Relativistic Geometry

Any application of the natural laws to a discrete portion of cosmos requires the definition of a system and its surroundings¹. A system, in this case, can be any object, any region of space, or spacetime set apart mentally from everything else, which then becomes the surroundings.

The general relativity theory describes the universe, the four-dimensional spacetime as a whole, without reference to the surroundings. The theory concerns mainly on the intrinsic properties of the spacetime and tacitly considers the latter as a system having no surrounding or a surrounding which is null and void. This intrinsic geometry takes no account of the distinguishing characteristics of spacetime as they might appear to an observer located outside the system.

Such a premise, i.e. four-dimensional spacetime without surrounding, could be premature since at the current stage of development the physicists have been forced to deal more with higher and higher dimensional hyperspaces. The curvatures of the four-dimensional spacetime, for example, could only take place if the spacetime is embedded in a much higher dimensional hyperspace; otherwise, it could only be flat.

The ten-dimensional surrounding hyperspaces

The concepts of the geometry of n-dimensional metric manifolds (hyperspaces) are straightforward generalizations of ideas of the study of surfaces embedded in the three-dimensional space. A generalization of the concepts of curvature and torsions to curves embedded in the n-dimensional manifolds is direct and straightforward, but matters become rapidly involved when one comes to consider hypersurfaces².

What we are interested the most, in this case, is about the circumstances in which an m-dimensional curved [Riemannian] hypersurface, R_m, can be embedded in the n-dimensional [Euclidean] manifold.  Concerning the global embedding of the whole of R_min E₃ almost no general results are known, however. It is possible to prove that a neighborhood of R_m can be embedded in E_n if at least n =  ½ m(m+1).

The curved four-dimensional spacetime of the general relativity theory can only be embedded at least in a ten-dimensional hyperspace. The ten independent components of the metric tensor in such a system are nothing but the dimensions of the surrounding hyperspace. The empty surrounding space (having zero dimensions?) as assumed in the general relativity is an oversimplification of the reality. We do not have to wait until the advent of the string theory only to be aware of the requirement of such a higher dimensional ambient hyperspace. Both macroscopically (the general relativity theory) and microscopically (superstring theory) require at least ten-dimensional space to preserve the proper applicability of physical laws. A question naturally arises: Do we need to have hypothetical tiny curled extra-dimensions?

Hypersurface vs. hyperspace

It is often more convenient to generalize the ideas of the study of surfaces embedded in the three-dimensional space depicting a group of hyperspaces. As the generalization of the idea, we depict a spacetime as a hypersurface embedded in a higher dimensional metric manifold (hyperspace) representing its surrounding. We conventionally define that in an n-dimensional framework, we have (n-1) dimensional hypersurface embedded in n-dimensional surrounding hyperspace, unless it is defined otherwise.

The advantage of using such a model is that we can better describe the dynamic of the system, for example, the rotation movement of the hypersurface around an axis located across its surface describing a colossal cycle of closed time-like curves. We can also describe the possible rotation movement of the hypersurface around an axis normal to its surface to explain the constant rotation of the solar system, galaxy, super-galaxy and so forth.

Another advantage we get is that we can take into account the geometry element that hitherto overlooked, i.e. the "thickness" of the space or hyperspace which is essential in revealing the quantum phenomena. The thickness of our 3-dimensional space, for example, was found to be equal to Planck distance of 10^-33 cm or is equal to the Planck instant of time which is 10^-44second. Nature abhors any object or shape to have zero thicknesses; otherwise, it will evaporate into thin air. Space and hyperspace have no exception.

We should bear in mind that what we are talking about the hypersurface here is not analogous to a piece of paper floating freely in the air (as in the case of "brane" theory), but more to an interface of an oil-water system. The hypersurface or more precisely hyper-interface locates in between two immiscible "fluids," which we refer as the opposing elements of the energy as a whole: the positive and negative energies. Here again, we can study the hyper-interfacial tension of the system as related to the gravitational constant (G).

References:

1.     Abbott & Van Ness: Thermodynamics, Schaum's outline series, Mc Graw Hill Co., New York, 1967

2.     Sokolnikoff, I.S.: Tensor Analysis, John Wiley & Sons, Inc., New York, Second Edition, 1964

Relativistic Geometry

Einstein's Biggest Blunder

Einstein always said that the biggest blunder he ever made in his life was his introduction of the cosmological constant in his general relativity theory. But in fact, it was not. The unawareness about his misconception on the nature of spacetime was instead his biggest blunder.

Let’s see what I mean by this. On the inextricability of the spacetime Einstein said1:” the non-divisibility of the four-dimensional continuum of events does not at all, however, involve the equivalence of the space coordinates with the time coordinate. On the contrary, we must remember that the time coordinate is defined physically wholly differently from the space coordinates“. With all due respects to Einstein, this is obviously nothing but a vicious-circle statement.

Multidimensional time

He rejected the belief on the division of the four-dimensional continuum into three-dimensional space and a one-dimensional time continuum. Indeed, we don’t have any problem with this, but what we concern most is that when we divide the four-dimensional continuum in two we will get a three-dimensional space taking place in between the four-dimensional time continuum. Hardly anybody is aware of such obvious logical fact. What we are used to perceiving the reality of one-dimensional time is, in fact, the resultant of these underlying four dimensions of time.

Indeed, the spacetime continuum as a whole should be both homogeneous and isotropic. It should have equivalent dimensions i.e. “pure” time dimensions, in the sense that they are undivided. As such, the spacetime is timeless; there is no past, now and future. It is also spaceless since the “now” is not yet present (a). There could be no matter or light to be created in such a proto-world, it is in total chaos.

The Slit Spacetime

The creation of material things can only be taking place if such a world is split. By splitting the four-dimensional world into two halves, a three-dimensional space (hypersurface) is created in between the two. The original time dimensions at such newborn space are transformed into different kind of dimensions (spatial dimensions) because of the effect of the hyper-interfacial tension (b) of this hypersurface (c). This spacetime’s split is analogous to the split of oil and water, a phenomenon we can observe in our daily life.

This split spacetime is what truly representing the actual world. Einstein and, alas, the mainstream physicists have erroneously taken this chaotic proto-world to represent the material world. No wonder they are obliged to set up light-cone frameworks at any point of such spacetime to preserve order and causality.

Einstein rejected from the outset the concept of absolute simultaneity. He developed his concept of relative simultaneity by giving a central theoretical role to the events and the propagation of light, in that it founds the concept of time upon the law of propagation of light.

But the notation of event is just another word of point coordinate, occurrence, happening or incident with no reference to the state of being or the existence of things. The good reference to define simultaneity is, therefore, not events measured through the propagation of light but the existence of fundamental (ephemeral) things such as those of quantum particles.

The simultaneity of particles’ creation and annihilation

So far we haven’t yet discussed how and why the spacetime is splitting. The special relativity theory has unified the fundamentals: space, time, energy and matter into two; the spacetime and energy. The general relativity reveals that the spacetime is not an independent reality as what we might think. The spacetime is not like the container independent to whatever fills it neither the energy something that fills the container. The grand unification of the relativity theory of those two leads us to conclude that energy is the only real and independent thing in nature while the spacetime is to fade away into merely the structural quality of energy.

Now, the energy by itself composes of positive and negative energies as formulated in the relativistic energy equation E2=m2c4+p2c2. Eventually, the energy as a whole breaks up into its opposite elements and splits accordingly the spacetime into two halves. The three-dimensional space is taking place in between the two as was discussed in the foregoing.

The interplays between the two opposite energies across through the 3-interface ignite quantum sparks (“quarks”) (d) which we perceive as the quantum particles. These “quarks” are short living (ephemeral) and appear to be persistent only because of the perpetual interplays between such opposite energies. We perceive these continuous interplays as perpetual creation and annihilation of quantum particles giving rise to the perception of the passage of time and the time direction: the past, now and future. The creations [and annihilation] of the quantum particles are taking place simultaneously giving rise to the universal now. The three-dimensional space and the universal now are the different aspects of the same thing.

This may answer to Einstein's worry about the nature of Now. Einstein himself denied the absolute character of simultaneity and thus the existence of the universal now. In the relativity theory that is currently conceptualized nothing corresponds to the experience of Now. Reporting a discussion, the philosopher Rudolf Carnap wrote3:”… Einstein explained that the experience of the Now means something special for man, something essentially different from the past and the future, but that this important difference does not and cannot occur within physics [… ] so he concluded that there is something essential about the Now which is just outside the realm of science”.

The reality of Now is nothing but space itself; it is not outside the realm of science. Referring the four-dimensionality of spacetime Minkowski did not explicitly deny the existence of space and concluded:” we should then have in the world no longer space, but an infinite number of spaces, analogously as there are in three-dimensional space number of planes. Three-dimensional geometry becomes a chapter in four-dimensional physics. Now you know why I said at the outset that space and time are to fade away into shadows, and only a world in itself will subsist” 2.

This space or spaces are not necessarily flat as in the case of Minkowski’s spacetime. This gives rise to local times along the different places along the curvature of the interface whose temporal dimension is different from that of the universal now. The current modern physics theories require higher and higher dimensions, in essence, are the endeavor to get the simultaneity of the existence of things (flat hypersurface) that might happen only at the higher temporal dimension. Only then the grand unification theory may be fully accomplished.

Notes:

1. Space and the universal now are different aspects of the same thing.

2. It is directly related to the nature of the gravitation constant.

3. We refer to concurrently space as hypersurface, 3-space, 3-surface, or 3-interface.

4. The notation of this “quark” (quantum spark) is more general than the conventional quark specifically representing the smallest element of the atomic nuclei.

References:

1. Einstein, Albert: The Meaning of Relativity, Princeton University Press, New Jersey, Fifth Edition, 1954

2. Einstein et al.: The principle of Relativity, Dover Publications, Inc., New York, 1952

3. Barbour, Julian: The End of Time, Phoenix, London, 2001

Einstein's Biggest Blunder

Genesis' Firmament

Hardly anybody is aware that the technical recount about the nature of the universe can be traced back far in time to the dawn of the history. Based on the scheme corresponding with the King Lists compiled by Sumerian and Babylonian scribed about 2000 B.C., we can find the respectable king under the name of En-me-en-dur-an-na (Enmenduranki) who was believed to know about the secret of heavens, as the seventh ruler in the Mesopotamian dynasties reigning before the Flood1.

Enmenduranki was believed to have ascended into heaven, learned all the secrets of divination and thereby became the source of all human knowledge. The legend tells that while he was in the divine assembly, he was set on a large throne of gold and shown how to observe oil on water …2

It seems so straightforward that nobody must spend so many endeavors only to get such a simple thing. It was elementary teaching indeed, but hitherto nobody comprehends it, and so the knowledge of heavens remains buried in secret.

Let us talk a little bit more technical. Having an oil-water mixture, we observe that the oil and water are separated by a very thin surface (interface) which is a little bit tenser than the surroundings because of the effect of something we know as the interfacial tension.

Now, let us concentrate on this interface and compared it with the Genesis' firmament:

"And God said, "Let there be a space within the water, and let it separate between water and water." So God made the space, and it separated between the water that was under and the water that was above the space" (Genesis 1:6-7)3.

Fig. 1. Interpretation of the firmament as a 3-dimensional interface in a 4-dimensional frame of reference

The interpretation of the notions of Enmenduranki’s interface and Genesis' firmament is the same which is the space. This particular interface is three-dimensional and, therefore, what Enmenduranki meant by the oil and water are nothing but four-dimensional fluids. By the same token, what Genesis means by water is nothing but four-dimensional fluid.

The word space or more precisely firmament in English is the translation of the Hebrew word of “raqia." The root of this word refers to how a goldsmith hammers gold leaf very thin3. This connotation further supports the “technical” depiction of the space as a very thin interface just like what Enmenduranki did.

It suggests that everything in nature, space or higher dimensional hyperspace included, must have a thickness; otherwise, it would vanish into thin air. The consequence of this is that space [or hyperspace] must always be embedded in the higher dimensional surrounding(s), or else the former would have no thickness at all and would be completely flat.

We can draw other information that is related to the geometry of Genesis firmament can from Summerians ideas about Cosmos. For the Sumerians, the universe was tripartite structure – heaven, earth, and the netherworld. It is unmistakably comparable to Enmenduranki’s oil, interface wand water or the Genesis' water above, firmament and water below, respectively.  The Sumerians also used the word 'tin' representing the "metal of heaven." It is the oldest version of the Genesis firmament. According to S.N. Kramer, it may be that the Sumerians thought that the floor of heaven was made of tin or some comparable metals2. It is the one example of inaccurate interpretations which downgraded valuable ancient knowledge.

Do we know how thin our space is? Space's thickness should not be zero but a little bit thicker. Based on the current knowledge, we may put forward the number of 10-33 cm, the Planck distance, as the minimum thickness, below which no tangible thing would exist, only energy does. No skillful goldsmith could hammer the gold leaf so thin, not even in Moses magical time.

Why did the ancients talk about a very thin gold, tin or another comparable leaf of metals? The answer is because this very thin interface is the only place where material things are created giving an appearance as though it is a large piece of thin solid gold.

So how about Genesis' four-dimensional water or Enmenduranki’s four-dimensional oil and water? What is their actual physical reality? At everybody surprise, the answer is that it is nothing but the [four-dimensional] energy. The water above the firmament and the water below the firmament is the ancient representation of the opposite elements of energy: the positive and negative energies.

This elucidation about the creation as the act of separating a preexisting thing ("water" or energy) instead of creation out of nothing naturally raises a critical question. Which one represents the truth: Genesis Cosmology or Big Bang? The answer is the former.

References:

1.  Barmachi, Faraj: Treasures of the Iraq Museum, Iraq Ministry of Information, Baghdad, 1976.

2.     Wright, J.E.: The Early History of Heaven, Oxford University Press, Oxford, 2000.

3.     Friedman, R.E.: Commentary on the Torah, Harper San Francisco, New York, 2001

Genesis' Firmament