RE: MDT & Kaluza Klein Theory:
MDT & Kaluza Klein Theory
The illustrations of Kaluza Klein Theory immediately bring MDT's fourth expanding dimension to mind. Imagine if every point of the fourth dimension is continually expanding, underlying all manifestations of Huygens' Principle throughout all of nature, as well as Kaluza Klein's extra dimension. A frozen snapshot of the fundamental fabric of spacetime in MDT's universe would look something like this:
http://jac_leon.club.fr/gravitation/images/kaluza-klein.gif
Again and again throughout nature we see MDT's fundamental action---local points becoming nonlocal spheres (underlying Huygens' Principle, entanglement, nonlocality, entropy), wherein every point on the surface, while distributed evenly throughout the three spatial dimensions, are yet at the same singular point in the fourth expanding dimension. dx4/dt = ic.
Or like this:
http://www.vcharkarn.com/uploads/17/17494.gif
Again and again throughout nature we see MDT's fundamental action--local points becoming nonlocal spheres (underlying Huygens' Principle, entanglement, nonlocality, entropy), wherein every point on the surface, while distributed evenly throughout the three spatial dimensions, are yet at the same singular point in the fourth expanding dimension. Dx4/dt = ic.
Wikipedia reports: http://en.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory
"In physics, Kaluza-Klein theory (or KK theory, for short) is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism. The theory was first published in 1921 and was discovered by the mathematician Theodor Kaluza who extended general relativity to a five-dimensional spacetime. The resulting equations can be separated out into further sets of equations, one of which is equivalent to Einstein field equations, another set equivalent to Maxwell's equations for the electromagnetic field and the final part an extra scalar field now termed the "radion"." --wikipedia
Kaluza-Klein theory posited that gravity and electromagnetism can be unified by adding a fifth dimension of compactified space, and MDT provides this compactified space. The novelty of MDT is that it proposes that this compactified space is not static, but that it is constantly appearing, as the fourth dimension expands. Thus MDT underlies all string theories. String Theory ignores Einstein's insight that space and time can move, warp, and bend. MDT acknowledges that dimensions can move, and that the fourth dimension is expanding relative to the three spatial dimensions.
Brian Greeene et al are always worrying about their calculations blowing up at small distances. Well, via the nonlocality of the fourth expanding dimenson, from where all wavelike behavior arises, all this is resolved.
Gravity, based on general relativity, freezes the fourth expanding dimension. Electromagnetism, based on the quantum mechanical properties of photons, treats the fourth expanding dimension as an entity in constant flux.
Below again is a figure depicting the compactified dimensions of Kaluza Klein--not that it is very similar to the earlier figures depicting MDT in terms of a dimension that is expanding in a spherically-symmetric manner.
http://jac_leon.club.fr/gravitation/images/kaluza-klein.gif
Again and again throughout nature we see MDT's fundamental action--local points becoming nonlocal spheres, wherein every point on the surface, while distributed evenly throughout the three spatial dimensions, are yet at the same singular point in the fourth expanding dimension. dx4/dt = ic.
Wikipedia reports on Kaluza Klein's limitations:
"As an approach to the unification of the forces, it is straightforward to apply the Kaluza-Klein theory in an attempt to unify gravity with the strong and electroweak forces by using the symmetry group of the Standard Model, SU(3) ﾃ-- SU(2) ﾃ-- U(1). However, an attempt to convert this interesting geometrical construction into a bona-fide model of reality founders on a number of issues, including the fact that the fermions must be introduced in an artificial way (in nonsupersymmetric models). A less problematic approach to the unification of the forces is taken by modern string theory and M-theory. Nonetheless, KK remains an important touchstone in theoretical physics and is often embedded in more sophisticated theories. It is studied in its own right as an object of geometric interest in K-theory.
Perhaps MDT, and the new, emergent picture of time, will provide a means for introducing fermions in a more natural manner. The exact nature and radius of the expanding fourth dimension could determine physical constants. Wikipedia reports:
Even in the absence of a completely satisfying theoretical physics framework, the idea of exploring extra, compactified, dimensions is of considerable interest in the experimental physics and astrophysics communities. A variety of predictions, with real experimental consequences, can be made (in the case of large extra dimensions/warped models). For example, on the simplest of principles, one might expect to have standing waves in the extra compactified dimension(s). If an extra dimension is of radius R, the energy of such a standing wave would be E = nhc / R with n an integer, h being Planck's constant and c the speed of light. This set of possible energy values is often called the Kaluza-Klein tower.
It is indeed interesting that M Theory and String Theory have 10 or 11 dimensions. MDT can account for these extra dimensions, starting with a three dimensional world. Every time a point expands in the fourth expanding dimension, it does so in an orthogonal manner, just as all radii of a sphere are orthogonal to its surface. Thus numerous dimensionalities can emerge; but always brought back to the reality and context of our three spatial dimensions.
Wikipeida reports: http://en.wikipedia.org/wiki/Why_10_dimensions%3F#Number_of_dimensions
Number of dimensions
One intriguing feature of string theory is that it involves the prediction of extra dimensions. The number of dimensions is not fixed by any consistency criterion, but flat spacetime solutions do exist in the so-called "critical dimension." Cosmological solutions exist in a wider variety of dimensionalities, and these different dimensions--more precisely different values of the "effective central charge," a count of degrees of freedom which reduces to dimensionality in weakly curved regimes--are related by dynamical transitions.[15]
Nothing in Maxwell's theory of electromagnetism or
Einstein's theory of relativity makes this kind of prediction; these theories require physicists to insert the number of dimensions "by hand," and this number is fixed and independent of potential energy. String theory allows one to relate the number of dimensions to scalar potential energy. Technically, this happens because a gauge anomaly exists for every separate number of predicted dimensions, and the gauge anomaly can be counteracted by including nontrivial potential energy into equations to solve motion. Furthermore, the absence of potential energy in the "critical dimension" explains why flat spacetime solutions are possible.
This can be better understood by noting that a photon included in a consistent theory (technically, a particle carrying a force related to an unbroken gauge symmetry) must be massless. The mass of the photon which is predicted by string theory depends on the energy of the string mode which represents the photon. This energy includes a contribution from the Casimir effect, namely from quantum fluctuations in the string. The size of this contribution depends on the number of dimensions since for a larger number of dimensions, there are more possible fluctuations in the string position. Therefore, the photon in flat spacetime will be massless--and the theory consistent--only for a particular number of dimensions.[16]
When the calculation is done, the critical dimensionality is not four as one may expect (three axes of space and one of time). Flat space string theories are 26-dimensional in the bosonic case, while superstring and M-theories turn out to involve 10 or 11 dimensions for flat solutions. In bosonic string theories, the 26 dimensions come from the Polyakov equation.[17] Starting from any dimension greater than four, it is necessary to consider how these are --reduced to four dimensional space-time.
http://en.wikipedia.org/wiki/Why_10_dimensions%3F#Number_of_dimensions
Picture a fourth expanding dimension. It would appear as a sphere in three stationary dimensions. Now if each point on the sphere is always expanding, just as each point on a spherically-symmetric wavefront is always expanding as a spherically symmetric wavefront, six new, linearly independent compactified dimensions will be introduced, as there are six degrees of freedom, resulting in a total of ten dimensions.
So it is that MDT easily accounts for the extra six dimensions, as the nature of the compactified, spherically-symmetric dimension is to be perpendicular to the point from which it emanated. That perpendicularity is implied by the that in MDT's fundamental equation:
Again, the expansion of a point into a sphere centered about that point results in a surface that as perpendicular to all radii. And that shows up in calculations whenever there is an inherent perpendicularity.
