The Crumple of the Spacetime

The General Relativity theory was developed based on the premise that the gravity force is the manifestation of the 4D-spacetime curvature ^a). This physical concept was derived from Riemann’s idea¹ that the force was nothing but a consequence of geometry, thus, banished Newton’s unnatural concept of “action-at-a-distance”.

However, Einstein mindset was to stick with intrinsic geometry in the sense that the spacetime as a system was regarded as having no surroundings or being embedded in nothingness ^b). In fact, when physicists talk about the wrinkle of the [4D-] spacetime they never care about which directions (dimensions) it wrinkles goes.

The strict concept of the geometry dictates otherwise ². The geometry concept of m-metric manifolds (hyperspaces) ^c) is a straightforward generalization of ideas of the study of surfaces embedded in 3D-space. However, it has been proven that in the circumstances in which an m-dimensional curved hypersurface can be embedded in the n-dimensional [Euclidean] ^d) manifold if at least n = ½ m(m + 1).

The crumple of the ordinary 2D-surface, a piece of paper, for example, requires a third dimension (3D-ambient space) for it to occur. But hardly anybody is aware that the crumple of 3D-space requires not only a fourth dimension but at least 3 additional dimensions (6D-ambient spacetime) ^e). Similarly, the crumple of 4D-spacetime would require at least 10D-ambient spacetime for it to occur without constraint in any direction (Figure-1). Does the nothingness have such properties for being able to embed something? What is nothingness anyway?

Nobody should blame Einstein on this negligence. People were just horrified about the idea of 4D-spacetime which had been introduced by Minkowski beforehand, not mention the 10D-spacetime. Had Einstein been aware of this higher dimensional surrounding requirement he would probably still prefer to take the surrounding as nothingness rather than 10D-ambient spacetime.

A more recent theory such as that of superstring requires 10D-ambient spacetime ^f) for its equations to be solvable.  Alas, being uncomfortable with such bizarre vast extension ^g) of the ambient spacetime, physicists blow down ^h) the majestic surroundings assuming the extra dimensions being curled leaving the ordinary 4 dimensions to remain intact.

Even after the superstring theory was established, Big Bang theory as the cosmological application of the General Relativity theory maintains its premise on the nothingness instead of 10D-ambient spacetime taken as the surrounding of the expanding 4D-spacetime. As such, Big Bang theory misses a bigger part of the “stage” that in no way it can explain the substantial missing dark matter and dark energy.

Having many defects in its premise Big Bang theory would eventually fall short except it takes among other the 10D-ambient spacetime as the surrounding of the universe (4D-spacetime) instead of nothingness.

Notes:

a)    Arthur Eddington expedition carried out to South Africa during the solar eclipse in 1919 verified the shifting of the position of a star within the field near the sun, thus, proving Einstein's general relativity prediction of the bending of light around a massive object.

b)   There is a vague definition of absolute nothingness or emptiness but we may guess that what most physicists mean by it is a sort of extension (spacetime) with an indefinite number of dimensions [zero or infinite dimensions?] having neither matter nor energy.

c)     The notation of n-spacetime is equivalent to n-hyperspace or n-hypersurface.

d)     This is the reason why the laws of nature look simpler in higher dimensions. If the dimensions of the surrounding spacetime are high enough then we might have a flat (Euclidean) surrounding where the physical laws become simpler.

e)    We may speculate that the existence of three generations of particles is the manifestation of 4D, 5D and 6D-particles abiding in the respective 4D, 5D and 6D-spacetime. The manifestation of the last two generations into our world could be only the cross-section of their whole body.

f)   The string theory accidentally derived the 10 dimensions mathematical requirement from Beta function originally dedicated to solving the strong force quest. The coincidence with the 10 dimensions geometrical requirement of the ambient spacetime embedding the 4D-spacetime is stupendous.

g)     Since they are not visible, the string theorists regard the extra-dimensions as spatial and being curled into tiny loops.

h)   Physicists assume that the 10D-ambient spacetime splits into a 4D-spacetime and a tiny curly 6D-metric manifold. This premise doesn't absolutely make sense just like the impossibility of splitting a 3D-cube into one 2D-plane and one 1D-line. The proper way to do it is successively splitting the 10D in two creating 9D as their interface and so forth down to 4D which we get as the interface of the two halves of the 5D split. We would, then, have a total of 7 spacetimes embedding each other in descending order of their dimensions.

The Crumple of the Spacetime

Hypersurface, the Extrinsic View of the World

Any application of the law to a discrete portion of the universe or even something else more prominent such as the possible multiverses requires the description of a system and its surroundings. A system can be any region of space, the whole universe itself or any region of multiverses selected for study and set apart [mentally] from everything else, which then becomes the surroundings.

There are two ways on how we can geometrically describe the world, i.e. intrinsically or extrinsically. Let us take an example on how we describe a lower dimensional object such as a surface. We can describe the properties of surfaces without reference to the space in which the surface is embedded as intrinsic properties. We can imagine that in this particular case the properties of the surface are analyzed from two dimensional flat being living in such surface, whose universe is determined solely by [two] surface parameters. In this intrinsic geometry a pair of isometric surfaces, a cone and cylinder, for example, are indistinguishable.

These surfaces appear to be quite distinct to an observer examining them [extrinsically] from a reference frame located in the space in which the surfaces are embedded. Geometrically, an entity that provides a characterization of the shape of the surface as it appears from the enveloping space is the normal line to the surface^[1].


Now let us turn to examine the concepts from the geometry of higher dimensional metric manifolds which are of our primary interest. Many of the concepts are straightforward generalizations of ideas introduced in the study of surfaces embedded in the three-dimensional [Euclidean] manifolds. For the visualization purposes, we wish to put forward the relative depictions of the n-dimensional world viewed intrinsically as standing alone n-hyperspace and viewed extrinsically as an n-hypersurface embedded in (n+1) or higher dimensional hyperspace (Figure-1).

An n-dimensional flat hypersurface can be entirely embedded in an (n+1) dimensional hyperspace, but as the hypersurface is curved, it needs much spacious ambient. Now, under what circumstances an m-dimensional variety (hypersurface) can be embedded in the n-dimensional Euclidean manifold (hyperspace)? The answer is that the m-hypersurface can be wholly embedded in hyperspace without restraint on whatever direction it might curve if and only if such a hyperspace has at least n = ½ m (m+1) dimensions. We call the latter as an allowable embedding hyperspace to a particular hypersurface.

It means that there are a total of [1/2m (m+1)-m] normal lines to the "surface" of such a hypersurface. In a dynamic condition where the hypersurface moves relative to the ambient hyperspace this-extra dimensions are identified as extra [temporal] dimensions, not the ones which curled in tiny looped as the string theory hypothesizes.

Einstein had developed the relativity theory both special and general relativity theories based on intrinsic geometry. It is the weakness of the relativity theory which ignores the surroundings representing more than 99% of the whole reality we are longing to recognize. No wonder the Big Bang theory, the derivation of the relativity theory, can only take into account 5% out of the total matter and energy affecting the known universe.


Intrinsically, we consider the independent variables of the metric tensor of Einstein four-dimensional spacetime as just mathematical variables having no physical significance. However, when we see the world extrinsically, those variables of the metric tensor are nothing but the manifestation of the underlying coordinates – the dimensions of the embedding hyperspace.

We can achieve the unification theory if we describe the world as a curves hypersurface at least in term of parameters of its minimum allowable embedding hyperspace.

The concept of brane, the representation of the world as a hypersurface is already in the right track except for the concept of its ambient hyperspace. The brane is not like a piece of paper floating around in thin air, but more like an interface of higher dimensional watery like substances – higher dimensional of positive and negative pure energies (Figure-2).

Hypersurface, the Extrinsic View of the World

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