The Spacetime Misconception and the Crisis in Physics

The mainstream physicists are still unable to recognize the true nature of space and time, albeit their recognition of the union of the two. The physicists have taken for granted the union which forms a four-dimensional continuum as such as representing the actual universe.

Physically, such a continuum should be homogeneous and isotropic in the sense that all of its dimensions are equivalent. However, physicists seem to lose their physical sense as they assume that the four-dimensional continuum has different dimensions. We can see this confusion from Einstein's statement on the inextricability of the spacetime:" 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 co-ordinate is defined physically wholly differently from the space co-ordinates"¹.

The dimensions of such spacetime continuum should be equivalent and their intrinsic nature is [undivided] time-like in a sense that there is no present, past, nor future ^a). It is precisely the condition which prevails in the world model  (spacetime) that mainstream physics have adopted. The notation of [unsplit] spacetime is better to be replaced by the 'eon' which is more appropriate to describe such undivided time which is quasi-eternity (Figure-1).

Einstein himself was worried about the absence of the concept of Now in modern physics as he said to his friend philosopher Rudolf Carnap. The latter wrote²:"… 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".

It is the background why physics is now in crisis.

Creation by Separation

Physicists are forced to set up a fundamental structure consisting of light cone at every point within the spacetime for the purpose to establish order within otherwise a chaotic world model.  The physicists have to establish such odd construction to preserve the causality because they miss identifying a critical step within the chain of the creation process, i.e., the act of separation, a common phenomenon in physics,  which is often called "symmetry breaking."

At school, we have learned this separation phenomenon, for example, in the electrical process where equal amounts of positive and negative electricity form if we rub a glass rod with a piece of silk. The glass rod becomes charged with positive electricity, and we find a precisely equal negative charge on the silk. This empirical fact shows that friction does not generate but only separates the two kinds of electrification d).

We may think this positive and negative electricity as two fluids that are present in all bodies in equal quantities. In non-electrical neutral bodies, they are everywhere present to the same amount so that their outward effects are counterbalanced. In electrified bodies, they separate. One part of the positive electricity has flowed from one body to another, just as much negative has flowed in the reverse direction ³.

Analogously, the same phenomenon happened in the cosmic creation. The four-dimensional spacetime, which physicists have assumed to be intact,  has spontaneously broken its symmetry as a result of the split of related energy into its positive and negative components (Figure-2). As such the spacetime was split in two, creating a three-dimensional [hyper] interface in between the two halves, transforming the dimensions along the interface into spatial ones.

It is just like the separation of oil and water where we can observe an interface taking place between the two.

Geometrically, we can imagine that the nature of dimensions along the interface is different from those within the bulk of oil and water because of the tension that arises at the interface.

Analogous to this three-dimensional oil-water system, we may posit that the nature of the gravity constant in our four-dimensional world is nothing but the interfacial tension of the 3-[hyper]interface.

Transversality of Light and Hypersurface

The concept of hyper-interface or more generally the hypersurface can be borne out based on the phenomenon of transverse waves. The weird phenomenon that hardly anybody thinks about is the transversality of light waves in which particles vibrate at right angles to the direction of propagation of the wave.  The transverse waves are taking place either on a surface of a liquid (water wave) or as the vibration of a stretch string, and not in the interior of a substance (body). However, as light waves propagate in the [interior of] space, there should be an explanation of this paradox.

Numerous experiments have proved the transversality of light waves. It should lead us to the conclusion that the medium wherein the light propagates should be surface-like. How come that it could be? We live in the interior of a body, not on the surface of something.

The answer lies in the concept of hypersurface that the mathematicians have introduced as a point of departure in the generalization of the concept of space, long before physicists surmise the multi-dimensionality of the spacetime. We may conceptualize the space as a 3-manifold  as a 3-hypersurface embedded in a 4-enveloping space. We can easily extend this concept to any higher multidimensional space (Figure-3).

Now, we have a proper place for light as a transverse wave to propagate on the hypersurface. It is a three-dimensional [hyper] surface in which photons vibrate at right angles (along with the time dimension) to the direction of propagation of the wave across the hypersurface. From this relativity point of view, we see the space as a 3-hypersurface vibrating to and fro in the time direction.

As the propagation of light waves indicates, we are dealing in this case not with waves in the interior of a substance but with phenomena on a surface (hypersurface or hyper-interface) or motions of whole configurations (like a vibration of strings). We have already a string theory which seems going nowhere and now tends to converge into a "brane" theory. We wish to suggest to shift the theory into a more proper hyper-interface theory.

It is in this context that we should develop the current brane theory.  The brane is more like a hyper-interface rather than like a piece of paper floating in the air or in the bulk of something that conceptualized in the current brane theory. Besides, the gravity force should act only along the surface of the brane and not out of it crossing through the higher-dimensional bulk.

Notes:

a.    The ancient creation myths referred to such condition as chaos.

References:

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

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

3.    Born, M: "Einstein's Theory of Relativity," Dover Publications, Inc., New York, 1962.

The Cosmic Inflation Never Happened

The Big Bang theory holds the premise that the universe originated from a singularity which came into being out of nothing through a single massive explosion. The concept of the minuscule size of such primeval singularity born from the thought projection of the current universe's expansion backward far in time.

The theory has at least two dubious primary grounds. The first one is the speculative concept of nothingness. The Big Bang theory presumes, violating the first law of thermodynamics, that energy (and matter) was created out of nothing. This idea came from the mindset that the creation of the universe (4-spacetime) was the beginning of everything. Notwithstanding, the theory takes for granted that the quantum fluctuation which stimulated the primeval explosion held in the nothingness before such creation.

The second speculative ground is about the size of the universe which can be shrunk indefinitely backward in time from the current size into a singularity. Close to the moment of creation, the size exponentially shrunk about 10⁶⁰ smaller just within 10⁴ seconds, from 10^-33 to 10^-37 second posterior to the explosion, the rate of which was exceedingly faster than the speed of light¹.

As we have elucidated so far, there can be no such thing as nothingness. The energy, as the only reality in nature, can neither be created out of nothing nor destroyed into nothing. The universe was born as the result of the interplay between the opposite (positive and negative) energies that created the universe and everything within, not out of nothing.

We can mathematically describe energy in its pure condition as waves' spectrum of different frequencies and amplitudes expressed in terms of Fourier series or its complex form, the Laurent series:

                         f(z) = F⁺(z) + c₀ + F^–(z)

It is a wave function expressed as the sum of its positive frequency (F⁺(z)) and negative frequency (F^–(z)).

Globally, we can depict this wave function in terms of Riemann sphere, the positive frequency F⁺(z) extends holomorphically into the southern hemisphere, and the negative frequency F^–(z) extends holomorphically into the northern hemisphere, where the equator represents the real coordinate and the longitudinal circles its imaginary time coordinate.

The domain of the positive and negative frequencies, however, does not fully extend to the poles, as the Riemann sphere has an annulus of convergence which excludes the domain around the zero points (singularity) as well as the infinity (Figure-1). 

 This pure mathematical analysis indicates that the split of the energy cannot create a stable interface (hypersurface) from the beginning when the energy started to split up to a certain period where it reaches the minimum size (represented by the Riemann sphere's inner ring of convergence).  The interface created in this period would instantly dissolve into energy.

It is only after reaching this limit that the hypersurface comes into being where it stabilized until it reaches its maximum size (represented by the Riemann sphere's outer ring of convergence).

The doomsday comes when the hypersurface reaches its maximum size.  At this particular time, the hypersurface becomes extremely unstable that makes it break down into pieces dissolving back into pure energy.  We illustrate these phenomena in Figures-2 and 3A.

This cosmology scenario avoids the need of either the concept of singularity to represent the beginning of the creation or the big crunch at the end of the universe's life as well as the everlasting expansion where the universe has no dead end.

This cosmology theory also excludes the need of the concept of cosmic hyper-inflation in the early period of the creation (Figure-3B), as the baby universe was born in an exceedingly larger size than that of the singularity which the Big Bang theory presumed.

The interplay between the positive and negative energies generates quantum fields across through the interface (hypersurface) ^a), located in between the two, perpetually creates quantum sparks (fundamental particles), the building block of the universe. As the split (hypersurface) area is enlarging with time, new matters are created in the expanding horizon keeping the average matter density per area almost constant.

This cosmology concept resembles the one of the continuous creation, steady-state expanding universe put forward by Fred Hoyle².

Notes:

a) We use the split area, interface, hypersurface, and space interchangeably.

References:

1.  Guth, A.: "The Inflationary Universe," Basic Books, New York, 1997

2.  Gregory, Jane: "Fred Hoyle's universe," Oxford University Press, New York, 2005

Multidimensional Time and Hypercomplex Numbers

Long before physicists embarked on the study of higher-dimensional spacetimes, 19^th-century mathematicians had firmly established the geometry concept of multidimensional metric manifolds. Many of these concepts were straightforward generalizations of ideas on the properties of surfaces embedded in the three-dimensional Euclidean manifold.

To simplify things, the mathematicians have introduced a multidimensional surface-like concept called hypersurface for modeling multidimensional space embedded in a higher multidimensional ambient manifold. A flat m-hypersurface can be appropriately embedded in an (m+1) space, but matters become more complicated when one comes to consider curved hypersurfaces. A curved m-dimensional hypersurface requires an ambient space whose dimensions are at least equal to or greater than ½m (m+1) 1.

Accordingly, a 4-dimensional curved spacetime requires at least a 10-dimensional ambient space. The spacetime's point position and, hence, the curvature of the spacetime is completely defined through a collection of numbers associated with the coordinate system set up in such 10-ambient space which we are more familiar with as the metric tensor's independent components of such 4-spacetime.

So what is so startling about it is when we explore the micro realm we would be confronting with the same 10-dimensional ambient space. Alas, in the later development, such as in that of the superstring theory, physicists made a blunder as they wrongly assumed the curly nature of the extra dimensions of such ambient space,  which made them going nowhere

The same fate happened to Big-Bang theory as physicists firmly exclude the existence of the universe's surrounding spaces. In doing so, physicists throw away the more significant part of the system, and this might be the reason why the theory incorporates only five percent of the total mass and energy that it actually should be.

Now the only option to cope with this impasse is jumping off the ship and abandon not only about the curly nature of the extra dimensions but also the one-dimensionality of time.

As the last article has deliberated,  those multiple temporal dimensions are the results of a series of successive symmetry breakings occur which had created different worlds, each of which had its respective temporal dimension (Figure-1).

Quaternion and Octonion

Now, how do we describe the structure and the geometry of such multiple temporal dimensions? To do this, we need to build a coordinate patch within such ambient space framework. To start with, let us deal with our 3-dimensional physical space embedded, as it should be, in a 6-ambient space. In such a case, we assign a coordinate patch consisting of three real space coordinates x₁, x₂, and x₃ and three imaginary time coordinates whose basis i, j and k.

If we denote x= x(x₁, x₂, x₃), then we can define any world point in such 3-physical space as:

q = x + ui + vj + wk,

expression is found to be nothing but the quaternion; a generalized complex number discovered a long time ago by Hamilton who established the geometry and the algebraic structure of this quaternion in 1843.

If we express the time variables u, v and w proportionally to the speed of light c_i of the respective temporal dimensions t_i then we can write:

q = x+ ic₁t₁ + jc₂t₂ + kc₃t₃,

This quaternion describes a general vector within a 6-dimensional space expressed as a function of space and time coordinates. Quaternions, therefore, describe a 6-dimensional vector space over the real numbers, depicting the dynamical geometry of 3-space embedded in 6-ambient space.

Similarly, we can define the 4-spacetime whose ambient space is ten dimensional through a coordinate patch consisting of three real space coordinates and seven imaginary time coordinates.

Again if we assign a space coordinates as x= x(x₁, x₂, x₃) and i, j, k, l,m, n, and o denote independent imaginary numbers as the coordinate basis representing seven different time coordinates, then we can define any point located at the 3-space in such coordinate patch as:

q = x + ai + bj + ck + dl + em + fn + go

, where x, a, b, c, d, e, f  and g are real numbers. Graves and Cayley had already discovered this expression, known as double quaternion or octonion, long time ago in 1845, although they did not know about the physical implication of it.

If we express the time variables a,b,c ... g proportionally to the speed of light c_i of the respective temporal dimensions t_i then we can write:

q=x+ ic₁t₁ + jc₂t₂ + kc₃t₃ +lc₄t₄ + mc₅t₅ + nc₆t₆ + oc₇t₇

Octonions form a 10-dimensional vector space over the real numbers, depicting a 3-physical space embedded in 10-dimensional ambient space.

In a later development, the original notions of quaternion and octonion are further modified and generalized through what so-called Clifford and Grassmann algebras applied to any higher dimensions framework which is found to have powerful implications in modern physics.
Many mathematicians and physicists wrongly perceived the quaternions and octonions as respectively describing 4-dimensional and 8-dimensional spacetime (having both one-dimensional time), which is inappropriate.

Penrose² regarded Hamilton's 22 year-devotion in his life in attempting to develop the quaternion calculus resulted in relative failure. On the contrary, we regard the Brougham Bridge's stone carved with the Hamilton fundamental equation would become a momentous milestone of the application of the hypercomplex calculus on the geometry of multidimensional time in both macroscopic and microscopic realms.

References:

1.    Sokolnikoff, L.S: "Tensor Analysis," Wiley Toppan, Second Edition, New York, 1964,   p. 205.

2.    Penrose, R.: "The Road to Reality," Vintage Books, London, 2005, p. 201

Symmetry and Symmetry Breaking

The asymmetry and its associated diversity that we observe today was the result of symmetry breakings which occurred in the early stage of the cosmos. In the beginning, ^a), the conditions were very different from those prevailing today, they were symmetric. The spatial dimensions as we know today did not yet exist; all dimensions were inherently temporal. However, as those temporal dimensions were yet undivided, there was no past, present ^b) and future.

At those conditions, the energy ^c) was unstable and tended to break into its positive and negative components. When it happened, the associated 4-spacetime (cosmos) ^d) was split into two parts creating an interface (3-hypersurface) in between the two. The dimensions across the interface transformed into spatial; leaving the dimensions outside it remained intact ^e).  Space, therefore, was born.

The two opposing energies ^f) perpetually generated sort of 4-lights (quantum fields) piercing through the interface (space) inducing secondary 3-(classical) fields which permeated and propagated across the interface (Figure-1A). As the quantum fields hit the interface, the strongest of them (Higgs fields) generated bright sparks which immediately disappeared as the opposite fields annihilated them (Figure-1B).

 The fundamental particles as we know are in reality nothing but these quantum-sparks which perpetually appear and disappear at the interface. As those quantum fields hit the entire surface of the interface and penetrate it only a short distance (across through the thickness of the space), they seem to us (who live in such interface/3-space) as eternal, omnipresent and invisible objects that can create and annihilate quantum particles.

Minkowski ^{1, g)} brilliantly fused the space and time into its undifferentiated state and brought back the spacetime into its original condition. However, then, something wrong happened. Instead of bringing the spacetime back into its symmetrical condition, Einstein²assumed that such unification did not make the temporal and spatial dimensions equivalent. Einstein failed to recognize that the asymmetry as we see today was the result of the spacetime symmetry breaking. This blunder has hampered the progress of physics for more than one hundred years now.

On the discovery of the four-dimensional spacetime, Einstein³ commented: “The non-mathematician is seized by a mysterious shuddering when he hears of four-dimensional things, by a feeling not unlike that awakened by thoughts of the occult.” No wonder, even after one hundred years of experience dealing with such spacetime, physicists are still bewildered and fail to recognize that their chaotic spacetime model does not represent the post symmetry breaking we observe today.

Supersymmetry Breaking and Multidimensional Worlds

The symmetry breaking of the 4-spacetime as we previously described was only one of the long series of successive symmetry breakings. It was the last of the long chain of a successive splitting of a higher-dimensional spacetime into its lower-dimensional parts.

To make it clear, let’s take the ambient 10-spacetime as a start. As this 10-spacetime broke its supersymmetry, a 9-hypersurface came into being along with its associated temporal dimension, t₇.  The latter, in turn, was split creating a smaller 8-hypersurface and its associated time, t₆ and so forth. This series of splits continued resulting in successive creations of the spacetimes in descending order of their dimensions and ended when the 3-space came into being along with its associated time t₁. A total of seven worlds ^h) have successively come into being with their own individual time, t_i, light and its respective speed, c_i, Planck constant, h_i, and “gravitational” constant, G_i.  

We can depict those seven worlds in term of their relative dimensionality (Figure-2.) or pictorially described as concentric spheres whose dimensions are larger outwards, in which the innermost layer is the 3-space with all of its solar system, stars, galaxies and super-galaxies (Figure-3A).

It is worthy to note that this picture may clarify the exact physical meaning of the ancient cosmology. For hundreds of years, people had wrongly considered this configuration as the geocentric cosmology in which the earth was at the center of the universe (Figure-3B). Even now, modern physicists fail to properly grasp the multidimensionality of the seven heavens described in the ancient cosmology ⁴.

Notes:

a.    It is the relative beginning, not the beginning of time.

b.   The notation of spacetime given for the cosmos at its original state is misleading as it gives the impression as it was asymmetrical from the beginning. It would be more appropriate if we use the notation world, cosmos or more technically [metric] manifold.

c. Energy in its entirety (4-energy); to avoid misunderstanding it would be more appropriate if we use the ancient notation: eon or simply eon.  The energy as we know is merely its superficial property (3-energy).

d.   There was no space, as space and the present time are different aspects of the same thing.

e.  This symmetry breaking is analogous to the phenomenon which occurs in the separation of two immiscible liquids, such as oil and water. In the body of the liquids, the cohesive forces are symmetric exerting equally in all directions. At the interface, however, such symmetry is broken because of unbalanced force exerting at the interface. As the system is in equilibrium,  the potential energy known as interfacial tension counter the net unbalance force. In terms of coordinate geometry, we may say that the interfacial tension differentiates the dimensions across the interface ("superficial" dimensions) from those of the original.

f.     The relativistic energy is composed of two opposite components as expressed in E² = m²c⁴ + p²c²

g.    Minkowski died one year only after the discovery, leaving confusion on his discovered object (spacetime)’s structure.

h.   The ancients called such worlds seven heavens.

References:

1.    Einstein, A. et al.: " The Principle of Relativity," Dover Publications, Inc., New York, 1952, p. 75.

2.    Einstein, A.: " The Meaning of Relativity," Princeton University Press, Fifth Edition, New Jersey, 1954, p. 31

3.    Einstein, A.: "Relativity," Crown Publishers Inc., Fifteenth Edition, New York, 1952, p.55

4.    Hawking, S.: "A Brief History of Time," Bantam Books, London, 1989, p. 3.

Space Thickness, Supermanifold and Multidimensional Time

We used to conceptualize the geometry elements such as point, line, surface and space as having, respectively, zeroed, one, two and three dimensions. There is nothing wrong with that as far as we are dealing with abstract objects such as a corner point between a floor and two walls, meeting line between ceiling and wall, table's surface or hall's spaciousness.

However, we cannot apply such a concept for real bodies, whatever the size is. A grain of sand is not a zeroed-dimensional object, but a three-dimensional cubic-like body, which has small length, width, and thickness. A string is a three-dimensional long cylindrical object having a small section. Similarly, a piece of paper is a three-dimensional surface object whose thickness is very thin (Figure-1). Had their thickness been reduced to zero, those objects would all have gone into thin air.

Nature does not seem to give any exception to natural bodies such as space or any other higher-dimensional spacetimes. For their existence to have physical meaning, all those bodies should have thickness.

It implies that space or spacetime, whatever its dimensions, are always be embedded in an ambient spacetime of at least one dimension higher. At the same token, the later is also embedded in turn in another much higher manifold (Figure-2).  This kind of infinite regress makes us believe that nature is vast and infinite, not only its areas but also its dimensions. 

System and Surroundings

Now, the formulation of the laws of nature depends naturally on which system we choose. Suppose we want to formulate physical laws within a system of an m-dimensional spacetime embedded in  N-dimensional ambient manifold, we get, then, physical laws of a system having (N-m) extra dimensions. The directions of these dimensions determine those of the spacetime’s thicknesses pointing outwards away from it.

We can also describe the same physical laws in a much simpler system where the same N-ambient space embedding (N-1)-hypersurface, instead of an m-spacetime. The thickness of such a hypersurface has the same direction as at the N^th dimension pointing outward away from it.

The laws of nature in the first system have very complex formulations and are difficult to resolve, as the system has too many extra-dimensions and, hence, fewer symmetries.  The laws of nature in the second system are relatively more straightforward as the system has only one extra-dimension and is highly symmetric.

We can best describe the laws of nature when the number of the dimensions of the ambient space embedding the system is large enough which "stretches" out the hypersurface to become completely flat and perfectly symmetric.

How do we determine the dimensions of the ambient space (N) vis-a-vis that of the embedded spacetime (m)? There is a minimum requirement for the number of the ambient space's dimensions in order that the spacetime can be “properly" embedded in the ambient space. The [non-flat] m-spacetime can be embedded in N-manifold only if at least N = ½ m(m+1) ¹⁾. The metric tensor of the m-spacetime dictates that the ambient space should have that amount of dimensions for all of its components can be properly defined.

Based on the above rule, the 2-surface requires  3-ambient space for which we do not doubt it. The non-flat 3-space, in our surprise,  requires 6-ambient spacetime, not to mention the 4-spacetime which requires 10-ambient manifold. It may indirectly explain why we have three generations of elementary particles and the 10-ambient manifold as revealed in the current theoretical physics. 

Multidimensional Time

Now, what are these dimensions all about? As we have discussed previously, the spacetime is the physical manifestation of energy. In its original state, the spacetime was perfectly symmetric. All of its dimensions are indistinguishable, and they are all "temporal." When the respective energy segregates into the positive and negative energies, the [temporal] spacetime's dimensions along the interface [separating those opposing energies] are transformed into spatial dimensions.  

For the classical 4-spacetime, the energy segregation transforms three of the spacetime's temporal dimensions along the interface into spatial (Figure-3). In a 6-spacetime, the energy’s segregation transforms the spacetime's five temporal dimensions along the interface into spatial dimensions. The same case also prevails for the 10-spacetime., where nine temporal dimensions along the interface become spatial.

The temporal dimensions  t1, t3 and t7 related to the 4-, 6- and 10-spacetimes, respectively, are different from each other. It is against the mainstream premise which tacitly asserts that there is only one temporal dimension in nature.

Based on the rule we have, a 4-spacetime requires a 10-ambient space for the physical laws to have solutions. However, as we have in this case 3 spatial dimensions and seven [imaginary] extra-temporal dimensions, the physical laws we get would be very complicated. It is imperative, therefore, to have the same laws applied to a system consisting of a 10-ambient space embedding 9-hypersurface, which are simpler as we have only one imaginary temporal dimension on top of the nine real ones.

 It is more or less what physicists have done in developing the string theory, except that the extra-dimensions were assumed to curl into tiny loops. Also, the temporal dimension of the system was assumed to be the same as that of ordinary time. Such wrong assumptions have been put forward because mainstream physics holds the premise that time is one-dimensional as previously mentioned. 

The relativity theory should rigorously hold the equivalence of space and time dimensions. The spatial and temporal dimensions should be transferable to each other depending on the system they become part. The extra dimensions are undetectable not because they curl into tiny loops but because they are temporal.

Supermanifold and Supersymmetry Generators

Physicists have many problems with their mathematical propositions as they used to conceptualize the spacetime as a standalone basis. Under such a concept they have taken the part of the reality out of the system. Such as is the case of the Big Bang theory, which is entirely Platonic, a system without any geometrical thicknesses, surrounding, nor even 3-space.

A reader of the Scientific American²⁾ once asked: "Where is the universe expanding to?"  The authoritative answer from the expert was: "... the universe's expansion does not push it into new territory - rather the spacetime grid itself is expanding".  The issue has arisen again and again since the Big Bang theory was put forward, as only a few people were satisfied with such an explanation. The excellent answer should be that the universe is expanding to at least the 10-dimensional ambient space, and not into nothing.

To make their model closer to the reality, some physicists artificially introduced what they called supersymmetry generators, replacing the thicknesses which they have “forgotten” to incorporate in their mathematical model. They call this manifold having thicknesses “Supermanifold”³⁾. The physicists should put forward the problems of embedding at the forefront of physical researches and develop a more holistic model instead of a piecemeal one.

References:

1.      Sokolnikoff, L.S.: ”Tensor Analysis," Wiley Toppan, Second Edition, New York, 1964, p. 205

2.      Kashlinsky, A.: "Where is the Universe Expanding to?", Scientific American, (Ask the Experts Forum), May 2007, p. 104

3.      Penrose R.: "The Road to Reality," Vintage Books, London, 2005, p. 879