suspected evildoer
  The Einsteinian Gravitation
For Poets and Science Teachers
by W. Jim Jastrzebski
Rev. 1.33
Our standard charm for diverting evil spells of evildoers. Click refresh when it stops diverting.
Note   For readers who are interested in physics only and want to skip sociology and politics of gravitation there is a thread through this article marked by skip the sociology links. You may also select from the following list only those sections that interest you. Each section ends with a back to top link that brings you back here.


Understanding the nature: science, religion, magic

To understand the real world it is necessary to ask a lot the same single question: how do we know that it is true? It is the most important question in science. If we don't know the answer to that question about something then our knowledge about this something is not scientific knowledge. It might be true knowledge as e.g. knowledge about facts usually is (e.g. "it's raining"), which is then called trivial knowledge. If it is not trivial (e.g. "the earth is round") and we don't know how we know that, this knowledge is not scientific. It is then a kind of religious knowledge, based on faith that what has been revealed to us by other people is true. This is only a religious knowledge even if those other people are scientists (skip the sociology).

Some kind of logical reasoning often supports such religious knowledge. The logical reasoning proved to be insufficient to rely on it exclusively and so, perhaps surprisingly to many, is not used in science to prove that something is true. It is not even used in religion, which relies on revelation rather than logic since theologians know that logic can't be used to support any religion. Religion can be supported only by faith. The reason that logic is insufficient is that it may be as easily applied to true assumptions as to false ones, and that we rarely know what is the case at the moment.

Logic is however used a lot in magical thinking and so by demagogues as a tool to convince people about legitimacy of demagogue's opinions. It is because humans are easily convinced by logic, very trustful towards charismatic figures, and very careless about the assumptions on which the seemingly logical reasoning is based.

This type of logical reasoning used by demagogues (magical thinking) we see around us every day unless we live far enough from any civilization, since humans, being social animals have this basic need to trust other humans. They trust especially their leaders and often those whom they consider wiser than themselves. This makes them susceptible to believe in any kind of nonsense and susceptible to being fooled by the folks on the right side of the IQ bell curve (but not far enough to know that fooling others is immoral).

Knowing all the above the scientists use logic very carefully not to fool themselves. Even despite that, from time to time they come upon a couple of ways of logical reasoning that seem to be perfectly legitimate but give opposite results. It is called a paradox then. A paradox always indicates that there is something waiting to be discovered, because there must be at least one wrong assumption in those various ways of logical reasoning. So through paradoxes we learn that there is something in our system of beliefs that is not true.

A widely known paradox with gravitation is one with a question whether an omnipotent God can create something so heavy that he couldn't lift. If we can't answer that question it means that something in our system of beliefs is not true. A simple solution of this paradox is that there is no omnipotent God. Yet only about 15% of humans accepts this solution. Remaining 85% prefers to believe that there is an omnipotent God who can arrange many things for them so they don't need to do them themselves. Luckily we don't need to worry about this paradox any more since it was already solved by George Hammond, and yes, there is a God (kind of).

Another famous paradox, partly about gravitation as we see later, is twin paradox, in which each twin should be younger from point of view of the other since both apply the same logic. The logic is that if something moves in relation to something else the time at this moving something runs slower in relation to the other, which is a statement of special theory of relativity. Then we see two twins moving in relation to each other which happens a lot in the real world. The question arises which twin is younger when they meet again?

Some people think that this is impossible that one will be younger than the other when they meet because if one moves in relation to the other the other moves exactly in the same way in relation to the first one. So there is no way that one becomes younger and the relativity must be wrong. But the statement of relativity about time running slower in a moving system is confirmed by experiments, so it is a trivial truth, a fact, and it is not logical at all to argue with facts. So there must be something else that is not true in the beliefs of those who see it as paradox. The paradox is solved (the false assumption discovered) by realizing that while absolute velocity has no physical meaning (and therefore does not exist), the acceleration, which used to be imagined to be just the rate of change of velocity, does have physical meaning. So the twin who accelerates while the other does not turns out to be younger.

It is still kind of strange to people, who think that acceleration is just the rate of change of velocity. How the rate of change of something that does not exist may have physical meaning? For those people, who by the way are thinking more rigorously than others (those others that are satisfied with the presented solution), this solution creates another paradox, and that's why the twin paradox baffles generations of humans for almost a century. Later in this article we'll see how this second paradox prompted Einstein to discover his theory of gravitation, and so how the second paradox (of physical existence of rate of change of something that does not exist) is solved.

For American students, who might read this article but who because of peculiarities of our educational system might have never heard about Albert Einstein, I should mention that he was the most prominent American scientist ever. He was known and appreciated around the world almost as much as Leonardo da Vinci who happened to be the most prominent not only a scientist but also a painter, sculptor, architect, and engineer. I might also add for the students who never heard about Leonardo neither that he was not an American but an Italian. He never even visited the US and if he liked to he wouldn't probably get the visa for the fear that he might have worked here illegally in one of his various professions. He couldn't be employed legally because he would be overqualified for any position in the US as even much less gifted people often are and then have to live at the expense of the US taxpayers. But let's not deviate into peculiarities of American culture, which is also an interesting subject but not of this article.

Science considers that something is true only if it can be reduced to trivial truths, the facts, through a type of logic that is called mathematical logic (math for short). And that's why there is so much math in science and why without understanding math it is often difficult to understand how the real world works. Being difficult to understand it is however not impossible and in many cases even quite easy as the author is trying to show below.

So since the only way to really know that something is true is to know how do we know that it is true, in the rest of this article the author is trying to answer that question in relation to Einsteinian Gravitation. He is trying to show how we know that it is true and how we know that Newtonian Gravitation is not.

A helpful thing in understanding the world is to know that the big part of what is called science is not really science but magic. The spells operating on fictitious things, producing right results but for wrong reasons. E.g. the Newtonian theory of gravitation (at least the brand with gravitational attraction, which was not necessarily proposed by Newton who turned to be wiser than that and didn't see any reason to assume that gravitational attraction existed) has been considered one of the truest theories around. It has been thought that it explained the mechanism of all gravitational phenomena as they really are. Yet it turned out that its right results were explained by wrong reasons. So it was pure magic.

The reasons for those right results became known after Einstein discovered his theory of gravitation and then those discovered reasons turned out to be not the same reasons that were assumed in Newtonian theory as ones that produced those results. That's how we have learned that Newtonian Gravitation was magic.

Before Einstein the Newtonian Gravitation was thought to be a physical theory, as is called a theory that delivers a true physical description of the mechanisms that are responsible for the phenomenon. If the theory does not describe physics but it presents only a mathematical description of the phenomenon it is called a mathematical theory (or phenomenological theory, a theory that just describes the phenomenon without explaining the reasons for it). After Einstein had discovered his theory the Newtonian theory of gravitation changed from physical theory back into a mathematical one (as Newton meant it to be), unfortunately into a false mathematical theory. What it says is only approximately true.

Many present theories in physics are just mathematical theories and nobody seems to have any idea why they work so well as they do. It of course creates an opportunity for many new discoveries in the future when physical theories are going to explain why all those mathematical theories work so well.

Some theorists of science (sometimes called Copenhagen School) maintain that it will never happen because God (see Hammond 1999) apparently does not want us to know too much and therefore learning certain things about the nature is impossible according to those gentlemen. They give quantum Mechanics as an example, because there is so called EPR paradox in Quantum Mechanics (invented by Einstein, Podolsky, and Rosen) that seemingly nobody knows how to solve and so the School tends to think that the solution of EPR paradox is impossible. This is by the way how humans always thought about all the other paradoxes and all of them were solved, sometimes after hundreds of years, after humans learned things that they hadn't known before. So we might have just another chance to learn something from EPR paradox as well. The paradox is unfortunately too complicated to describe it here but its description can be easily found on the internet.

The EPR paradox presents also a chance for those who communicate with God directly to ask God what is the solution of that paradox. But this is a slippery subject with a potential of offending feelings of a lot of religious folks. Especially when those folks believe that there are really three Gods and all of them should be consulted on the subject, so none would be revealing to humans the truth behind the backs of the others. Or those for whom number of Gods is not determined exactly yet and so consulting all of Them wouldn't be practical. Needless to say that offending anyone is not the author's purpose. The purpose is to promote the truth about Einsteinian Gravitation, so let's leave the religions alone, and return to the safer subject of magic that seems to be non offensive to anybody since magic is always what other people are engaged in.

A mathematical theory is of course the same thing as magical spell that has been used by magicians in pre-scientific times to produce desired results. The ancient magicians, similarly as physicists that were using Newtonian Gravitation a century ago, didn't have any idea why their spells worked. When an ancient magician pointed at the right time to the eastern part of the horizon uttering three times "abracadabra" the sun started rising from below the horizon (the horizon was a straight line then because the earth was flat then). The spell worked every time. The formula that three "abracadabras" at right time equal sunrise was tested and worked every day. It is the same with calculation of a trajectory of a missile using Newtonian formula for the nonexistent, as we know now, gravitational attractive force between the missile and the earth. The Newtonian formula works every time and only those who understand Einsteinian Gravitation know why. A group that might soon include the reader (if he/she does not fall asleep before the author gets to the point).

Both, the ancient magician and contemporary ballistic officer, might have their ideas (theories) why their respective spells worked. At least the ballistic officer had a wrong idea if he believed that it was because the earth attracted the missile. He was applying magic in his calculations. It might have been much less naive and much better tested magic than one used by the ancient magician, but it was the same type of action. Applying mathematical formulas (spells in older terminology) without knowing why they work. Many people call those formulas science. I'll try to be more rigorous and call it magic.

The most prominent other magical theory in contemporary physics is the mentioned already Quantum Mechanics. We can use it to predict things with astonishing accuracy yet we don't seem to know why all those predictions come out right. It is very frustrating situation for physicists and so many of them change their profession and become e.g. financial specialists to utilize in the financial world their allegedly vast knowledge of mathematics of the real world. Apparently they conclude that if they are to work as magicians anyway let it bring them more money. So we may observe less and less people interested in science and more and more in making money which might easily result in new dark ages.

The author hopes to contribute to reversing that trend by showing that it is not that difficult to understand the nature even if someone is allergic to math like e.g. some artists are. The example of Leonardo may show us that to be a great artist one has to understand the nature. In fact only the understanding makes one a great artist and it is impossible to do any art really well without understanding its subject at least to some degree.

Coming back to the subject and to make a more graphic comparison between physical and mathematical theories we might imagine a physics of any phenomenon as a snail, and math of it as the snail's shell. The shell is a rigid thing around the snail, which fits the snail and tells us something about his/her properties, but not much, if anything, about the reasons for them. The shell may be also completely empty and we wouldn't know about it if we didn't look inside. So let's try to look inside the Newtonian Gravitation to see if there is any physics in it.

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Alleged Gravitational Attraction

People who are not familiar with Einstein's discoveries about gravitation tend to think that the reason for gravitational phenomena has been the universal gravitational attraction. It has been thought to be one of the four fundamental forces of nature. Indeed it looks like the objects in the universe attract each other. Yet they look like that only if we don't look too closely.

It is like with all magic tricks. If we observe everything that makes the trick it turns out that really it is not what it looks like it is. It turns out that it is really something else.

The same thing happens with the alleged gravitational attraction. If we measure everything with high accuracy it turns out that there is no way of fitting the gravitational attraction into the picture. The gravitational attraction is a fake explanation for what's going on. The objects in the universe just happen to move in a fashion that simulates attraction but there is no force that forces them to do that. It is just their natural free movement. The explanation why this free movement is so strange is the subject of Einstein's theory of gravitation.

In times of Newton such free movement in absence of forces was thought to be possible only with constant velocity (meaning with constant speed along a straight line). It has been thought that any other movement requires a force to make it different than movement with constant velocity. Einstein proposed, and precise measurements confirmed what he proposed, that all objects when left alone move not along straight lines in space but along straight lines in certain more general "super space" that is composed of space and time together and called spacetime for short. It is like we call a man and a woman who live together a couple rather than "a man and a woman living together" because "couple" tells us more about their relation. Many of their actions depend on each other (e.g. they don't have any more that much of a freedom of choice as to whom they marry; they may be even already married to each other). It is similar with the spacetime in which behavior of space depends on behavior of time and vice versa. As if space had been married to time. Something that was not known in times of Newton when it was thought that time and space don't form a couple in which behavior of one is related to behavior of other in any way. In Newtonian theory the time and the space are totally independent from each other. At the times of Newton's it was not known that the time at some circumstances may run slower and in other faster. Einstein knew this already (especially after discovering his earlier theory called special relativity) and applied it to gravitation. It turned out that in places where the time runs faster (more time) the space shrinks a little bit (less space) and v.v. As if the nature tried to keep the same the total amount of both all the time.

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Alleged Expansion of the Universe

One of the results of that mutual dependence between time and space is that space looks as if it were expanding. At this point the author feels that it is proper to warn the reader that it is not the opinion of the majority of cosmologists or even astrophysicists. If the reader rather believes the majority she shouldn't believe the author. The author does not believe the majority for the reason that he discovered a very neat "proof" that the universe is not expanding. The proof wasn't falsified yet, which makes it still valid leaving the author with no choice but to believe that the universe is not expanding and that the curvature of space is the reason for the illusion of expansion.

This illusion of expansion can be roughly explained by the curving of the space of our universe by the presence of matter in it. The relation between rate of time and the curvature of space, which is such that the more space there is the slower the time runs (and as we know the presence of matter increases the amount of space), makes the time running slower the farther we look.

It simulates the expansion of the universe because when distance between us and something else increases then it looks as if time in this something else was running slower than our time. It is called Doppler effect and it comes from the fact that as the distance between this something and us grows the signal from this something (e.g. the light that it emits) needs more and more time to reach us. This additional time that the signal needs to reach us causes that we see events at this something (e.g. in a distant galaxy) happening with certain delay and so as if they were happening slower. So the slowing of time for any reason, looks exactly as increasing distance.

This effect of slowing time is observed in the universe in the amount predicted by Einstein's theory of gravitation but despite that the astrophysicists don't use Einstein's theory to explain this slowing of time as being real. They use Doppler effect with which they are much more familiar to explain that slowing of time as apparent, and so they interpret it as an indication of increasing distance between the galaxies in the universe and us. So they think that the universe is expanding. It leads them to rather silly results but they don't mind. Which of course might be interesting to know why they don't mind.

       (if you want to follow physics only skip the sociology).

The reasons for many astrophysicists ignoring Einstein's Gravitation are too complex to describe them fully in this article, and they are not always related to gravitation. So let me present only a short and true as far as I know explanation why it happens.

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Why astrophysicists act silly

The short explanation is that when Einstein discovered how gravitation works he described the physics of it with a simplest equation that he could come upon, called Einstein's Field Equation. The general form of this equation is

         Rik - (R/2) gik + /\ gik + Tik = 0

It is not necessary to understand the equation in order to understand Einstein's Gravitation. For the really curious I might just say that in this equation all the terms with subscripts ik, are four-dimensional second rank tensors. R and /\ (Greek letter lambda) are single numbers (so called "scalars"). /\ is a number called cosmological constant, which value is assumed by cosmologists to match their pet theories since various values imply various behaviors of the universe. Those tensors might be imagined as four by four tables (and so each is composed of 16 single numbers, since 4 x 4 = 16). Those tables specify how properties of time, space, energy, and stress, change along all four directions (three in space and one in time). The equation is shown here only to show how simple Einstein's Field Equation is and where is the problem with astrophysicists.

In this equation the shape of the spacetime is described by gik, so called metric tensor of the spacetime, which specifies how properties of time and space change along all four directions.

The simplest of all metric tensors is symmetric tensor. It is the simplest since being symmetric it has the smallest amount of independent components. It is like any symmetric shape has to be the same on one side as it is on the other so half of it is determined already by the first half so only half can be independent.

It turned out that symmetric metric tensor sufficed to describe all the gravitational phenomena that were known while Einstein discovered his Field Equation. It was not known yet then that the universe looks as if it were expanding so Einstein assumed symmetric metric tensor in his equation. Mathematically the symmetry of a tensor is expressed by gik = gki which is a symbolic representation of the fact that the number in i-row and k-column of the 4 x 4 table that makes a tensor is the same as the number in k-row and i-column (the first subscript denotes row, the second denotes column).

It turns out that to apply Einstein's description of gravitation (his Field Equation) to explain in a sensible way that doesn't contradict any other physics, the illusion of the expanding universe it is necessary to assume that metric tensor is non-symmetric. Such tensor has more independent components, which is enough to provide for the illusion of expansion.

In 1950 Einstein changed his opinion about symetry of metric tensor. In his article "On the Generalized Theory of Gravitation" he maintained that a non-symmetric metric tensor has to be assumed for gik in his equation. A quote from Einstein's article: "The answer on which the theory under discussion is based is that the symmetrical tensor field must be replaced by a non-symmetrical one. This means that the condition gik = gki for the field components must be dropped". It was for different reason than expansion, but the fact that he proposed a non-symmetric metric tensor for his equation remains.

This non-symmetry of metric tensor provides for an effect not yet discovered by astrophysicist, which might be called general time dilation. Except explaining the illusion of the expanding space this effect is also needed to satisfy the conservation of energy while there are any moving objects in the universe. Without this so called general time dilation any gravitational interaction with any moving object could be used to create energy from nothing in a similar way that energy is created in the tidal power plant, by the earth's oceans moving in relation to the moon. In case of the earth-moon system the part of rotational kinetic energy of the earth is converted into electricity in a tidal power plant, and part is transferred to the moon making it flying higher and higher.

So in general the effect puts a small quasi Newtonian drag on any moving object in the universe. "Quasi Newtonian" since this would be a Newtonian counterpart of this relativistic effect that is observed in the real world as reddening of light and called cosmological redshift or more properly Hubble's redshift.

Since the Newtonian Gravitation relates to the same thing as Einsteinian Gravitation (to gravitational effects), each Newtonian effect has to have relativistic counterpart however not vice versa because Einsteinian theory is more general and so it explains more. If we apply that drag to particles of light (photons) it turns out that they really look as if they encounter such "drag" in space since we observe an apparent loss of their energy (their reddening that is producing the mentioned cosmological redshift). This is the effect that is taken by astrophysicists for the simplistic Doppler effect, and then for the evidence that the universe is expanding. It is "apparent loss of energy" since we should be aware that physically this effect is caused only by the time running slower the farther we look as required by relation between time and space in the spacetime. By non-symmetric (and possibly also non-Riemannian, to be more detailed) metric tensor that describes that relation.

The effect has been also observed in the behavior of space probes Pioneer 10 and 11 but since astropysicists still think that metric tensor is symmetric they can't explain the behavior of these space probes and call the effect "anomalous".

It is basically all that there is to the allegedly great mystery why the universe looks as if it were expanding that the astrophysicists can't explain for about three quarters of a century. The math explaining details of that phenomenon one may find on different pages in this web site. Because of simplicity of Einsteinian Gravitation the knowledge of high school math is sufficient to understand this math. Everyone who knows high school calculus can calculate the value of Hubble's constant for a non expanding universe (that astrophysicists consider the constant of expansion) and have the fun of noting that this is exactly the value that is observed in our universe.

If this is so simple why astrophysicists haven't adopted that solution?

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Problems with Science

When it turned out that non-symmetric metric does the trick it was too late. All cosmologists, a.k.a. gravity phisicists (as they call themselves) who adopted Einstein's equations got so much attached to symmetric metric tensor that they didn't want to change anything and to learn more complicated math. Instead they declared that the universe is really expanding, and that there is no possibility that it is not since the symmetric metric tensor, which they believe is the right solution (against Einstein's opinion), requires the universe to expand, and so they don't need even to discuss it.

The Einstein's idea of non-symmetric metric tensor of the spacetime has been dismissed without any investigation of its value and the Einstein's theory of gravitation has been converted into a mathematical theory, in which, because of symmetry of metric tensor, the principle of conservation of energy is not valid (while all the evidence so far says that it is valid suggesting a serious problem with contemporary gravitation physics).

gravity phisicists maintain (without any proof) that energy gets created from nothing and than destroyed to nothing practically every time when something moves. And they also refuse to discuss the issue. E.g. suggestions that maybe the principle of conservation of energy is valid after all, are not allowed on internet in moderated science newsgroups that are controlled by gravity phisicists (as e.g. sci.physics.research).

The gravity phisicists still don't want to discuss the issue after many decades that passed since those times. They became probably the only scientists in history of science who don't like to discuss their subject and when asked about it by a physicist they become immediately irritated and aggressive. This was what Prof. Roy Glauber, physics professor from Harvard University, Cambridge, Massachusetts, told the author when he asked him whether he knew any gravity phisicist who would be interested in discussing gravitation since the author couldn't find one himself. Apparently there is none. Prof. Baez (who calls himself a gravity phisicist), a moderator in sci.physics.research news group on internet said to the author in response to a question about conservation of energy (in particular about the source of kinetic energy of a free falling object):

"It is always surprising when it happens, but sometimes to learn more about the world we must stop asking certain questions...
... namely, those based on false assumptions."

Almost all scientific theories are based on false assumptions. A lot of questions were asked based on those false assumptions, and yet the humanity progressed from the caves to the trips to the moon. E.g. the whole astrophysics has been built on a false assumption of existence of "gravitational attraction" and yet it lead to the understanding of the universe that we have now which seems much better than it was only hundreds years ago when people thought that the earth is flat and located at the center of the universe. If humanity followed Prof. Baez's advice we would be still sitting in caves wondering whether a question if stones should be split isn't accidentally based on a false assumption.

Yet gravity phisicists need some rationalizations for blocking questions about their ideas from public forums, not being able to defend those ideas with sensible arguments. The above quote shows how gravity phisicists rationalize blocking questions about viability of their ideas. They declare that those questions are based on false assumptions and therefore (contrary to the history of science) not leading to understanding of the world. It's another example of replacing science with magic and replacing arguments with opinions of men who allegedly know which assumptions are false. It is also the way in which political systems get converted from democracy to fascism as it happened e.g. in Russia, Italy, and Germany, where the men who allegedly knew all the truth, and therefore knew who should be killed and with whom to start a war to save humanity from villains, were respectively Lenin, Mussolini, and Hitler.

The gravity phisicists prevent publishing papers on the subject in scientific journals in which only they advise editors on which paper fits the print (calling printing of inconvenient papers the waste of valuable journal space, and of course who is better to evaluate paper on gravitation than a "gravity phisicist", right?). They prevent observations that might demonstrate that the universe is not expanding (calling them the waste of valuable telescope time) because they decide who is going to get telescope time.

They have enough power to do all that since as it turns out many VIPs in science administration are gravity phisicists. It is not accidental. It is because those physicists who are interested in discovering how the nature works are not interested in gravitation where there are no experiments and everything is already explained by Einstein's theory. So all the nonsense in gravitation physics that happens today "is not that the subject is hard; it is that good men are occupied elsewhere. Remind me not to come to any more gravitation conferences!" (Richard P. Feynman, What Do You Care What Other People Think, fragment of a letter to his wife from a gravitation conference).

So gravitation is the field of choice for administrators of science that makes them free from any hassle of scientific work. Nobody bothers those scientists with expectation of some new achievements in this area, and cares if they produce any scientific work. Their whole time may be dedicated to traveling to various scientific conferences around the world at tax payer's expense and administrating the flow of those billions of tax dollars that government spends on science.

This might be a good thing that those scientists don't need to do any scientific work and may concentrate on administration but it has also some bad effects. To protect their access to large sum of money they tend to think that they have to make an impression that they are real scientists who also contribute to science. They feel pressed to make an impression that they posses vast knowledge of physics and mathematics and also enormous intelligence, which might be not the case in some cases, and not even needed for administrative functions, as proved by about all our Presidents. But people who appoint those scientists to those high positions might not know that. So this explains why gravity phisicists are so reluctant to talk about their science. It wouldn't be politically right to make fools out of themselves.

Those people are forced therefore to try to make Einsteinian Gravitation look complicated enough to discourage anyone from trying to learn it. For that reason generations of astrophysicists (the only physicists that really need to know exactly how gravitation works), are taught only Einsteinian Gravitation with symmetric gik tensor. Or even just Newtonian gravitation with correction for curvature of space (called Post-Newtonian Approximation). It leaves many of them completely confused about gravitation, especially Einsteinian, and with a firm belief that it can't be understood. By the way, that it can't be understood, was also the impression of all the teachers who taught Einsteinian Gravitation with whom the author had a chance to talk. None of them claimed understanding it, they "just taught it", as they all said. So astrophysicists and astronomers in general are the primary victims of gravitation physics, which otherwise would be totally harmless aberration in human culture.

The interesting part of it might be that possibility of non-symmetric gik tensor is not even mentioned in any of the textbooks on gravitation physics. Neither is Einstein's proposition about it. Even in detailed bibliographies of the subject. E.g. it is not mentioned in the over 1200 pages "Gravitation" by Misner, Thorne, and Wheeler. The result is that contemporary astronomers don't know that Einsteinian Gravitation explains the expansion of the universe as an illusion. Neither that it requires the space to be a kind of viscous medium, which is purely relativistic (non Newtonian) feature of gravitation because Newtonian gravitational field is a conservative field and so no energy could be lost through any kind of viscosity if the world were Newtonian. And this viscosity is impossible to come upon using only math with symmetric metric tensor.

So gravity phisicists, to protect their comfortable position in science, which in itself is not much different what people in other professions do, unfortunately also mislead astronomers and astrophysicists into believing that the universe is really expanding. And they in turn mislead us.

The gravity phisicists also declared that Einstein's theory is the math that they are using (one with symmetric gik tensor), and not the math that Einstein recommended in 1950 (with non-symmetric gik tensor). In this way they made out of Einstein's physical theory, which is able to explain much more than even is presently discovered, a mathematical theory that is not even able to explain the illusion of expansion of the universe.

The phenomenon of viscosity of space is not a problem in contemporary astrophysics since there are no observed phenomena, except the illusion of expansion that would need it as an explanation. For the time being it is just a theoretical phenomenon predicted by Einsteinian physics (now different from it's official math; the snail proved to be bigger than her shell). It will show up itself only when spaceships sent to distant planetary systems start missing their targets because astrophysicists didn't know that Einsteinian physics predicts slight deceleration of those spaceships (about 10-9 m/s2, ten billion times weaker then the earth's "gravitational field"). We are still many years or maybe centuries form those events that would require more careful application of Einsteinian physics. It is however nice that what Einstein discovered almost a century ago has such far-reaching implications. In fact it might be even valid forever as are discoveries of Ancient Greeks. To bad that those implications are kept from being published in scientific journals.

I hope it all explains why astrophysicists act silly so now let's return to gravitation. The interested readers who want to learn more about sociology and politics of science might want to read "Quasars, Redshifts, and Controversies" by Halton Arp. Arp has been a famous astronomer who was fired from his position in the US for making unauthorized observations of quasars that contradicted some opinions of gravity phisicists about the universe. He had to publish his own book to publish those observations since scientific journals don't publish stuff that contradicts beliefs of VIPs.

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The difference between Einsteinian and Newtonian theories of gravitation

The straight lines that all the freely moving objects (ones on which no forces act) take in spacetime are called geodesics in spacetime. It happens that the projections of those spacetime geodesics on space, the paths of freely moving objects that we observe in space, are in general curved because of certain properties of time and space (or certain geometric properties of spacetime).

The properties of time explained exactly why Newtonian theory reflected so well the behavior of the real world. The idea of attractive force produced exactly the same results as those properties of time produce. Since the hypothetical gravitational attraction fits only the properties of time but not the properties of space (its curvature), there were tiny differences between the Newtonian theory and the real world due to the curvature of space of the real world.

Now we know that there is no gravitational attraction, and that those properties of spacetime are all that is needed to cause all the gravitational phenomena that we observe. The understanding of gravitation has been simplified. And now, while starting with those properties of the spacetime, we see how gravitation works. Now we are sure we have a physical theory of gravitation because we are able to answer how do we know, by going from the observed phenomena like falling objects or planets revolving around the sun (trivial truths) to the measurable properties of space and time (other trivial truths). There are no unknown things left in between.

We may still not know why the spacetime has those properties that we may measure and explain gravitational phenomena with, but this part does not belong to the theory of gravitation. The same as explanation why gases are composed of particles is not the subject of corpuscular theory of gases. Once we know that gases are composed of particles, and of what kinds of particles, we can have a physical theory that explains the behavior of gases. It does not explain nor needs to explain the origins of particles. This is the subject of different theory which may be still a mathematical theory (or magic).

The same happens with gravitation. We know how properties of spacetime cause gravitational phenomena, but we don't know why spacetime has those properties. Why the energy and stress generate those properties as Einstein's Field Equation describes them. We don't even know if it is really energy and stress that cause this behavior of space and time. We might be wrong as physicists before us were wrong about the existence of gravitational attraction. This is something for future generations to discover.

At present time there is much more about physics that we don't know than that we know. We are just at the beginning of discovering how the nature works and even not all the humans believe that it is all happening for natural, identifiable, reasons. In our country alone 97% of the population still believes that there are supernatural forces acting behind the scene. There is a long way to some meaningful civilization in this country alone. So we should be rather glad that at least we know that much as we do and be grateful to Einstein for his work.

       (if you want to follow physics only skip the sociology).

Einstein's theory explains how those properties of space and time generate gravitational phenomena and provides an exact description how they do it. But before we get to that let's take a look at the old theory of gravitation (Newtonian theory) to see how easy it is to make a mistake even in exact sciences when one is not careful enough. And those even very cautious exact scientists make those mistakes, and ones that are not cautious enough even subscribe to quite foolish ideas.

One of such spectacularly foolish ideas of our time is big bang hypothesis. It is a hypothesis that the whole universe, all its matter that is now contained in about trillion galaxies, was created out of nothing (allegedly it was "the beginning") in one instant about 10 to 20 billion years ago (opinions are divided) as a tiny dot, trillions times smaller than a dot above this i, and that it is expanding ever since.

The main reason for that hypothesis is the fact that, as it was mentioned above, the universe looks as if it were expanding. If we assume that it is really expanding then by imagining what happened before our time one may conclude that the whole universe was smaller and then before that even smaller. Until finally it had to start from that small dot and before that from nothing. This is an example of how through quite logical reasoning one can get a silly result when one is reasoning logically but some of assumptions are wrong.

The initial wrong assumption in this case is of course that the universe is expanding. As it was mentioned this idea of expansion would had never shown up if astrophysicists paid more attention to Einstein's theory of gravitation. They could easily see why the universe has to looks as if it were expanding even if it does not. Yet they didn't. They preferred the "elegance" of "big bang". To understand this rather sociological but related to gravitation phenomenon let's start with the history of human perception of gravitational phenomena to see how humans in general and even the wisest of them in particular are able to believe in even more silly things than "big bang".

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What humans thought about gravitation throughout their history

First the gods gave humans the earth that was flat and situated horizontally so the humans wouldn't fall off of it. There wasn't any question why they can't rise up towards the heavens and always fall down towards the hell. It was obvious that to rise up towards gods humans had to earn that privilege by good morals that were defined by those gods and communicated to humans through priests who communicated directly with gods.

The good morals included giving gifts to those priests (money was not invented yet). Everybody saw that it was good.

It was the first intellectual achievement of humans and it was called religion (or rather religions since there was as many religions as tribes of humans). Humans were spiritual beings then and had clear understanding which way is up and which way is down (at least when they were not drunk). They were lead by priests who in turn listened to direct instructions from gods. Many of those instructions were written down so we can now read them in various documents as dictated by various gods to various priests.

Then it turned out that the earth is not flat but round like an orange and still not a thing falls off of it, not even water which was known to flow always down. It was kind of mystery and as each mystery that humans encountered during their short history of intelligent life it was quickly explained by their wise men. In this case the explanation was an attractive gravitational force that attracted everything to the earth showing this way which way is down.

It seemed a good explanation and it turned out that it can be described mathematically by a very compact formula discovered by Newton.

After Newton discovered his formula it turned out that not only the earth attracts everything but everything attracts everything else. E.g. the moon and the sun attract oceans on the earth causing tides, and the sun attracts the earth forcing it to travel a whole circle around it once a year and a lot of other things like that.

The formula for that attractive force is of course F = G M m / r2, where F is this attractive force, M and m are masses of the objects that are supposed to attract each other, and r is the distance between centers of these objects. The assumption that such attractive force really existed made an impression that the Newtonian theory of gravitation is a physical theory.

Then it turned out that there are two gravitational phenomena that couldn't be explained by Newtonian formula. One was the behavior of planet Mercury, the other the bending of the light ray in vicinity of the sun. They couldn't be explained by the existence of attractive gravitational force only. It created a suspicion that physics of Newtonian theory might be different than the physics of the real world. When Einstein came along with his theory and explained those two phenomena (and all others as well) with new physics the suspicion about invalidity of Newtonian physics turned into certainty and Newtonian attractive gravitational force disappeared from physics with a slight "poof", which unfortunately very few people heard.

However before it happened the Newtonian formula worked so well that it allowed to predict movements of all celestial bodies (other than planet Mercury) that were known to humans at that time. So from that time on, everybody who ever got any education was told that everything in the universe is attracted to everything else.

There were still those who believed that the earth is flat because otherwise the water would flow off of it. They were trying to impose their views on others via blowing them up or shooting at them with various weapons to let them know which religion is the only true one. Those behaviors survived to present time so we might still investigate how the mentality of those humans works by interviewing them, and discover all the mechanisms of their thinking, making a physical theory out of it.

We might even interview our own Presidents about the bombings that they ordered (if they still remember the reasons or have ever known). Explaining the reasons for human behavior is an interesting science called sociology, but only the theory of gravitation, which helps with those bombings, is the subject of this article, so let's continue with gravitation.

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Practical applications of theory of gravitation

After Newton discovered his formula for the "attractive gravitational force" the humans were very happy that finally some of their formulas, unlike many other magical formulas, worked real well and that now they can calculate the trajectories of their bombs with high precision. Now they could drop their bombs on any pharmaceutical factory in the world (e.g. if they wanted to keep prices of medicine at sufficiently high level) precisely as they wanted, even in Africa. Or e.g. drop them on a Chinese embassy elsewhere. Because of that precision they didn't need to worry that they hit a British, French or German embassies instead, risking an international controversy. Of course even if the controversy had arisen the bombing could be blamed on error in the theory of gravitation or that the bomber didn't know it was an embassy. He might just have thought that it was a school or a hospital or a church or even an empty parking lot with the latter being the least credible because who would like to bomb an empty parking lot.

Anyway regardless of potential danger of controversies it was a great thing that precise bombing. It was supervised by ballistic officers that every army had to have to calculate those trajectories, and so humans started to investigate their theory of gravitation with greater and greater accuracy to confirm that this Newtonian formula is valid absolutely and so precision of their bombing will be absolute. It was a nice feeling to be able to scare sense out of the opponent and be able "to kick some butts" when required by the "national interest".

The other application of the theory of gravitation, the prediction of movements of celestial bodies didn't have that many practical applications, if any. The bombing seemed to be the main cause for governments financing the research. Possibly as much as today the biotechnology that promises even greater advantage than precision bombing as it might make possible to wipe out selectively the whole unfriendly races of humans once all the particularities of their genetic code are known.

This confidence in accuracy of Newtonian formula lasted a few hundred years and then, when the accuracy of observations became much higher than Newtonian formula could stand the astronomers noticed that about 5% of the movement of planet Mercury didn't fit the Newtonian formula. Mercury happened to be the planet that is closest to the sun and therefore it moves faster than any other planet. It was thought that it moves fastest to counter with its centrifugal force the attraction of the sun that because of that close distance to the sun it was for Mercury greater than for any other planet.

There was a suspicion that either strong force of attraction or high speed of the planet makes it deviating from Newtonian prediction about its behavior. Soon it turned out that also a ray of light that passes very close to the sun bends twice as much as Newtonian gravitation would allow for particles that had gravitational mass and velocity of particles of light, so called photons.

It was a chance for Einstein to present his solution to this problem. Einstein, who a few years earlier explained why magnetic force is a mathematical fiction by his special theory of relativity, proposed what physics might really cause all the gravitational phenomena and why attractive gravitational force is a mathematical fiction as well. He called his new theory "general relativity". Einstein blamed the curved space in vicinity of the sun (a thing unheard of in Newtonian physics) for anomalies in Mercury's movement and for the excessive bending of the light ray. It turned out that the blame has been properly placed.

It turned out that half of the deflection of light ray is caused by the curvature of space. This curvature means that there is a little more space in some directions than in others (i.e. if this curved space gets encolsed by a perfect sphere there are different distances between opposite points on that sphere when measured through this space inside the sphere in different directions) which happens more in close vicinity to large masses than farther away from those masses. It is something that was not reflected in any way in the Newtonian theory with its simplistic attractive force. It made the main difference between the Newtonian theory and the real world.

If there were no curvature of space it wouldn't be possible to find out that the Newtonian theory was false, and Einstein's theory wouldn't be even published. In science a new theory is adopted only when the old theory starts contradicting some trivial truth like Newtonian theory started when it said "1" and the nature said "2". Since Einstein's theory also said "2" it replaced the Newton's theory. Later in this article, in the section titled Conservation of Enegy in Einsteinian Gravitation we'll se why the space has to be curved if the rate of time changes from place to place.

So far Einstein's proposition holds. So far there is nothing known to humans that wouldn't agree with predictions following from both Einstein's theories of relativity (special and general). So we may hope that our present knowledge about gravitation is not magic like Newtonian formula turned out to be but it is real science, i.e. something that exists really in the real world.

In the process of explaining small differences between predictions of Newtonian gravitation and the real world Einstein found that the old universal attractive gravitational force is a fiction. One force was dropped from the set of fundamental forces of nature.

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Cultural Change

The physicists who comprehended it, all three of them (just kidding, but less than it seems) had the surprise of their lives. It transformed the scientific culture. It showed clearly a general truth that even things that look obviously real like the universal attractive gravitational force or God may be actually totally fictitious. In other words it has shown the ability of humans to fool themselves without limits.

All of it has been hardly noticed by lay people and also, oddly enough, probably by majority of astrophysicists, creating an odd situation that all those people know that Einstein has done something important for science but don't really know what it has been. They tend to think that it is too complex to be comprehended by the common folks or even by the common astrophysicists so they don't even think about it. Most of them still believe that the existence of the universal attractive gravitational force is one of the most real things in the universe.

One may read a lot of stuff about the universal attractive gravitational force, and how important role it has in the universe, in popular science articles in practically any popular science magazine like e.g. "Scientific American". Not that Americans are more backward than other nations since ignorance seems to be spread relatively uniformly throughout the world. However statistics that show 97% of our fellow citizens still believing in supernatural might not get such high numbers in any other country with possible exception of interior of New Guinea.

Those popular science articles become especially silly when the author, who overlooked disappearance of gravitational attractive force from the nature in the first place, worries whether this force will stop the expansion of the universe, adding that it worries also real scientists.

There were many attempts to save the Newtonian theory (and still are similarly like there are attempts to save the flat earth theory) by introducing additional features into it or replacing it by another magical theory like Quantum gravitation trying to repeat the success of Quantum Mechanics but with no similar luck so far. There is also a half way theory containing both attractive force and curvature of space. This theory is taught to those students who are considered by those who teach them too feebleminded to comprehend Einsteinian Gravitation as it is. In this theory the gravitational attraction is allegedly so strong that it curves the space.

This is how many astrophysicists imagine the reason for the curvature of space. For the details the reader might want to see the web pages by Prof. Ned Wright who unfortunately didn't have time to answer the question why he thinks that there is such a thing as attractive gravitational force (that is bending this space) in the first place.

None of those improved theories with gravitational attraction, despite that they produce formulas that accurately describe the real world, can produce any viable physics (the mechanism behind the math). So they are, as Newtonian theory, only mathematical theories that work as any good magic does. None of them is as simple as Einstein's theory though. Since scientists prefer simplicity to sticking to old ideas, the Einstein's theory became generally accepted as a true as far as we know theory of gravitation at least among regular scientists.

Science uses a principle known as Occam's razor to get rid of all fictitious stuff ("entia non sunt multiplicanda praeter necesitatem", meaning roughly that beliefs shouldn't be multiplied beyond necessity). Because of that principle the belief in attractive gravitational force disappeared from physics. Today this force exists mostly as a fictitious entity. In such a form it is very handy while doing calculations for relatively slowly moving objects as planets, spacecrafts, bridges, etc. Of course it still exists also in minds of many of those who neglected to familiarize themselves with Einstein's gravitation, a defect that this article is meant to repair.

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Benefits of Faith

It should be mentioned that belief in universal gravitational attraction is not unlike many other beliefs in other fictitious things in human cultures that in some situations are treated as if they were real since it simplifies many complex matters.

Many humans may believe, without ever in their lives getting into any trouble, that all those fictitious things are real. Those humans may even benefit from those beliefs, also for biological reasons, by improving chemical processes in their bodies since many of the chemical processes in our bodies are controlled by our brains, which in turn are controlled by our superstitions and prejudices, a.k.a. spirituality. Which may be the reason for the evolution keeping such big part of the population of this planet as compulsory believers rather than thinkers who as various studies seem to indicate are limited to only about 5% of general population.

Other examples of such fictitious things that simplify our lives may be magnetic force, American Democracy, or Santa Claus, each handy in certain more or less complex situation.

E.g. an assumption that magnetic force is a special force that is separate from Coulomb attraction and repulsion between electric charges simplifies math of electrical engineering.

Great majority of electrical engineers don't even suspect that magnetic force is just a mathematical fiction and the only real thing seems to be the Coulomb force with the rest of the picture being just relativistic effects due to time dilation and length contraction in systems of moving charges. Those relativistic effects simulate the existence of special magnetic force that looks like separate from common attraction and repulsion between electric charges. The reality of magnetic force is so compelling to some people that they seriously are looking for "magnetic monopoles" (hypothetical particles that if existed would generate magnetic forces without help of any electric charges - a nonsense according to relativity).

So basically for those of us who are perfectly happy with using magic the gravitational attractive force is a very handy fiction however, for those who are curious even about things that they don't need to know, mechanics of gravitational phenomena should be finally explained.

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Gravitational phenomena

To understand how gravitational phenomena arise without gravitational attraction let's imagine that we are inside of a big container with a slippery floor, carried by a truck and the truck is just negotiating a tight turn on the road.

If we are someplace in the middle of the container we start sliding toward the container's wall. We might think that the wall attracts us. Similarly as when we fall toward the earth we tend to think that the earth attracts us.

Physicists, who analyze such things more carefully, thought that two different forces act in those two cases: inertial force in the former and gravitational in the latter.

The modern physics recognizes only one force in both cases, an inertial force called also a pseudo force. The name comes from the fact that e.g. in the case of the turning truck we are really not pushed against the wall with any force. It is the wall that is moving toward us when the truck is turning while we are still moving in a straight line as the truck had been moving before getting to the curve. From the point of view of the wall that does not know about its acceleration it looks as some force pushes us against it.

When, in the process of the truck making a turn, we finally collide with the wall, it starts pushing us with certain force depending on how tight turn the truck makes and with what speed. Therefore depending on how fast the wall accelerates towards us, now together with us pressed against it. If there were an open door in that wall we might never collide with the wall, just flew out through the door and never feel any force despite that from the wall's point of view there might have been a  force that has thrown us out of the container.

So pseudo forces are forces that seem to exist from the point of view of one system and may not exist at all from the point of view of another system. In this case the one system would be container's wall that is seeing us moving faster and faster toward her (presumably pushed by some force), and the other system is us who just keeps moving along a straight line as we did before the tuck got to the curve.

In the case of falling on the earth the situation may seem not identical but it is surely a similar one. Like in the case with a moving container we don't feel any force while we are falling toward the earth (we are weightless then). When we finally collide with the earth, presumably survive the collision, and then lie pressed against it we are feeling a force pushing us up. If there were a hole in the earth leading to the other side of the earth so that we wouldn't collide with the earth, we would never feel any force as in the case of flying out of the container.

The difference between the two cases is that we think that the earth's surface does not accelerate towards us when we are lying on it, as the container's wall has been. Einstein has proposed that it actually does accelerate towards us all the time and we accelerate with it when we stay put on it. And that's why we feel the force that we used to call gravitational attraction that still is a gravitational force but not attraction anymore, just plain inertial force caused by acceleration.

We don't feel any force when we are falling on the earth, ignoring negligible gravitational effects generated by our own body called tidal effects. Those effects arise from the change in the time rate generated by the earth mass around itself that is still called gravitational field but does not mean field of force any more. Neither we feel any force (except small tidal forces about which later) when we happen to be on an orbit around the earth. We feel a force that looks like attraction only after we are pressed against an object whose surface then accelerates together with us, creating an inertial force that presses us to that object. The nature of that strange acceleration seemingly without any movement is the main point of Einstein's discovery.

To understand Einstein's discovery and why the gravitational force is the same inertial force we feel at other occasions e.g. when we are tossed around in a moving vehicle, we should first understand what the acceleration really is as a physical phenomenon and how it is generated.

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As time is something that is measured by a clock, the acceleration is something that is measured by an accelerometer.

It may appear strange to some readers who are accustomed to old definition of acceleration (that acceleration is how fast velocity of something changes) but there is really no other sure way to tell whether something accelerates or not.

We used to say that acceleration is when something moves faster and faster (the old definition of acceleration). The acceleration is then the amount by which the velocity increases during e.g. one second.

E.g. when something is falling on the earth its speed is growing by about 10 meters per second or about 20 MPH) during each second. A question immediately arises: the velocity in relation to what? It is a question that has never been answered satisfactorily. Just to have some answer people agreed that it is in relation to something that does not accelerate however ridiculous the answer may be because how do we know that is does not accelerate. It has been assumed that there must be something like this in the universe. It was even named absolute space.

Newton discovered a formula, valid to this day, that inertial force acting on an object is proportional to the mass of the object and the acceleration of that object in relation to something in which there are no inertial forces felt (F = ma, where F is force, m is accelerating mass, and a is acceleration).

Such a system where inertial forces are zero has been named inertial system. All systems that would move with constant velocity in relation to absolute space (to a non accelerating system) were automatically inertial systems. In times of Newton the acceleration was still in relation to absolute space.

In contemporary physics the idea of absolute space has been given up as something that can't be measured and therefore does not exist as a physical object, and so, by Occam's Razor, at all. The physics adopted a view that the force in Newton's formula for inertial force is just the measure of the acceleration, which in that way became measurable directly regardless of any movement in relation to anything or any space. Acceleration became a real, measurable thing, while absolute space something fictitious. We don't need anymore to relate acceleration to a movement in relation to any space however we still can if it is convenient for some reason. Now there is no ambiguity, and no guessing what is that strange thing called absolute space. There is no absolute space. If the accelerometer fixed to an object shows something, the object accelerates. If shows nothing if the object does not accelerate or the accelerometer is out of commission. That's basically all what there is to it. It still does not explain why an accelerometer on the surface of the earth accelerates (shows about 10 (m/s)/s directed up, towards the sky) and one that is in free fall shows nothing. The explanation requires a further investigation what happens when objects accelerate.

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Gravitational force

The thing that is relevant for our purpose is time running faster in space at direction in which the object accelerates and slower in the opposite direction. An accelerated observer is seeing clocks ticking faster than normal in the direction of acceleration and slower than normal in opposite direction. The difference is proportional to the acceleration and the distance to the clock. It is rather easy to figure out why it has to be so:

Let's imagine ourselves sitting at the rear end of a long room comparing a clock on the front-end wall with our wristwatch. If the room is at rest or moves with a constant speed (in relation to something that does not accelerate as verified by an accelerometer fixed to it) the clock on the wall has to be set a little ahead to look synchronized with our wristwatch. It is because the light from clock on the wall needs time to reach us across the room. When the room starts accelerating, we will see images of the clock getting to us a little earlier than when the room moved with constant speed, because we encounter those images of the clock moving against them with higher and higher speed. This way the clock looks running ahead of our wristwatch proportionally to the acceleration and to the distance to the clock.

When we stop accelerating the clock on the wall will be running at the same rate as our wristwatch again, but it won't be in sync anymore because we've seen it running faster when we were accelerating and it can't just jump back to "right time" when we stopped accelerating. It will show more ticks than our wristwatch. It shows that during the acceleration the time was running faster at the place where that clock is (at the wall in front of us).

This is one of facts that people have hard time to believe. It turns out to be a physical fact: rate of time in respect to us is not fixed but depends on various conditions. E.g. it depends on our acceleration and the distance to the place when the rate of time is measured.

The opposite will happen to the clock behind us. It will show fewer ticks, which confirms that it has been running slower. Or that time has been running slower there where this clock is.

Those effects have been noticed by the physicists before Einstein but they were considered to be too mysterious to think about since it was not considered possible that time may run faster or slower than at its usual rate. It was left for future generations to explain, which Einstein did by demonstrating that if the speed of light is always the same then the time actually has to run faster or slower depending on various physical conditions. It was a simple explanation and a basis of Einstein's special relativity that deals with systems that move with constant velocity.

Then, when Einstein started considering also accelerated movements this variable time rate became a basis for a new theory of gravitation called general relativity.

All the above is the explanation of the second paradox in the twin paradox. This second part was that the if rate of change of velocity was causing one twin aging faster than the other then the velocity of one twin with respect to the other was the same as the other in respect to the first and so were their mutual "accelerations". So it is not the rate of change of velocity with respect to something, that causes those effects, but real acceleration that is a physical fact measured by an accelerometer regardless whether it is related to any velocity with respect to any reference point.

The traveling twin who flies to a distant star has to accelerate toward the earth when she starts her return trip and her accelerometer will show it. During this acceleration caused by the change of direction of the movement from away from the earth to towards the earth the time on the earth runs much faster in relation to the accelerating twin. It runs faster proportionally to the change in velocity and to the distance to the earth. This faster running time has never a chance to wind back so on her return to the earth the traveling twin meets her twin that is now older than the traveling twin.

It would be the same if the traveling twin were just sitting on a more massive planet then the earth (while her twin was sitting on the earth).

While sitting on a planet one feels acceleration away from that planet and so time away from that planet has to run faster. So again, after her return from vacation on a more massive than the earth planet the traveling twin would found her twin older too even ignoring all the acceleration effects arising from the change of direction of movement.

So once we know how acceleration influences time rate, and we know that velocity is there only as a purely abstract mathematical thing (an integral of that acceleration) the twin paradox disappears. It disappears because one twin really accelerates (her accelerometer shows something rather large) and the other, who stays on the earth, does not (her accelerometer shows about zero). Now we know the difference between the twins' movementsand so we know which one will be younger when they meet: the accelerating one.

To see how this acceleration causes gravitational force let notice that we have here acceleration causing a difference in the time rates along direction of acceleration. There is a strict relation between the two that may be expressed by a simple formula involving only rate of time, acceleration, distance, and the speed of light. Einstein thought that if acceleration is causing a change in the time rate then perhaps the change in time rate would cause an acceleration (and the unavoidable inertial force connected with it). It has been called the principle of equivalence (of acceleration and gravitational force) and it turned out that the nature really works that way.

The presence of mass (or energy, to which this mass is equivalent by famous Einstein's formula E = mc2) causes that time runs slower in the vicinity of that mass, as verified independently by very accurate clocks that became available only recently. It is all that is needed to create acceleration detected by accelerometers and cause inertial forces proportional to that acceleration. It is all without any movement of anything. This way the origin of gravitational force has been explained by the mass of the earth causing the time running slower in its vicinity causing all the conditions that show up when something accelerates. Therefore, on Einstein's principle of equivalence it causes the acceleration, which in turn causes inertial force.

This slowing of time in vicinity of a mass is called gravitational time dilation. This time dilation causing inertial force is what people used to take for the old attractive gravitational force. It may still be called gravitational but there is no way of making it attractive any more since it disappears immediately after the "attracted" object loses physical contact with the "attracting" one. So it does not act at the distance as an attractive force would and as it was imagined that it does.

There is certain popular misconception, even among physicists who are not too familiar with Einstein's theory so it is worth of explaining.

This misconception is that a hypothetical "gravitational attractive force" causes gravitational time dilation and all other relativistic gravitational effects (like e.g. curvature of space). Obviously it can't be so since time dilation by itself causes acceleration and so it causes the inertial force that is exactly equal to that hypothetical "gravitational attraction". If there existed also this hypothetical "gravitational attractive force" then the gravitational force that we observe on the earth would be twice as big as it is: one part caused by the "gravitational attractive force" and one by the time dilation. We observe only one of those parts and since time dilation is verifiable fact (trivial truth verifiable by precise clocks) the other part (the "gravitational attractive force") must be a fiction.

This is how the gravitational attractive force disappeared from physics. That it still stays in minds of some physicists is kind of mystery, which can't be explained by asking those physicists questions since as it was mentioned they claim the lack of time for answering them.

In any case this gravitational time dilation effect (sometimes called time curvature) is the first half of the Einsteinian Gravitation. It explains fully the same things that Newtonian gravitation managed to explain and gives exactly the same numerical results.

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Tidal forces

--- under construction ---

If it were all than we would have the same rule for behavior of physical objects and the same problem that we have with Newtonian gravitation: why Mercury doesn't follow the rule and why light ray bends twice as much as predicted by the rule. Fortunately there is more to Einsteinian Gravitation.

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Curved space

As the first part of Einsteinian Gravitation is about what happens to the time in the vicinity of masses (which explains all those things that Newtonian gravitation explains but without introducing any magical "gravitational attraction") the second half is about what happens to the space around masses.

Basically Einstein's theory states that there is more space around masses than there would be without those masses present. It means that if we make a spherical shell having volume of e.g. 1,000 gallons when empty then we pour into it water then the space inside the sphere becomes bigger than before with the outside shell ever changing its size. It is difficult to believe yet it seems to be what actually happens. The diameter of the shell as measured from outside will be the same as it was before but the space inside will be bigger (a measured with the same rulers that we used to measure the shell from outside).

If we measured the diameter form inside moving along a straight line through the center from one side of the shell to the other, we would find that the diameter of the shell inside is bigger than when measured from the outside. This is what is called "curved space". The space inside the sphere is bigger than the space taken by the sphere itself.

We may pour into a 1,000 gallon sphere (where empty) e.g. 1,001 gallons of some (very) heavy liquid which means that the space inside became bigger by one gallon.

To understand this strange thing with diameter of the shell being greater inside than outside (which happens to be a physical fact) we may think about it as rulers becoming shorter when put inside the shell containing mass inside. As clocks run slower in vicinity of mass because the time runs slower there (which is called time dilation) the rulers become shorter in vicinity of mass (which is called length contraction). So if our rulers are shorter inside the shell the diameter of the shell measured from inside will be bigger.

But physically the clocks run all the time with the same speed (the same as the time). It is the time which slows down in vicinity of mass as seen from outside. The rulers are also all the time the same, just there is more space in those areas in vicinity of mass than it would be in regular (flat, Euclidean) space, so the rulers when compared to the diameter of the shell look shorter to us who look from outside.

The space with such strange properties is called "curved" because the increase of volume inside our shell that contains mass is similar to increase of area of the surface inside a ring when the ring is placed on a surface of a ball instead of on a flat table. The area of the surface inside a ring on a ball will be greater than the area inside the same ring placed on a flat table. We say that surface of the ball inside the ring is curved as opposed to the surface of the table inside the ring that is flat. Similarly we say that space inside our sphere is curved after mass (or energy, since every energy has mass as famous E=mc2 tells) showed up in it, as opposed to flat empty space.

Another interesting thing about the above is that the space in vicinity of mass gets bigger (or "rulers shrink") by the same relative amount (by the same percentage) as the time slows down.

This relation of time to space causes that light ray bends twice as much as predicted by Newtonian gravitation. As it was mentioned earlier the Newtonian gravitation only predicts accurately all the gravitational effects caused by gravitational time dilation and none caused by curvature of space. That's why it predicts only half of deflection and Einsteinian Gravitation predicts the whole angle as it is observed in the real world. And this is also how we know that time dilation causing half of deflection is the same as increase in amount of space that causes the other half of deflection. This equality of spatial and temporal effects shows that the spacetime is more complex creature than space and time separately and which are considered separately in Newtonian theory. That somehow the time depends on the space and the space on the time. It is similar like in the mentioned couple where the woman and the man depend on each other and make more complex structure than two of them being considered together but independent form each other.

This dependency of time on space explains why the universe looks as it were expanding. There are masses in the universe as planets, stars, galaxies and all the other observed and not observed yet junk between them. Each of them curves a space a little bit (makes more of it in its vicinity) when we look through that increased space deeper and deeper into space we see time slowing down more and more. As it has been mentioned, such slowing of time simulates Doppler effect that looks like all the sources of light in the universe were moving away from us with velocities proportional to their distance from us. As before Einstein the behavior of time in the universe simulated the existence of the universal gravitational attraction, now it simulates the universal expansion however now it is after Einstein.

The astrophysicists prefer to insist that it is not possible to propose explanation of the observed phenomena other than Doppler effect, which according to them forces everybody to believe that the universe is expanding. And since it is not possible to propose explanation all papers on that subject are rejected by all scientific journals without even stating the reason for rejection other than the papers don't support the idea that the universe is expanding (small wonder). And it lasts already for almost two decades. It will probably last well into this century until some Very Important Person, whose paper nobody will dare to reject, discovers that Einstein's theory explained all of it already. Then the big bang will disappear also from minds of astrophysicists.

Those of the readers who are interested in this subject enough to read to this point might have noticed that this illusion of the expansion of the universe is caused by behavior of time, and that behavior of time is reflected in Newtonian gravitation. It might give them an idea that therefore it should be possible to demonstrate just with Newtonian formula that a non-expanding universe should look as if it were expanding. That is indeed so, and this is what the author has done to convince astrophysicists (without much success though) that the expansion of the universe is an illusion. It has been shown with simple Newtonian math what should be the observed rate of the apparent expansion if the universe didn't expand and it turns out that the result is as it is really observed. It is also the same result as would be predicted by just following strictly Einstein's gravitation and the fact that time dilation is the same as curvature of space. Both produce the value of Hubble's constant of apparent expansion equal speed of light divided by Einstein's radius of the universe. The details of the derivation of Hubble's constant for our universe are on this site in  Physics of Illusion of Expanding Space. It is also an explanation for general public but more detailed then those two paragraphs above and with full mathematical support for those who want to see how it is derived.

The above basically explains all the gravitational phenomena that are observed up to date. The are also some predictions of what we might to encounter when we gather more data about the universe. All of them are quite interesting.

One such thing is that the more mass is placed inside the shell the more additional space there will be inside it but the amount of space increases by a greater amount than the mass that creates that additional space. The math of this mechanism described by Schwarzschild through solution of Einstein Field Equations (that describe Einsteinian Gravitation) indicates that for any sphere there exists a certain amount of mass that makes that additional space infinite. If there were such a mass inside our sphere we could pour infinite amount of water into it. Such an object, with an infinite amount of space inside is called a black hole. The name comes from a fact that if there were infinite space inside it the light would need an infinite amount of time to travel though it to get out of it and come to us. So practically we would never see that light regardless how long we looked at that object. We would see something that does not emit light at all, and so it is perfectly black.

Another interesting feature of such an object would be that it would be at infinite distance from us. It would be so because when there is more space around a big mass and also the time slows down in this area then the light needs more time to get to that mass and back. If we use radar to measure the distance to that heavy object the photons we send to it come after longer time than they would if the mass of the object were small. In case of a hypothetical black hole that time, and therefore the distance too, would be infinite.

It is not known whether such objects as black holes exist in nature. Some scientists maintain that they can't exist (Einstein was one of them) because of those infinities they produce like the effect of time slowing to halt at the surface of the sphere and so no more objects can fall into such a sphere in a reasonable time. So a real black hole couldn't be formed during the lifetime of the universe (regardless how long it were, unless it were infinite as well). The surface of the sphere which contain enough mass to stop the flow of time is called event horizon since at that surface the time stays still and so nothing can ever happen. No events are possible on and beyond that surface. Some scientists believe that black holes exist but no reasonable hypothesis telling how the objects may fall into them through the event horizon to form them has been proposed yet (however unreasonable hypotheses were proposed and many scientists, not understanding Einseinian gravitation, believe that those who proposed those hypotheses did check that they make sense). Consequently the black holes are more popular in SF texts than in science. And often this SF is presented to the public as science by naive astrophysicists who don't understand Einsteinian Gravitation but believe what they are told by experts many of whom don't understand it neither. Since we want to keep this text as close to science as possible let's forget about the black holes. There are even stranger and observable things in the nature so we don't need to get into SF to be baffled.

It should be mentioned that the curvature of space has very little influence on gravitational phenomena in our solar system and next to none in vicinity of the earth. It applies mostly to deep space of the universe. To understand what is really going on in the universe it would be good to learn what happens to its space when each mass curves it only a little bit.

Main thing that happens to space is that if the space behaves this way it has to be closed. It means that from whatever point in the universe one starts to travel in a straight line in whichever direction, if one travels long enough along that straight line, one is bound to return to the starting point.

It seems unbelievable, perhaps much more than to the people who believe that the earth is flat, that going all the time West one may return one day to the staring point from the East never even getting to the edge of the earth. Of course we know that the two dimensional surface of the earth is closed via the third dimension. The three dimensional space of the universe can't be closed via the fourth dimension since it is easy to notice that the fourth spatial dimension does not exist (it is impossible to make four mutually perpendicular directions). So let's try to explain how our space can still be closed despite that the lack of fourth spatial dimension.

While reading popular science books about the universe one may find a quasi explanation of the closed three-dimensional curved space. That "it is the same" as the surface of the earth (or of a balloon), which is also closed, and that space has "just one dimension more" being three dimensional, while surface of the earth is two-dimensional. This is a good example of  magical thinking which might be explained on this example, so the reader may be aware of it while reading other popular texts about science which may contain other instances of magical thinking. The magical thinking is very natural to humans and showed up in human civilizations most likely just after invention of the language. It is thinking rather about the names of things than about the things themselves. Analogies get created based only on similarity of words or ideas. And those analogies are very often false ones (another name for magical thinking is false analogy), and therefore they don't really explain anything.

Since there are only three dimensions, the curvature of space can't be explained with analogy to a two-dimensional sphere that is curved into third dimension. To explain the curvature of three-dimensional space one has to do it within tree dimensions. This might seem to be possible if there is a way of explaining spherical geometry in two dimensions, by using only two dimensions, just on the flat surface. It shows up that there is. Besides, for many curved geometries (as e.g. for Lobaczevski's geometry) a three-dimensional model can't be even constructed: there is no three-dimensional surface in Euclidean space that would have the Lobaczevski's geometry and yet there are phenomena in nature that this geometry describes. So we have to find a way of understanding curved geometries in different ways than geometries of some curved surfaces in Euclidean three-dimensional space. E.g. a simple geometry of a surface of a ball or even simpler one of a surface of a cone that may be unrolled into a flat Euclidean surface. The mathematicians say that geometry of the surface of a cone or a cylinder is flat despite that to a lay person it looks curved. It is flat because all distances between points on such a surface are the same as on a flat surface of a table. Bending of that surface into a cylinder does not change those distances while bending it into a sphere would.

So to understand how three dimensional space can be curved despite that there is no fourth dimension it might be good to understand first how two dimensional surface may be curved without being curved into third dimension as if the third dimension didn't exist at all.

The model of such a curved surface is quite simple. Let's imagine a big flat disk e.g. of 20,000 km radius that has such a property that whatever moves from its center towards its edge keeps its size in direction towards the center of the disk but gets a little bigger in the perpendicular direction. It gets back to its original size when moves back to the center. If it is a man walking on that disk, if he makes a circles around the center of the disk, the circle that is twice as far form the center wouldn't have twice as long circumference but a little less than that.

Lets assume that the relation between getting bigger and distance form the center is such that sizes of those circles are exactly the same as sizes of parallels on the earth at the distances from the earth's pole the same as radii of those circles. In such a case the man drawing circles and measuring them may conclude that he is not on a flat surface of a disk, but on a curved surface of a sphere of the size of the earth. For him, for all practical purposes the surface would be a curved surface with the same geometry as the surface of the earth. So he wouldn't be surprised when at the distance of 10,000 km from the center his circle would have length of only 40,000 km instead of about 62,832 km as it would on a flat surface. Or, that inside this circle there is more area than it were on a flat surface within the circle of the same circumference 40,000 km. And even that all circles with radii greater then 10,000 km become smaller instead of getting bigger. And yet he would be on a "flat" two-dimensional surface of a disk. A surface that is not "curved" into any third dimension.

The trick with changing size of a ruler in direction perpendicular to the line from the ruler to the center of the disk would make the geometry looking as if the surface were a surface of a sphere. The last circle that he would make about 20,000 km from the center would be so small that he could just slide around it and walk beyond it for another 20,000 km getting back to the center of the disk from the opposite direction. So we see here a model of a curved two-dimensional surface without necessity of introducing the third dimension (this model is know as hot plate model in literature of curved surfaces).

It is good to add that since the geometry is the same as the geometry of a surface of a sphere there is really no center in it despite that the disk has a center. The trick with changing sizes makes impossible to tell which point of the disk is the center. All the points look the same, as they look the same on the surface of the earth. The trick with sizes makes out of flat two-dimensional space a curved two-dimensional closed surface of a two-dimensional sphere.

Now this two dimensional model can be changed to three dimensions in the way that instead of walking and making circles the man can fly in any direction and build spheres around some center. If sizes change in the same way as before, the surfaces of his spheres will be growing a little slower than their radii, and the biggest of them all will have a circumference of 40,000 km. Then circumferences of the next spheres will be smaller and smaller until the last one, at 20,000 km from the center will be so small that that he could just pass by it. He could fly beyond it for another 20,000 km getting back to the center of the disk from the opposite direction. For all practical purposes he will be in a closed tree-dimensional space with its weird properties as going east along a straight line and coming back from west along the same straight line. The trick with sizes makes out of a flat three-dimensional space a curved three-dimensional closed space of a tree-dimensional "sphere". And all of it happens without introducing any fourth dimension.

The above shows that it is possible by playing with sizes to change a flat space into a curved space. This is what seems to be going on in our universe except that it is not the size of the objects or rulers that the movement is changing. The distances in that space change so that those rulers look bigger in some places and directions than in other like there looked smaller in direct vicinity to some particular mass. Also the radius of that tree-dimensional sphere into which the space is curved is much bigger than the radius of the earth. This radius is called Einstein's radius of the universe and it's size is about 20 billions light years (add or subtract a few billions). It happens to be the same radius called Einstein's radius of curvature of space mentioned as the number that shows up in derived via Newtonian formula value of Hubble's constant of the apparent expansion of the universe. It shows how many additional questions about the universe, except why Mercury moves differently than predicted, and why light ray bends more than predicted, the Einsteinian Gravitation explaines.

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Conservation of Energy in Einsteinian Gravitation

One issue that confuses many people is a question what is the real world equivalent of the old Newtonian potential energy (a.k.a. "gravitational energy").

Some of us may remember from high school that we were told that (according to Newtonian gravitation) when we lifted a weight, we did work against attractive gravitational force with which the earth attracted the weight. That the work we did got converted into potential energy of "gravitational field". This potential energy was thought to be able to be recovered when this lifted object was falling down either doing work or just acquiring kinetic energy.

Now we learn from this article that there is no such thing in the real world as gravitational attractive force. The attractive gravitational force turned out to be a mathematical fiction. And so there is no physical field. "Field" is only a name of the space where the phenomena under consideration occur. And obviously a name canít absorb any energy.

Seemingly there is nothing that could be transformed into kinetic energy or absorb the kinetic energy converting it in "potential energy of the field". And yet all lifted objects need energy to be lifted, and when they fall they acquire kinetic energy and this kinetic energy is of course real as we can verify if the object falls on our foot. So where this kinetic energy comes from if no force acts on a free falling object? Obviously it can't be created out of nothing since the principle of conservation of energy prohibits such extravagance.

Those of you who read this article carefully might have guessed already where the energy comes from. Those who didn't might be still baffled by the fact that despite that no forces act on a freely falling object it falls faster and faster and so its energy is seemingly increasing. But where is this energy coming from? Those baffled people need some simple explanation.

The explanation is of course simple as most things in nature. It is only people who make them complicated, mostly because they learn mathematical descriptions of those things without understanding physics behind this math, and the math that describes simple physical phenomena is often extremely complicated.

So when a gravity phisicist is asked how a falling object acquires kinetic energy that seemingly is coming from nowhere, he might say: "don't worry, energy is automatically conserved since divergence of stress-energy tensor vanishes identically at any event" (?!).

It might be good explanation to some but most likely won't be understood at all by poets and not even by some science teachers. Interestingly enough even many gravity phisicists who deliver such "explanation" do not understand it and more or less privately think that this energy is created from nothing.

They don't understand it because it is just a mathematical description of what's really going on and the whole education of gravity phisicists in physics is limited to memorizing the math. I'm not kidding this time. This is what's really going on in gravitation physics. So let's explain in plain English what Einsteinian Gravitation says about how it works.

The thing that you could guess already is that total energy of any object is, as Einstein discovered, its inertial mass multiplied by the speed of light squared (the famous E=mc2). When the object is in free fall, e.g. falls straight "down", its velocity "v" increases and so its kinetic energy increases even faster than its velocity (when velocity doubles the kinetic energy quadruples as one may see from the formula for kinetic energy mv2/2, where m is the mass of the object, if one still remembers it from school).

It also happens that inertial mass of the falling object increases since energy has mass (again the same E=mc2) and so mass of the object depends on its velocity. Yet while falling down, the object gets into space where, as we already know, the time runs slower, and there is also more space (the famous Einsteinian "curvature of space"). This slower running time, and increasing amount of space that the light has to cover, causes the speed of light being slower in relation to a remote observer who is observing what happens and calculating energy of the falling object by multiplying its mass "m" by the speed of light "c" squared.

And so the observed speed of light drops by a tiny bit. Obviously it does not drop "locally" (as seen by the object itself) where it is always the same "c" as required by the basic principle of relativity that local speed of light, one that is measured by the object itself (if it could do such a thing) is always the same "c". But of course the object does not see its own kinetic energy neither so for the object (locally) its mass and speed of light, and therefore its energy is always the same (E=mc2). Only the remote observer is wondering what's going on and where the kinetic energy comes form if there is no force attracting the falling object to the earth.

It must be that increased mass has the same influence on energy as decreased speed of light so the amount of the energy of the falling object (as observed by all observers) stays the same while the object is falling. Actually it stays the same whichever way the object goes unless the object is pushed by some force. It only looks like it changes its energy when itís in a free fall because we don't notice the changes in rate of time nor changes in the amount of space and so the tiny change in the speed of light. So the fact that total energy of this object remains constant is difficult to notice.

Being difficult to notice itís not impossible to calculate and the calculations show that "lost" (due to smaller speed of light) internal energy of the object, and its "gained" (due to increasing inertial mass of the object) kinetic energy, are equal. There is no net change of the total energy of a object in a free fall as observed by any observer. The principle of conservation of energy is still working as it has always been and the mysteriously disappearing Newtonian "potential energy" is found as a part of the total energy of the falling object. It is hidden there in the falling object, perfectly localized, contrary to ideas of those who still explain the world with Newtonian gravitation with its "universal gravitational attraction" and for whom the energy is "somewhere", in some unspecified limbo (called "field").

It turns out that energy is in a limbo only in mathematical theories like Newtonian gravitation or Maxwell's electromagnetic theory. In the real world however since energy has mass (as E=mc2 shows) this energy has to be in a concrete point in space at any concrete time because this is where this mass is located. So it is located within the free falling object itself. And so the "potential" energy is not "gravitational". It is just the internal energy of the object itself.

So we see that according to Einsteinian Gravitation the "gravitational energy" does not exist at all.

It turns out to be the same kind of mathematical fiction as "gravitational force", and many other mathematical fictions that are slowly explained as such by physical theories like Einsteinian Gravitation.

The magic of fictitious entities working (almost) as predicted by mathematical theories that only say what is going on and when, is being slowly replaced by science, by physical theories that not only explain what is going on and when, but also why.

For mathematically oriented poets and science teachers there is an appendix showing how Newtonian potential energy is recovered from the internal energy of the object while the object is falling. When the object is rising the process works of course in the other direction: the disappearing kinetic energy is accumulated as internal energy of the rising object.


After writing this article in which the author hoped to be able to explain all the Einsteinian Gravitation for all those who are intimidated by its math he sees that to do it well he'd need to write a book. But since we have now internet to communicate, the book may be not needed. If those few people who will read this article have some questions about it they may send them via e-mail to the author and he'll try to answer all of them just by adding more explanation to this article or by adding new pages to the web site which might benefit possible other readers as well.

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Appendix: Calculation of the change of total energy of a free falling object

For brevity the author doesn't make here distinction between vectors and scalars, assuming that the movement is along a straight line and all vectors are parallel to this line. There is no loss of generality since the same reasoning may be repeated separately for each spatial axis with the same result.

The total energy of any object is

  E = mc2 (1)

where m is inertial (a.k.a. relativistic) mass of the object and c is speed of light) and so the differential of total energy while the object is falling is

  dE = c2 dm + 2mc dc (2)

Inertial mass of the falling object is

  m = m0/sqrt(1 - v2/c2) (3)

where m0 is rest mass of the object and v is its velocity.

Since at free fall in "gravitational field" with acceleration g, velocity v2 = 2gx where x is the distance by which the object fell (in direction of its acceleration g) then from eq. (3)

  m = m0/sqrt(1 - 2gx/c2) (4)

Replacing for brevity m0 by m (since m = ~m0), and ignoring small higher order terms, for the object that fell by dx

  dm = (mg/c2)dx (5)

The observed change in speed of light (as demonstrated later via the principle of equivalence) is

  dc = -(g/2c)dx (6)

After substituting (5) and (6) to eq.(2), we get the change of total energy of the free falling object as

  dE = mgdx - mgdx = 0 (7)

which shows that there is no change in the total energy of a free falling object.

Eq. (6) may be derived from an example with a rocket in space, sufficiently far from all masses to feel no "gravitational field", accelerating letís say as much as the objects that fall on the earth. If there is a light ray that enters the rocket perpendicularly to the direction of acceleration of the rocket the observer in the rocket will feel the gravitational field (the same as on the earth with the accuracy to the inhomogenity of the earth 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. The accelerating with the rocket observer however will see the light ray bent towards the rear end of the rocket (assuming that the rocket accelerates forwards). In the relation to the rocket 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 (8)

where t is the time that take the light to cross the rocket.

In relation to the accelerating with the rocket 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 crosses halfway through the rocket the tangent to the circle at the point of leaving the rocket. It makes the angle between these tangents (the angle of the deflection of the ray)

  Q = h / (tc/2) (9)

Substituting h from (8)

  Q = g t / c (10)

According to Einstein's principle of equivalence of acceleration and "gravitational field" this case is identical (with the mentioned above accuracy) to the case when the light ray moves across a rocket that is standing on the earth, and so the ray bends in the "gravitational field", the same as the ray seen by the observer in the accelerating rocket. Since, as we know from the observations of light bending in the vicinity of the sun, half of the bending comes from the changing speed of light (the time dilation) and other half from the curvature of space, we have to take the half of Q and find the change of the speed of light with the distance in field g. From elementary optics the angle of deflection

  f = (dc/dx) t (11)

where dc/dx is change in speed of light per unit of distance in direction along g and, as before, t is the time that take the light to cross the rocket. Comparing (10) and (11) in which f = Q / 2, and restoring the sign according to direction of acceleration g of the objects falling inside the rocket ("down") as we had it in (6) we see that the same as in (6)

  dc = -(g/2c)dx (12)

quod erat demonstrandum.

While doing the above calculation one may notice how the inability of the nature to produce energy from nothing (expressed by the principle of conservation of energy) forces the relative increase in amount of space to be equal to the relative decrease in the time rate. Which is one more interesting feature of physics of Einsteinian Gravitation.

Another interesting feature of Einsteinian Gravitation is the ease with which it produces equation for Newtonian gravtational force F that acts on any mass m restrained in a gravitational field g. It's obtained by simply differentiating the gravitational potential energy contained in that mass, that turned out to be expressed by eq. (1) along a vertical path x and substituting into it dc/dx as expressed by eq. (12):

  F = -dE/dx = -2mc(dc/dx) = mg (13)

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