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Wednesday, April 14, 2010

On The Nature Of Spacetime

Albeit their great success in developing a strong foundation of modern physics, physicists are still unable to reconcile the two most fundamental but contradictory theories: the relativity theory and quantum mechanics. Any attempt to reconcile those two theories was simply doomed to failure. Quantum theory requires "weird" things such as discontinuity, non-locality, and non-causality as its key features, whereas relativity theory implies the ordinary.
There have been many discussions on the correctness of those fundamental theories, and yet we are still far away from the end. In the famous grand debate between Albert Einstein and Niels Bohr in the late of 1920s, Einstein refused to accept the Copenhagen interpretation on the quantum reality which Bohr was the proponent. Einstein stubbornly maintained his standpoint until he died. Physicists continue at trying to unify those theories until recently, but surprisingly enough they spent most of their time on abstract mathematics and hypothetical objects which probably has no physical significance at all.
Scientific progress can lead to a kind of forgetfulness. We often take for granted the premises that prevail in our time as the starting points for our thinking. We must remember that the foundations of those premises get covered over by a series of answer and question built upon them. The new questions are built from the answers that were given to the old questions which are now no longer asked1.
The special theory of relativity is one of such foundations we like to revisit. Most physicists believe that the special relativity theory is a finish theory and there should be no more questions about it.

Energy, the Primordial Substance
The special relativity theory concerns the fundamentals namely space, time, energy, and matter which nature has puzzled people for hundreds of years. The theory has unified those fundamentals into two distinct entities: the spacetime and energy.
Einstein unified the energy and matter into a single substance: the energy. A small amount of matter can be converted into a vast amount of energy that we can tap as we wish, for good or evil purposes. This discovery, expressed in Einstein famous equation E = mc2, becomes one of the basic foundations of the progress in our modern life.
Another remarkable achievement in the relativity theory was the discovery of the spacetime. Hermann Minkowski (1908) unified the space and time into a single entity called spacetime and demonstrated that as an individual, space and time are incidental to the four-dimensional spacetime. He bluntly stated2:" space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."
Einstein failed to grasp the opportunity to put these two into unification and blending them to build a grander theory. He leaped, instead, to study the effect of energy [and matter] on the curvature of spacetime to solve the problem of gravity which was his primary obsession.
So the rest is history.
Spacetime and energy are not in an equal footing as we think. The spacetime is not like a sort of container and energy something that fills the container. The energy and spacetime, on the contrary, are inextricable just like water substance and its spherical form in a drop of water. In plain English, the energy is noun and spacetime adjective, i.e. the geometrical quality of energy.
The spacetime does have no existence on its own. We would rather say in Minkowski's style:" the spacetime by itself and energy by itself are doomed to fade away into mere shadow and only a kind of union of the two will preserve an independent reality.”
As a complete theory, the special relativity should assert that energy as the only independent reality in nature. It is primordial; neither can be destroyed nor created, omnipresent; permeating throughout every part of an object or a place. It is the source of everything else. It is the underlying reality of the energy conservation principle, the most fundamental law of nature.

The Spacetime’s Dimensions
The dimensionality of the spacetime reflects the degree of freedom or potency of energy. The special relativity theory reveals that energy is four-dimensional a). The ancients called this four-dimensional primordial substance: water. Apparently, what they meant was watery like substance, not the ordinary one. We may call it "eter" b) (energy-water) to distinguish it from the conventional notation of energy as we know it in our daily life.
Nobody can perceive four-dimensional energy as a whole. What we know about energy in our daily life is only superficial. We know from the school3 that energy seems to be a mere mathematical abstraction that has no existence apart from its functional relationship to other variables or coordinates. We are used to be trained to see the world upside down.
Some advanced theories demonstrate that quantum interactions require higher dimensions. The superstring theory demands ten dimensions. Why ten dimensions? When Einstein postulated that the spacetime could be curved under the effect of energy, he saw it only intrinsically and just ignored what kind of surroundings which allow this to happen. He did not bother that for such a 4-dimensional spacetime to be curved, it needs at least ten-dimensional embedding expanses. It is demonstrated in the corresponding metric tensor which requires ten independent variables. 
Viewing extrinsically these independent variables are nothing but the dimensions of the embedding surrounding space. If the surrounding expanse is just five-dimensional, then the embedded 4-spacetime should be nothing but flat. As a general rule4, we can extend this phenomenon to the spacetime of infinitely many dimensions. The following question naturally arises: Under what circumstances an n-dimensional manifold (expanse) can embed the m-dimensional hypersurface? The answer is that the surrounding expanse should have at least n = 1/2 m(m+1) dimensions.
It certainly opens to a new paradigm that may lead us to the infinite realm; not in the sense of infinite of its expansion but its dimensions. There is in nature a broad spectrum of energies of various potencies and degree of freedoms a). They superimpose each other with their relative position subsequently analogous to the position of a point to a line, line to surface, surface to space and space to hyperspace and so on (Fig. 1A and 1B). The point in this illustration (Fig. 1B) is our dear little universe where all of its planets, stars, galaxies, cluster of galaxies and so forth embedded in much higher dimensional worlds.
Figure 1. The relative position of  multidimensional worlds
The Undivided Spacetime

In its originality, as a single undivided whole, the spacetime by itself and from its nature is shapeless and formless. It is timeless in the sense that there is no clear cut between the past, present, and future. It is also space-less in the sense that there is no locality for world events to take place. As such space and time have no physical reality. Not even a single light or matter can exist; it was in total darkness. The ancients referred to this primordial unity a cosmic egg and its respective condition watery chaos.
Within such an undivided spacetime no spatial dimensions yet to be present. As such, all dimensions are temporal c). The modern physics has taken for granted this undivided whole four-dimensional spacetime to represent the real physical world. It is the root of the troubles that we have now in physics. Strangely enough, most physicists seem quite comfortable and believe that nothing in relativity corresponds to the universality of the experienced now. For physicist, the universal now is against the basic principle of the relativity of simultaneity.
Einstein seemed a little bit uncertain about his concept of spacetime but was reasoning in somewhat vicious circle5: “the non-divisibility of the four-dimensional continuum of events does not at all, however, involve the equivalence of the space coordinate with the time coordinate. On the contrary, we must remember that the time coordinate is defined physically wholly differently from the space coordinates.”
So physicists used to feeling comfortable to assign the differentiated temporal and spatial dimensions at any point throughout the spacetime. What they do is assigning the coordinate of events, not the state of being. Having done that, they obliged to set up a different frame of reference in every point throughout the spacetime which can be thought of in terms of a light cone to avoid disorder and preserve the causality.
In his book, Barbour6 underlines Einstein’s concern about the problem of now: “… that this experience [of now] cannot be grasped by science seemed to me a matter of painful but inevitable resignation. So he concluded that there is something essential about the now which is just outside the realm of science.”

Yes, Einstein himself who did put the reality of now outside the realm of science. He was not aware that the spacetime could not remain intact. The highly symmetric spacetime is extremely unstable. Eventually, it breaks its symmetry as a result of the segregation of its composite: the repulsive positive and negative energies. The spacetime splits in two and creating in between a hyper-interface which is nothing but the 3-space, the universal now d).

The Split Spacetime
The creation of our material world had only begun when the spacetime became divided. The split of the four-dimensional spacetime creates a three-dimensional space just like the separation of oil and water creates an interface in between - a simple phenomenon that we can observe in our daily life.
The 3-interface (space) is not necessarily flat but could be quiet rough, locally more or less curved, having small and large holes here and there and accordingly its corresponding time’s direction at every point in its surface is also quite irregular. The “thickness” of the space is the instant, an infinitely small fraction of time (10-44second): the experience now. The whole space is the universal now, and the now is nothing but space as a whole d).
It is noteworthy to mention that the ancients used the same phenomenon to explain the creation of the universe. The ancient people believed that through the division of primordial unity (cosmic egg), space was created and order was restored. Through the separation of heaven and earth (the metaphor of something which is above and below), cosmos was created out of chaos. Genesis refers to the division of waters which were under and waters which were above that created a firmament in the midst of those waters. Apparently, both stories refer to the split of higher dimensional primordial unity or "waters," and the firmament - the interface in between those waters - is nothing but space itself – our universe.
The root of the word firmament in Hebrew “raqia”7 refers to how a goldsmith hammers gold leaf very thin. Had biblical scholars known about how thin the firmament is they will be speechless. Indeed, the thickness of the firmament - the thickness of the space, our universe - is the Planck length unit which is 10-33 cm [equivalent to 10-44 second], the smallest universal unit that still has physical meaning. No skillful goldsmith in the entire world can hammer the gold leaf so thin, not even in Moses magical time. However, it should not surprise anybody. Had our universe has no thickness at all it would completely vanish into thin air. 
Symmetry Breaking
We did not yet discuss in detail why and how the spacetime split on itself. The spacetime in itself seems like a sort of substantial false vacuum. However, it was not an absolute void, but energy; otherwise, there will be no spacetime at all. The special relativity theory reveals that energy consists of the positive and negative energies, as derived from the relativity energy equation E2= m2c4+p2c2. These opposing energies are required for the creation of a physical world to happen without which the world would remain intangible, static and chaotic.
In the beginning, there was potentially as much positive as negative energy such that the spacetime was perfectly symmetric.This highly symmetric condition was precarious and eventually led to a catastrophic symmetry breaking. As it happened, the spacetime turned up to become polar with its opposing properties segregated in its two opposite poles (Fig. 2a to 2b).

Figure 2. The segregation of the positive and negative energies

As the opposite energies are repulsive, sort of currents of energies circulates in both sides of the divided spacetime. It makes the 3-interface, located in the middle of the two opposite energies, rotating in the direction normal to its surface, i.e., the fourth [temporal] dimension. The ancient depicted this rotating universe (3-interface) in the direction of the temporal (fourth) dimension as symbolized as a rotating globe representing the dynamic character of yin and yang. We will give a little secret to understand the real physical meaning of it that nobody knows for thousands of years. This symbol, the ancient Chinese diagram representing the Ultimate Supreme could only be understood if it is viewed from the four-dimensional perspective (Fig. 2c to 2f). Niel Bohr takes this symbolism as its coat of arms14, but it seemed that he did it only intuitively and did not really know the reason why he did it.
As the result of interplays between positive and negative energies, different sorts of [quantum] fields and forces are created piercing across the interface (at the distance of 10-33 cm). A sort of 4-"quantum electric" currents piercing through the interface (space) inducing [three-dimensional] "magnetic" and electromagnetic fields propagating across space. The ancients marked this happening with an imperative expression: "let there be light!"
Other sorts of quantum (Higgs?) fields generate [quantum] sparks (quarks d)) at the 3-interface giving birth the fundamental particles, the building blocks of the universe. The interplays between the two opposing energies produce constant transmutation and changes at the rate equal to the speed of light.
Nothing at the interface stands still, and nothing remains constant; nothing persists and exists on its own. Everything is under perpetual creation and annihilation that happens non-locally as a single whole. Fundamental particles that perpetually appear and disappear at the interface analogous to the pixels appearing and disappearing at a giant TV screen, which together form the whole display. The display is continually changing following the change of the pattern of the pixels as though they are entangled as a single whole. It is the underlying reality of the non-locality in the quantum realm.
Although the opposing properties of spacetime have segregated on two different sides at an equal amount, its distributions on each side are not perfectly even. As a consequence, the interface (space) in between the two is not perfectly flat but rough and lumpy, and possibly there are some holes e) here and there and locally more or less curved depending on the agglomeration of matters created at that particular locus. 
Fundamental particles that are “simultaneously”d) created as a single entangled whole, under the influence of those mentioned above three-dimensional "magnetic" fields, attractively interact with each other that keeps them from going asunder. It is just like the interfacial tension (the gravity constant) that keeps the interface of oil and water from breaking apart. It is precisely the phenomenon of gravity that Einstein had elegantly embodied in his general relativity theory.
When the spacetime has broken up in two, the split (crack) might create the interface in two possible ways: either the interface was instantaneously created across the whole spacetime, or it was gradually created starting from a certain point then "stretched out" across the spacetime. The first case corresponds to the concept of a static universe, while the second case the expanding universe as the alternative to the Big Bang theory. The so-called starting "point" in the latter case is more like a little bubble expanding in the "water" rather than a singularity expanding ex nihilo. The split (crack) that propagates in an accelerating speed across the spacetime gives the perception to the existence of some extra energy driving the expansion of the universe. The mystery of dark energy and it sorts may be revisited in this perspective.

Ephemeral Particles – The Missing Link
We used to imagine that when we continuously divide a chunk of matter into its smaller and smaller parts, we will eventually come to the level where we cannot divide the smallest parts anymore. The logical reason for this is because those fundamental particles have no more internal structure that can be divided any further. These tiny particles are like billiard balls resting or moving around across space just like what happen in the macroscopic world.
Here again, we call for a paradigm shift. We can repeat the process of the continuous split but with different logical thinking. When we continuously divide a chunk of matter into its smaller and smaller parts, we will eventually come to the level where we cannot divide the smallest parts anymore. The logical reason for this is because those fundamental particles are so small and unstable and immediately disappear into thin air. A fundamental particle is spatially and temporally discrete (ephemeral).
The fundamental particle - whether it is a point-like particle, string, membrane or else – should not only be the smallest but also ephemeral. Nothing subsists in two different instants. Everything exists only instantaneously. In every brief moment, there exists a gap, an abysm of non-existent, between two consecutive existents. At every instant everything is entirely new, wholly cut from before and after. The continuity and persistence of particle at a large scale can be regarded as an underlying appearance reality of sub-atomic reality which is discrete and ephemeral. In the macro world, the persistence of matter is the appearance of perpetual creation and annihilation of the quantum particles.
In the quantum world, no such a single particle permanently exists. A single particle that we believe exists within a specific lap of time more or less long, is, in fact, a series of different but similar particles which are created one after another. In the quantum world we can not throw the same dice many times, but we throw myriad ephemeral dice one by one. What we thought as a single particle taking place simultaneously in two different places (as in the case of the double slits experiment) was, in reality, two similar but different particles created and annihilated almost instantaneously. A quantum measurement is a snapshot of just only one particle out of myriad particles which rapidly appear and disappear one after another at the rate equal to the speed of light.
It is the dice that the physicists are looking for, the dice that Einstein was not aware of. When Einstein played dice with Bohr, just near the end of the game, Einstein unintentionally put the only ephemeral dice in his pocket and while leaving said: "God does not play dice."
Defying our common sense we should make up our mind that space is somewhat being built up from the particles instead of particles inhabiting a previously assigned spatial geometry. Roger Penrose11 remarkably described the Galilean dynamical concept of space similarly as the dynamical concept of ephemeral particles. He remarked that according to Galilean relativity:" the notation of the physical space seems to be of something that evaporates completely as one moment passes, and reappears as a completely different space as the next moment arrives. There is no single space to be as an arena for actions of the physical world evolving with time; there is instead a different space for each moment in time, with no natural identification between the various spaces. There is no dynamical meaning to saying that a particular point in space is, or is not, the same point as some chosen point space at a later time. Every succeeding moment represents a new universe, similar to preceding (vaporized) universe, but also different."

Alas, Roger Penrose rejected such a beautiful dynamic picture of perpetual creation and annihilation of the ephemeral universe as a physical reality. By adding the feature of curved space to that dynamical space concept, it would become in total agreement with the ensemble relativity concepts from the time of Galileo up to modern physics.

The Flow of Time
We are unaware of those constant changes since the lap's time of the individual sequence of creation and annihilation is at the order of 10-44 second - the Planck time unit, the smallest universal time unit that nature allows corresponds to the changes of spaces that take place at the speed of light. The ordering of time, the essential ingredient of the dynamical concept of time is exactly the appearance of underlying sequences of creation and annihilation that take place at the speed of light. When new particles come into existence, it seems to us as the unformed future is transforming itself into the reality of now, and when they vanish, it seems to us that the reality of now is dissolving out to the past.
This perpetual creation and annihilation are underlying the reality of the flow of time. The direction of the said electromagnetic waves and thus the irreversibility of the constant change and transformation sequences determine the direction of the time flow. It is the fundamental phenomenon that explains why the speed of any given object in (three-dimensional) space cannot exceed the speed of light which is the basic foundation of the special relativity.
As stated in the preceding because of the repulsive force of the positive-negative energies the 3-interface (space) is rotating in the direction normal to its surface (the fourth dimension). Roger Penrose11 strongly ruled out the closed time-like direction curves as unphysical for it will violate causality. We wish here to emphasize, however, that a rotating universe around a closed time-like direction does not necessarily violate causality and it is incredibly plausible.
However, one question is still open. If the universe does rotate around a closed time-like direction, then how could be that nature allows time goes forward on one side of the universe while moving backward on the other side? The only explanation to this is that only matters are created on one side and antimatters are equally created on the other side of the universe as such that matters go forward in time on one side and antimatters move backward on the other side (Fig. 3). The effect is that times will pass forward on both sides.
Figure 3. A closed time-like rotation

As we can see, the above elucidation answers altogether the mystery that long puzzled the physicist about the absence of antimatters in this corner of the universe where we are around and the possible existence of "dark" antimatters on the other remote side of the universe. The concept of the revolving universe seems to have been known since time immemorial as schematically depicted in the revolving globe of "yin and yang" that we are all familiar with but without understanding about the real meaning of it.
It is advisable to abandon once for all the futile debate about the time machine and with all due respects put Stephen Hawking's chronology protection12 in a time machine museum for there is no more worry about someone who wishes to travel back in time to kill his or her grandmother.
Multiple Time Dimensions
So far we have been dealing with the 4-spacetime as a stand-alone system with no consideration of its surroundings. However, the current findings led us to a theory that requires larger dimensions - many more than meet the eye. Theodore Kaluza (1919) first proposed a new concept that describes interactions that occur in the spacetime whose dimensions increase to five. By doing that Kaluza and Klein had effectively incorporated electromagnetism into a geometric framework, just like Einstein (1915) did similarly for the gravitational field into the geometric of the four-dimensional spacetime. Werner Heisenberg (1929) indicated that the quantum theory required a space of infinitely many dimensions, although he believed that the space was not real but imaginary.
So today most of the physicists have to deal with higher-dimensional spacetime and are used to working comfortably with, such as the ten dimensions that are required by string theory. Moreover, yet, hardly anybody can fully grasp objects having more than three dimensions and tend to treat them as a mathematic object rather than tangible things.
However, as did Kaluza and Klein who assumed that the additional fifth dimension had curled up into the folded fabric of cosmos, the string theory also includes such speculation for the extra dimensions into tiny loops. So this is the source of trouble. We do not need to adopt the concept of curled up dimensions just only because we can not see them. This misleading premise leads us to so many bizarre things that we have to confront, such as the hypothetic Callabi-Yau little things that so much exhausted our precious energy to study it.
We wish to emphasize, instead, that all of those extra dimensions are well extended, and there are as many as extra extended dimensions available whatever required by the theory. However, why then we can not see all of them? Again, the relativity invariance principle rigorously maintains the space and time symmetry and allows the transformation of space and time from one to another depending on the frame of reference we are taking on. When we view it intrinsically (from within), the extra dimensions look like temporal dimensions. However, if we view it extrinsically (from outside) then all dimensions look like spatial dimensions except the highest one remains as a temporal dimension. Under such premises where we have multiple temporal dimensions, we can explain the non-locality, the entanglement phenomenon, the universal simultaneity which takes place at a time of higher dimension.
However, why and how nature has such ten-dimensional spacetime in addition to the ordinary four-dimensional spacetime? The "official" explanation to this question is that the ten-dimensional spacetime cracked in two, creating two separated worlds: a four-dimensional- and a six-dimensional spacetime. The four-dimensional spacetime expanded explosively, while the six-dimensionally spacetime contracted until it shrank to a tiny size. This mode of splitting is very odd just as odd as splitting a three-dimensional cube in two part to get a plane and a line. One should do. Instead, a series of splits starting from slicing the cube to get planes and then cut the plane to get lines and cut the line to get points.
The initial conditions of the ten-dimensional spacetime were similar but more intense than the ones of the four-dimensional spacetime. When eventually this ten-dimensional spacetime broke up in two parts, a nine-dimensional spacetime came into being in between those two — the split of the later created in turn an eight-dimensional spacetime. The same process continued successively until finally the four-dimensional spacetime was born. In the end, we have a total of seven spacetimes ("heavens") coexist and overlap one within another at a different level of dimensions. Each heaven has its dimensional system - its own space and own time - and, hence, its corresponding light. Had we told the above process in a little bit different way one would believe to have been told a biblical genesis with its seven days of creation.
It is now possible to geometrize and visualize all interactions and forces under the multidimensional framework that we have established so far. We can depict them as an interface of a 4-dimensional oil-water system. The geometry relationship between space and time: space can be depicted as the 3-interface and time the 4-continuum (the continuum above the interface represents the future, the continuum below represents the past or vice versa depending on the rotational direction normal to the surface of the interface). Matter and energy: matter can be depicted as created (and annihilated) at the 3-interface, energy as the 4-continuum (the continuum above the interface represents negative energy, the continuum below represents the positive energy or vice versa again depending on the rotational direction of the interface). From those analogies, we can deduce the relationship between the strong and weak forces, the weak and electromagnetic forces, the electromagnetic and gravity forces and different kind of quantum (Higgs fields?), or else as you wish.
A Journey to the Frontier
The superstring theory and the relativity's 4-curved spacetime require a surrounding of ten dimensions that bring us to the seven heavens' realm. However, at its current status, the superstring theory is advancing far enough that it demands an additional dimension – the eleventh. The later could be just the tip of an iceberg, the pseudo-dimension which represents the remaining undetected [mathematically] higher [temporal] dimensions.
With the discussion that we have so far, we should say that our imaginary journey to the frontier together with "our ancient friends" will soon come to an end. We have left far behind us our beautiful (3D) space with its myriad stars and galaxies majestically whirling around. Our old friends simply named it the world of "solid." So far we have been passing rapidly through the seven heavens (4D to 10D worlds) that our dear friends symbolized by "waters." They explained further that what Genesis meant by the seven days of creation was nothing but the seven different time dimensions with their corresponding seven heavens. Beyond our 4-world, the time became longer as the dimensions of heaven became higher. The same cases were also valid for their corresponding lights (the ancients call light upon lights); the light in higher dimensional heaven become faster, much faster than our ordinary light.
Heading toward the eleventh dimension we were emerging from “water” and pass across the water’s surface (10D) where we saw the celestial sacred “lotus” was floating around. Above and around us, we have the blue sky - the vast “air” - the eleven-dimensional spacetime and beyond (11D to 26D?). There, we feel like a carp emerging from little Michio’s water pond with its beautiful water lilies13. We are all speechless looking at vibrant lights shining above. In this exciting moment, we take note that these "water" and the vast "air" above should be the first heaven and earth which had been created in "the beginning" of Genesis account. So our friends confirmed that the "beresith," Genesis' beginning, was indeed relative since we still had in front of us the world of "fire" far away beyond the sky above (26D? and higher) which existence preceded Genesis creation.
The last remark that our friends wish to convey to us is about the constant size ratio of those heavens from one to the next. Our oldest friend asserts that the constant relative size ratio between one heaven the next is just like the ratio between the size of a grain of sand as compared to the size of the universe. "So my sons," he concludes "now you know how big the biggest is." Our dear wise old friends are indeed very good at using metaphors to describe reality. They use a pure substance of different phases such as solid, water, air and fire to symbolize the degree of existence comparable to our technical notions of actual, probable, possible and potential. So beyond all is, of course, the impossible.
Having said that we decide to make a detour; otherwise, we will speed up straight into the burning worlds of "fire." To meet your curiosity, you can join later on if you wish to have a one-way ticket mission assigned with only one simple task: going ad infinitum to observe the source of the manifested worlds: the enormous waves of the infinite dimensions of the gigantic ocean of pure consciousness. However, we quite sure that even are your only imagination will be prohibited to pass through; otherwise you will directly know the mind of God.
Notes:
a.  The energy is not only 4-dimensional. It is the lowest among the variety of higher energies in a broad spectrum of energy of infinitely many dimensions.
b.  Eter; it should be distinguished from classical “ether” whose existence was rejected by the relativity theory.
c.  More precisely undifferentiated time dimensions (eon), there is no differentiation of the past, present, and future. Time, like space, is multidimensional. This space-time symmetry is one of the fundamental relativity principles which most physicists take no notice.
d.   The loci of simultaneous events are unnecessary to happen in the fourth [time] dimension; they could take place in a higher time dimension. The universal simultaneity is the basis of the grand unified theory, without which the theory will remain fragmented.
e.    The small holes could be physically sorts of what we know as a hypothetic black hole. 
References:

1.   Simpson, Thomas K: "Maxwell on the Electromagnetic Field," Rutgers University Press, Fourth paperback Printing, London, 2003
2.    Einstein, Albert et al.: "The Principle of Relativity," Dover Publications, Inc., New York, 1952
3.  Abbott and Van Ness: "Thermodynamics," Schaum's Outline Series, McGraw-Hill Book Company, New York, 1967
4. Sokolnikoff, L.S.: "Tensor Analysis," John Wiley & Sons, Inc., Second Edition, New York, 1964
5.  Einstein, Albert: "The Meaning of Relativity," Princeton University Press, Fifth Edition, Princeton, N.J. 1954
6.   Barbour, Julian: "The End of Time," Second Impression, Phoenix, Orion Books, London, 2001
7.  Friedman, Richard Elliott: "Commentary on the Torah," Harper Collins, San Francisco, 2001
8.   Born, Max: "Einstein's Theory of Relativity," Revised edition, Dover Publications, Inc., New York, 1962
9.    Randall, Lisa: "Warped Passages," Harper Perennial, New York, 2006
10. Einstein, Albert: "Relativity, the Special and General Theory," Fifteenth Edition, Crown Publisher, New York, 1952.
11. Penrose, Roger: "The Road to Reality," Vintage, Random House, London, 2005
12. Hawking, Stephen, et al." The Future of Spacetime", W.W. Norton & Co., New York, 2002
13. Kaku, Michio: "Hyperspace," Anchor Books, Doubleday, New York, 1995
14. Capra, Fritjof: The Tao of Physics, Fontana/Collins, Six Impression, Suffolk, 1980


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