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,

where x, u, v and w are real numbers. This 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

Multidimensional Time and Hypercomplex Numbers

Energy Reality and the Creation of the Universe

Everybody knows about energy but rarely is aware of its true nature. People used to regard energy as something abstract, the same way as they do for the power, force or its kinds. What they concern most about energy is its function, not its reality. We learn from the school that energy is a mathematical abstraction that has no existence apart from its functional relationship to other variables or coordinates that do have a physical interpretation and which can be measured¹.

The kinetic energy of a given mass has no other reality than as a function of its velocity. We know the potential energy from its position and the rest energy from its mass. What people know about energy is only superficial. They only see ripples moving across the surface of the ocean, not the water beneath. As we will see the energy is not at all an abstract thing. On the contrary, it is the only real substance from which everything is derived. Energy is omnipresence; eternal (in relativity term), neither can be created nor destroyed.

Energy and Its Bodily Form

From the relativity theory, if it is adequately understood, we can know the true nature of energy. Einstein's equation E = mc² [and the energy conservation] pointed out that matter was nothing but a different aspect of energy. The relativity also unified other fundamentals, i.e. the space and time into a single entity called spacetime. The relativity, therefore, reduces the fundamentals in two namely the spacetime and energy.

Einstein ingeniously connected the two through the influence of energy on the spacetime resulting in the curvature of the latter. From this idea, he revealed the nature of gravity. It was a success story, but that was not enough. What Einstein supposed to do was taking a little step further unifying the spacetime and energy.

What Einstein failed to grasp was a fundamental [metaphysical] principle: a substance does not come into existence without its bodily form. Nature abhors nudity.  Just like the three-dimensional spherical structure is the bodily form of water in a drop of water, the [four-dimensional] spacetime is the bodily form for the energy.

There is no energy without spacetime and no spacetime devoid of energy. Science fails to recognize this very nature of  energy because physicists are more familiar with the energy's manifestation on the surface, not the underlying energy. Energy in its actual reality is more like an [four-dimensional] ocean of energy. As a three-dimensional being, we can only perceive the manifestation of energy on a three-dimensional hypersurface (space) ^a) in the form of fields and forces.

The spacetime is nothing but the structural quality of energy. The dimensionality of the spacetime is not its intrinsic property but the manifestation of the energy's degree of freedom. The higher the energy's potency is, the higher its spacetime's dimensions are. There should be a broad spectrum of energies of different potencies exist out there in nature, as indicated by the discovery of theoretical higher dimensional spacetimes in some more recent theories.

Another energy's feature, which physicists seem to be reluctant to deal with, is its opposing components which are revealed by the relativistic energy equation E² = m²c⁴ + p²c². As we will see, these two opposite energies, the positive and negative, play an essential role in the creation of material things and space.

Multidimensional Time

The physicists seem to have been forced to deal more and more with higher dimensional spaces. Kaluza and Klein were the first who believed that the gravity and the electromagnetic waves put together could be adequately described if the spacetime had five dimensions instead of the standard four. Under the superstring theory, physicists even proposed higher dimensions: ten and more recently eleven.

As did Kaluza and Klein, physicists presumed that all the extra-dimensions beyond the fourth curled into tiny, undetectable loops. This idea has brought us into many bizarre things such as doughnut-like and Calabi-Yau spaces² which give us a hundred thousand possible solutions. One should have used Oscam razor to cut off those ugly curly dimensions long time ago and put back more elegant extended extra-dimensions in their places.

We cannot perceive the extra-dimensions because they are time dimensions. The term spacetime given to the unification of the space and time is misleading. The term gives us the impression that such a union has already different space and time dimensions since the beginning. On the contrary, in its originality the dimensions of the spacetime were equivalent. The union of the space and time at that stage is more like four-dimensional time (its dimensions were time-like).  The time dimensions were not yet divided. Neither space nor "Now" was there, nor was a material thing and even light. It was in total darkness and chaos.

At this condition, we better name the union of space and time 'eon,' the "eternity" (Figure-1a). It was only when the opposing energies had segregated from each other (the spacetime split in two) that we can call it spacetime (Figure-1b). As the energies separated, an interface (space) naturally appeared. As such, the [time] dimensions along the interface were transformed to become spatial.

The physicists totally missed this scenario and jumped instead using the undivided whole spacetime for representing the real worlds. The following statement reflects Einstein's standpoint³:” 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". This kind of vicious-circle thinking has brought us to the troubles that we have now in physics.

Grand Relativity Theory

In order to preserve causality and order, physicists have been forced to assign "artificial" light cones, as the fundamental structure, at every point within the spacetime. Within such a world, the simultaneity of Nows, even for two neighboring events, is totally beyond the grasp of the system.  Einstein was worried so much about this, as his friend Rudolph Carnap recounted ⁴:” Einstein said the problem of the Now worried him seriously …  That the science  cannot grasp the experience of Now seemed to him a matter of painful. So he concluded that there is something essential about the Now which is just outside the realm of science”.

The reason why physicists denied the existence of [universal] "now" is because it was against the principle of the relativity of simultaneity. We can avoid this dilemma if we are considering it under a multidimensional time framework. Events which happen non-simultaneously in a particular time dimension could be simultaneous in another higher dimensional time. We can always find a higher time dimension in which events happen simultaneously (flat hypersurface) whereas they were not so in a lower time dimension (Figure-2).

For example, a ten-dimensional spacetime can be equally described as a three-dimensional space having seven-time dimensions or as nine-dimensional spacetime having one-time dimension. The former has its physical law broken into many while the latter has less. We can always try to explore a higher and higher dimensional spacetime until we get one unified law.

The spatial and time dimensions are transformable to each other depending on how we regard the spacetime as a system. Whether we consider the system as a lower dimensional hypersurface having many time dimensions or as a higher dimensional hypersurface having one temporal dimension, the physical laws remain the same. We can only unify of physical laws can only be done at such a higher dimensional [flat] hypersurface (spacetime) where all events are taking place simultaneously. We may call this postulate the grand relativity principle.

Ghostly Rotating Universe

Under the general relativity theory, the arrangement of light cones can be quite irregular rather than uniform. In an extreme situation, they may form closed time-like curves. Penrose⁵ ruled this situation out as simply unphysical as it violates causality. However, his worry was unfounded. The two opposing energies do not create matters [and space] permanently but perpetually create and annihilate those  particles. Particles are created and annihilated perpetually at the pace equal to the speed of light as expressed by E = mc² or dynamically by E² = m²c⁴ + p²c^{2  b)}.  The interactions between the two energies are manifested as a flux of [four-dimensional] electrical currents (Higgs fields?) across through the surface generating quantum sparks (“quarks”) at the surface, analogous to the flashing pixels appear and disappear on a giant TV screen.

As the two energies are repulsive, they are "flowing" in the opposite directions to each other, forcing the surface rotates around an axis lying on its plane (Figure 3a). The electric currents which flow across through the surface induce [three-dimensional] magnetic fields which propagate traversing the surface.  It makes the surface rotate following the right-hand rule, around an axis perpendicular to the surface (Figure 3b). This rotation causes circular movements of heavenly bodies i.e., the solar system, galaxies, super-galaxies and spin of both macroscopic bodies and quantum particles.

The interface, the boundary which separates the two energies, is volatile ^c) like a mirage having virtual existence. The constant interactions of the two energies make the existence of the interface seem persistent. Penrose was very close to this idea when he discussed the Galilean relativity. However, he considered this idea as unphysical ⁶⁾:” It may seem alarming that our very notion of 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."

What he stated was, in fact, correct except that space does not appear and disappear in time but, on the contrary, it is the appearance and disappearance of the space which creates time passage.  Had he done otherwise, he would lead us on the right track to reality.

The separation of the spacetime does not take place instantaneously but gradually. The "area" of the separation is stretching out across the spacetime just like a big crack propagates in a solid body. The former represents a static universe while the latter represents the expanding universe (Figure-4). It is the underlying reality of the expanding universe as against the Big Bang theory.

Notes:

a) We use the notations of space, hypersurface, and hyper-interface, or surface and interface interchangeably.

b)     At the quantum level these equations are completely reversible

c)   The relativistic expression (m²c⁴ + p²c²) involves complex numbers so that there is no clear cut separation (interface) between those two separated energies, in the sense that the interface appears only instantly and disappears immediately after.

References:

1.   Abbott and Ness, V.: Thermodynamics”, Schaum’s Outline Series, Mc Graw-Hill, New York, 1967, p. 1

2.     Greene, B.: "The Elegant Universe," Vintage Books, New York, 2003, p. 207-208.

3.   Einstein, A.: "The Meaning of Relativity," Fifth Edition, Princeton Science Library, New Jersey, 1988, p. 31

4.     Barbour, J.:" The End of Time," Phoenix, Second Impression, London, 2001, p. 14

5.     Penrose, R.:" The Road to Reality," Vintage Books, London, 2004, p. 408-409.

6.     Ibid, p. 387

Energy Reality and the Creation of the Universe