion of space, providing calculation of Hubble constant of apparent expansion of space and its (apparent) acceleration (confirmed by observations after 1998 with data from SN Project), predicting the density of the universe, estimated already by astronomers within a fraction of order of magnitude.

The calulations narrowed the uncertainty of the density of space to 8% standard deviation, the same as uncertainty of the Hubble constant since density of space is derived from Hubble constant. Even the average size of pieces of non luminous matter of the universe has been provided. And all was done from first principles, with no adjustable parameters, just from the principle of conservaton of energy and Einstein's general realtivity.

It has been shown that restoring the principle of conservation of energy as a valid physical principle and with it restoring the dynamical friction for photons allows to drop the assumption that the universe is expanding, however not, as it might have been expected, through restoring the Newtonian idea of tired light proposed by Zwicky. It has been done by demonstrating that in a world where energy is conserved the dynamical friction of photons is a relativistic effect of general time dilation. An effect of the rate of time dilation compensating for the curvature of space for the reason of inability of nature to produce energy from nothing.

Restoring the conservation of energy in gravitation might invalidate the (pseudo) Riemannian geometry as a description of geometry of spacetime and might require introduction of more general geometry in which a degenerate non symmetric metric tensor might be possible. Consequently it requires an introduction of the mentioned effect of general time dilation as a valid physical principle being a necessary consequence of the more general principle of conservation of energy.

This way Einstein's theory, by separating itself for good from the magic of expanding space, becomes a physical theory explaining all the controversial or not understood elements of Einsteinian physics.

It shows the location of gravitational energy, and by this the origin of gravitational force as the minus derivative of the internal energy of particle with respect to displacement, (-d/dx)[m c2(x)] (where m is the mass of particle, and c(x) is coordinate speed of light) which makes the gravitational energy the same as the internal energy of the particle. It shows the reasons for the Hubble redshift, and so it explains the illusion of accelerating expansion of space. Possibly it explains also the surprisingly high redshifts of quasars that, as Halton Arp has insisted, are associeted with galaxies of much smaller redshifts and so not even located at the distances assumed by the theorists.

Because of all those things there are several differences between Einstein's GR and Big Bang GR. They are in assumptions about the real world and necessarily in conclusions from these assumptions. These assumptions and conclusions are specified separately in the two tables below:

Differences between two theories

ASSUMPTIONS Big Bang GR Einstein's GR
reason for gravitation geometry of spacetime
conservation of energy invalid valid
speed of light constant throughout the whole space local is constant c, non local (coordinate speed of light) depends on displacement within reference frame: c=c(x)
[see Basics ...]
accelerating expansion of space fact due to dark energy see
conclusions
metric tensor of spacetime expanding, symmetric, Riemannian

CONCLUSIONS Big Bang GR Einstein's GR
accelerating expansion of space see
assumptions 		illusion due to conservation of energy predicting Hubble parameter as H0 = c / RE where c is speed of light and RE is Einstein's radius of curvature of space
[see Appendix 2]
metric tensor of spacetime stationary, non symmetric and degenerate, (non Riemannian)
[see Illusion ...]
reason for Hubble redshift Doppler shift due to recession of galaxies conservation of energy
[see Appendix 2]
reson for CMBR redshifted light from Big Bang absorpsion of redshifted starlight by non luminous matter of universe and re-emission at temperature of termal equlibrium 2.7oK
[see Appendix 3]
reason for high redshift of quasars speed of recession conservation of energy in cluds of dust, Z = exp (r sqrt(4pGr) / c) - 1 where Z is redshift, r is radius of cloud, G is gravitational constant, r is density of cloud, c is speed of light
[see paper (in PDF format) The general time dilaton: relativistic redshift in stationary clouds of dust]
reason for gravitational force "acceleration of space" Fi = - (d/dxi) E0(x), where E0(x) is internal energy of particle depending on position in space (x)
[see Basics ...]
gravitational (potential) energy ? the internal energy of particle E0(x) = m c2(x), where m is mass of particle and c(x) is coordinate speed of light depending on position in space (x)
[see Basics ...]
location of gravitational energy ? location of gravitating particle: x
[see Basics ...]
density of space ? 6x10-27 kg/m3
[see Appendix 2]
acceleration of expansion in terms of dH/dt and H0 ? dH/dt = - 0.5 H02
[see Appendix 2]
acceleration of space probes ? - 7x10-10 m/s2
[see Appendix 2]
average size of pieces of non luminous matter ? order of one meter
[see Appendix 3]
angular size of galaxies as function of distance from observer ? should have minimum at redshift Zo < exp(π / 2) because of the apparent accelerating expansion of space that is closed into a 3-sphere

The areas that a theory has no answers for are marked with question marks. All of them are in the Big Bang GR and it suggests that Einstein's original theory is OK while the Big Bang GR is wrong. Therefore we might assume that gravitational force and energy may be explained accrding to the old Einstein's theory based on conservation of energy and explain them as such. Therefore we seem to be justified in explaining gravitation as it is explained in the following section.

Observational evidence for Einstein's GR as opposed to Big Bang GR

Variable Einstein's GR predictions Observed value σ(sigma) Units
theoretical numerical
Radius of curvature of space RE 4.3 ? ? Gpc
Hubble parameter at Earth Ho = c / RE 69.6* 69.6 2.8 km/s/Mpc
Acceleration of space probes Pioneer 10 and 11 ao = c2 / RE 7 8.7 1.3 10-10 m/s2
Density of the universe
(of "gravitational energy") 		ρ = c2 / (4πGRE2) 6 5.5 4.5 10-27 kg/m3
apparent acceleration of expansion of space in Ho2 dH/dt = 1 / 2 0.5 0.45 ? - Ho2
redshift of minimal angular size of galaxies [verify] Zo < exp(π / 2) < exp(π / 2) 1.6 ? 1
* assumed value
    Notes
  • Most of the observed values are functions of the radius of curvature of space a.k.a. Einstein's radius (RE) that is adjusted so that Ho comes out exactly as observed.
    In the equations for theoretical predictions the unspecified variables are: G is Newtonian gravitational constant, c is speed of light, t is coordinate time.
  • The acceleration of the apparent expansion of space is made a funcion of Ho for easier comparison with the observed value given as function of Ho.
  • Five of the above values are predicted by Einstein's GR. None of the observed values is predicted by the Big Bang GR.

Appendix 1:

Relation between time dilation and gravitational field

In any accelerating system (e.g. an accelerating rocket ship) the time in direction of acceleration a runs faster and slower in the opposite direction. The effect might be understood as photons radiated from places in the system in direction of acceleration gaining energy while they are passing distance dx towards the detector. The increase in energy is
  dE = (dx)a m (1.1)
where m is inertial mass of the photon. Substituting E / c2 for mass of the photon and dividing both sides by E we get relative change in energy of the photon, which is the same as the relative change of its frequency (because of Planck's relation E = hn) or in other words relative change of the time rate at the source of the photon since photon may be considered to be a clock ticking as fast as time runs at the place where it is coming from:
  dE/E = dn/n = dt/t = (dx)a / c2 (1.2)
Expressed as a ratio of proper time (t) to coordinate time t it looks like
  dt/dt = 1 + (dx)a / c2 (1.3)
where x is the distance from the observer to the observed place in space.

This way we have a convinient way of translating time dilation expressed by dt/t in (1.2) into gravitational field g that works in opposite direction to acceleration (g = - a) and gravitational field into time dilation by the principle of equivalence. Whenever we have a certain amount of acceleration a causing certain amount of time dilation we must have also the same amount of time dilation causing acceleration. One translates into the other. Acceleration <=> time dilation (in opposite directions).

To get field in vicinity of mass M we may simply stand on the shoulders of a giant and apply Newton's equation since it is accurate enough for our purpose:
  g = GM r / r3 (1.4)
where G is Newtonian gravitational constant, G = 6.673x10-11m3/kgs2, and r is the displacement from the point under consideration to the center of gravity of mass M.

Appendix 2:

Derivation of Hubble constant of Einstein's Universe

In the simplest case of homogeneous space the ralation between proper time in deep space and the coordinate time of the observer turns out to be dτ/dt = exp(-r / R), where τ is proper time in deep place, t is coordinate time at observer, r is distance from the observer to the observed place in deep space, and R a constant that turns out to be equal to "Einstein's radius" (RE) and so the "Hubble constant" of "Einstein's Universe" is H0 = c / RE where c is speed of light. A simple derivation of Hubble constant of Einstein's Universe, from first principles (like e.g. the principle of conservation of energy), in only 6 lines of high school calculus, is presented in this document.

The obvious application of this effect is the calculation of density of space of our universe from the value of observed Hubble constant. The value of Hubble constant 70 km/s/Mpc implies density of space 6x10-27 kg/m3 with the same accuracy as the Hubble constant is determined. Another application might be calculations of densities and sizes of clouds of dust, or galaxies, that quasars are located in from the redshifts of those quasars.

Appendix 3:

Average size of pieces of non luminous matter of universe

Assuming that the spectral distribution of energy radiated by a star may be presented, with an accuracy to the absorption lines of its atmosphere, by equation

I_o_(nu) = c_1 nu^3 / [exp(c_2 nu / T_s) - 1] (3.1)

where c1 and c2 are constants and Ts is the temperature of the star's surface (with the peak value at n = 2.82 Ts / c2), according to ν(r) = ν(0) exp(- r / R) and (3.1) the distribution at distance r from the source is

I(nu, r) = (c_1[nu exp(r/R)]^3)/(exp[c_2 nu exp(r/R) / Ts] - 1) (3.2)

therefore for any observer, the spectral distribution of radiation from all the stars is

I(nu)=c_3 Integ 0..oo ((p(r)[nu exp(r/R)]^3)/(exp[c_2 nu exp(r/R)/T_s]-1))dr (3.3)

where c3 is a constant and p(r) is probability of light passing distance r without hitting any obstacle on its way, which is

p(r) = exp(- r A / L^3) (3.4)
where A is the average area of an obstacle and L is the average distance between obstacles, assuming that rA << L3. Combining (3.3) and (3.4) and making substitution z = c2 n/Ts, x = z exp(r / R), and a = AR / L3 one gets the spectral density of the radiation from all luminous sources as

I(nu) = c_4 z^a Integer from 0 to infinity of (x^(2-a)/(exp(x) - 1)) dx (3.5)
where c4 is a constant. It is visible from (3.5) that this distribution is not a black body distribution, and therefore the background radiation is not just the redshifted starlight. Therefore the background radiation must be a radiation from the non-luminous matter of the universe, matter that is in thermal equilibrium with the redshifted starlight. For a << 1 the peak value of this distribution represented by (3.5) is at z = 1.55 a1/2 and therefore the temperature of a black body having the peak value of its distribution of radiated energy at the same frequency is

T = 0.55 T_s A R / L^3 (3.6)

The average distance between the obstacles L may be determined from the relation r L3 = ro D3 where r is as before the density of the universe, ro is the density of the obstacle, and D is the diameter of the obstacle (assuming that the obstacles are roughly spherical objects for which approximately A = D2, and that almost the whole matter of the universe is composed of such obstacles). R and r can be determined from H = c / R = sqrt(4 π G ρ). After all the substitutions the average diameter of the obstacle is

D = 0.04 c H T_s / (G T rho_o) (3.7)

Assuming value of Hubble constant H = 10-18 1/s, average temperature of stars Ts = 104 K, temperature of thermal equilibrium of the universe 2.7 K, and the density of the matter of obstacles ρo = 103 kg/m3 (H2O), the average diameter of the obstacle is of order of 2 m. It is large enough size to make the non luminous matter of the universe responsible for the absorption of light in the millimeter wavelength range.

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