FOR SOME REASON, THESE EQUATIONS APPEAR CORRUPTED. I AM CHECKING WITH WORDPRESS TECHNICAL SUPPORT AND HOPE TO HAVE THIS FIXED IN THE NEAR FUTURE — JAY.
I am posting here a further excerpt from my paper at Kaluza-Klein Theory, Lorentz Force Geodesics and the Maxwell Tensor with QED. Notwithstanding some good discussion at sci.physics.relativity, I am coming to believe that the intrinsic spin interpreation of the compacified, hypercylindrical fifth dimension presented in section 4 of this paper may be compelling. The math isn’t too hard, and you can follow it below: The starting point for discussion equation is (3.2) below,
which is used to connect the ratio from the Lorentz law to geodesic motion in five dimensions, and is a numeric constant of proportionality. Section 4 below picks up from this.
Excerpt from Section 4:
Transforming into an “at rest” frame, , the spacetime metric equation reduces to , and (3.2) becomes:
For a timelike fifth dimension, may be drawn as a second axis orthogonal to , and the physics ratio (which, by the way, results in the material body in an electromagnetic field actually “feeling” a Newtonian force in the sense of due to the inequivalence of electrical and inertial mass) measures the “angle” at which the material body moves through the “time plane.”
For a spacelikefifth dimension, where one may wish to employ a compactified, hyper-cylindrical (see [Sundrum, R., TASI 2004 Lectures: To the Fifth Dimension and Back, http://arxiv.org/abs/hep-th/0508134 (2005).], Figure 1) and is a constant radius (distinguish from the Ricci scalar by context), . Substituting this into (3.2), leaving in the ratio obtained in (4.1), and inserting into the first term to maintain a dimensionless equation, then yields:
We see that here, the physics ratio measures an “angular frequency” of fifth-dimensional rotation. Interestingly, this frequency runs inversely to the mass, and by classical principles, this means that the angular momentum with fixed radius is independent of the mass, i.e., constant. If one doubles the mass, one halves the tangential velocity, and if the radius stays constant, then so too does the angular momentum. Together with the factor, one might suspect that this constant angular momentum is, by virtue of its constancy independently of mass, related to intrinsic spin. In fact, following this line of thought, one can arrive at an exact expression for the compactification radius , in the following manner:
Assume that is spacelike, casting one’s lot with the preponderance of those who study Kaluza-Klein theory. In (4.2), move the away from the first term and move the over to the first term. Then, multiply all terms by another . Everything is now dimensioned as an angular momentum , which we have just ascertained is constant irrespective of mass. So, set this all to , which for , represents intrinsic spin. The result is as follows:
Now, take the second and fourth terms, and solve for with , to yield:
where is the Planck length. This gives a definitive size for the compactification radius, and it is very close to the Planck length. (more…)