Multidimensional Reality (Part I)

The mathematicians used to say that there is no branch of science in which the tyranny of authority has been felt more strongly than in geometry1. We would instead say that this statement is no longer valid but in physics. The relativity theory under the giant name as of Einstein together with its derivative, the Big Bang, dominated the thought and shaped the development of physics and cosmology for around one hundred years up to now.

However, its endless conflict with quantum mechanics has put the physics in crisis as we see today. There were a few brave physicists to whom the relativity theory did not seem convincing, but the hands of authority were so heavy that it is almost impossible to put forward their different ideas to fix up the theory. Notwithstanding, let us scrutinize the fundamental concepts of the relativity theory aimed at improving the theory in concordance with quantum mechanics.

Spacetime as the Geometrical Quality of Energy

The special relativity theory deals with an idealized four-dimensional spacetime whose energy hidden behind the scene. As such, everything is in rest or steady motion forever. There is no force or friction which might accelerate or decelerate the motion. Even gravity is abhorred to exist. Such a world should be completely flat. In this particular circumstance, because of the energy's passive role, the spacetime appears as though it is an independent reality.

The general relativity theory, on the other hand, deals with a more real-world whose energy is lively on the go. The spacetime can be no longer flat but somewhat curved here and there due to the effect of gravity and forces exerting in those particular parts. The spacetime is not independent of energy. The spacetime is not like a container and energy something that fills the container. The energy and spacetime are respectively more like water substance and the spherical form in a drop of water. Undeniably, the spacetime by itself does not have the existence on its own; it fades away into shadow to become merely the geometrical quality of the energy. Einstein2 has inaccurately interpreted that the spacetime was the geometrical quality of the fields instead of energy.

Geometrical Intrinsic View of the General Relativity Theory

The fault of the relativity theory is that it treats the spacetime's geometry properties intrinsically, without due reference to the surroundings in which the spacetime might be embedded. It ignores the majority part of the reality: the surrounding. Take, for example, Einstein's metric tensor, wh Intrinsically, this tensor is mathematically explained as a function of ten independent variables without further explanation about what these variables physically could be.

As we may recall, a curved m-dimensional spacetime (m-hypersurface) can only be embedded in the n-dimensional [Euclidean] manifold if the embedding manifold has at least n = ½ m (m+1) dimensions. We know that a curved two-dimensional surface can be easily embedded in three-dimensional space, but a curved three-dimensional space can only be freely (no constraint in any direction) embedded in a hyperspace if and only if the latter has six dimensions. Had the embedding hyperspace been four-dimensional, space would be completely flat.

A further generalization is straightforward. A curved four-dimensional spacetime requires at least 10-dimensional surrounding hyperspace, and so on up to infinity, the Absolute realm, whose surrounding has no meaning. Only then, we can talk about a system without surrounding, not the one which the general relativity assumes. Even when the general relativity assumes that the spacetime's surrounding is an absolute void, the following question naturally arises: how many dimensions the void has for it could embed the four-dimensional spacetime? Are they none, ten, infinite or else?

You know now that even long before physicists formulated the string theory, the general relativity theory has tacitly demonstrated that the reality was at least ten-dimensional, which the theory has, alas, overlooked it. However, the so-called "extra" dimensions are well extended, not curled into tiny loops such as prematurely hypothesized in the string theory. How come, then, we cannot see those extra dimensions? The bold answer is that those extra dimensions are temporal. To everybody's amazement, time is indeed multidimensional.

(to be continued)

References:

1.     Sokolnikoff, L.S.: "Tensor Analysis," John Wiley & Sons, Inc., Second Edition, New York, 1964

2.  Einstein, Albert: "The Meaning of Relativity," Princeton University Press, Fifth Edition, Princeton, N.J. 1954.

Multidimensional Reality (Part I)

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