Lab Notes for a Scientific Revolution (Physics)

March 29, 2008

Stepping Back from Kaluza-Klein: Planned Revisions

Those who have followed my Weblog are aware that I have been putting in a lot of work on Kaluza-Klein theory.  This post is to step back from the canvas, lay out the overall picture of what I am pursuing, and summarize what I plan at present to change or correct in the coming days and weeks.  This is in keeping with the concept of this Weblog as “Lab Notes,” or as a public “scientific diary.”

There are really two main aspects to this Kaluza-Klein work:

First, generally, I have found that 5-D Kaluza-Klein theory is most simply approached by starting with (classical) Lorentz force motion, and requiring the Lorentz force motion to be along the geodesics of the five dimensional geometry.  I am far from the first person who recognizes that the Lorentz force can be represented as geodesic motion in a 5-D model.  But I have found, by starting with the Lorentz force, and by requiring the 5-D electromagnetic field strength tensor to be fully antisymmetric, that all of the many “special assumptions” which are often employed in Kaluza-Klein theory energy very naturally on a completely deductive basis, with no further assumptions required.  I also believe that this approach leads to what are perhaps some new results, especially insofar as the Maxwell tensor is concerned, and insofar as QED may be considered in a non-linear context.   The latest draft of this global work on Kaluza-Klein may be seen at Kaluza-Klein Theory and Lorentz Force Geodesics.

Second, specifically, within this broader context, is the hypothesis that the fifth-dimensional “curled” motion is the direct mainspring of intrinsic spin.  More than anything else, the resistance by many physicists to Kaluza-Klein and higher-dimensional theories, rests on the simple fact that this fifth dimension — and any other higher dimensions — are thought to not be directly observable.  In simplest form, “too small” is the usual reason given for this.  Thus, if it should become possible to sustain the hypothesis that intrinsic spin is a directly-observable and universally-pervasive outgrowth of the fifth dimension, this would revitalize Kaluza-Klein as a legitimate and not accidental union of gravitation and electrodynamics, and at the same time lend credence to the higher-dimensional efforts also being undertaken by many researchers.  The latest draft paper developing with this specific line of inquiry is at Intrinsic Spin and the Kaluza-Klein Fifth Dimension.

Now, the general paper at Kaluza-Klein Theory and Lorentz Force Geodesics is very much a work in progress and there are things in this that I know need to be fixed or changed.  If you should review this, please keep in mind the following caveats:  

First, sections 1-4 are superseded by the work at Intrinsic Spin and the Kaluza-Klein Fifth Dimension and have not been updated recently.   

Second, sections 5-7 are still largely OK, with some minor changes envisioned.  Especially, I intend to derive the “restriction” \Gamma^u_{55}=0 from F^{{\rm M} {\rm N} } =-F^{{\rm N} {\rm M} } rather than impose it as an ad hoc condition.   

Third, sections 8-11 needs some reworking, and specifically: a) I want to start with an integration over the five-dimensional volume with a gravitational constant G_{(5)}  suited thereto, and relate this to the four dimensional integrals that are there at present; and b) I have serious misgivings about using a non-symmetric (torsion) energy tensor and am inclined to redevelop this to impose symmetry on the energy tensor — or at least to explore torsion versus no torsion in a way that might lead to an experimental test.  If we impose symmetry on the energy tensor, then the Maxwell tensor will be the J^{\mu } =0 special case of a broader tensor which includes a J^\mu A^\nu + J^\nu A^\mu  term and which applies, e.g., to energy flux densities (Poynting components) T^{0k} , k=1,2,3 for “waves” of large numbers of electrons.  

Fourth, I am content with section 12, and expect it will survive the next draft largely intact.  Especially important is the covariant derivatives of the electrodynamic potentials being related to the ordinary derivatives of the gravitational potentials, which means that the way in which people often relate electrodynamic potentials to gravitational potentials in Kaluza-Klein theory is valid only in the linear approximation.  Importantly, this gives us a lever in the opposite direction, into non-linear electrodynamics.  

Fifth, I expect the development of non-linear QED in section 13 to survive the next draft, but for the fact that the R=0 starting point will be removed as a consequence of my enforcing a symmetric energy tensor in sections 8-11.  Just take out all the “R=0” terms and leave the rest of the equation alone, and everything else is more or less intact.

Finally, the experiment in Section 15, if it stays, would be an experiment to test a symmetric, torsionless energy tensor against a non-symmetric energy tensor with torsion.  (Basically, metric theory versus Cartan theory.)  This is more of a “back of the envelope” section at present, but I do want to pursue specifying an experiment that will test the possible energy tensors which are available from variational principles via this Kaluza-Klein theory. 

The paper at Intrinsic Spin and the Kaluza-Klein Fifth Dimension dealing specifically with the intrinsic spin hypothesis is also a work in progress, and at this time, I envision the following:

First, I will in a forthcoming draft explore positrons as well as electrons.  In compactified Kaluza-Klein, these exhibit opposite motions through x^5 , and by developing the positron further, we can move from the Pauli spin matrices toward the Dirac \gamma^\mu and Dirac’s equation.

Second, I have been engaged in some good discussion with my friend Daryl M. on a thread at sci.physics.relativity.  Though he believes I am “barking up the wrong tree,” he has provided a number of helpful comments, and especially at the bottom of post #2 where he discusses quantization in the fifth dimension using a wavelength n \lambda = 2 \pi R .  (I actually think that for fermions, one has to consider orientation / entanglement issues, and so to secure the correct “version,” one should use n \lambda = 4 \pi R  which introduces a factor of 2 which then can be turned into a half-integer spin.)  I am presently playing with some calculations based on this approach, which you will recognize as a throwback to the old Bohr models of the atom.

Third, this work of course uses x^5 = R\phi to define the compact fifth dimension.  However, in obtaining dx^5, I have taken R to be a fixed, constant radius.  In light of considering a wavelength n \lambda = 4 \pi R per above, I believe it important to consider variations in R rather than fixed R , and so, to employ dx^5 = Rd\phi + \phi dR .

There will likely be other changes along the way, but these are the ones which are most apparent to me at present.  I hope this gives you some perspective on where this “work in flux” is at, and where it may be headed.

Thanks for tuning in!

Jay. 

March 28, 2008

Further Considerations on the Energy Tensor: Metric versus Torsion

After reviewing some very helpful discussion on sci.physics.foundations and sci.physics.relativity in the threads I initiated regarding non-symmetric energy tensors and a suggested Kaluza-Klein experiment, and going back to Misner, Thorne and Wheeler’s “Gravitation,” I am starting to shift my viewpoint to be in opposition to the idea of using a non-symmetric (Cartan / Torsion) energy tensor because of the adverse impact this has on formulating a metric theory of gravitation.

There is a non-symmetric energy tensor in equations (15.11) to (15.13) of:

Kaluza-Klein Theory and Lorentz Force Geodesics Rev. 6.0

which is based upon the *non-symmetric* energy tensor of trace matter derived in (11.6).  What I have been turning over, is whether I ought to be comfortable with this result, and my sense runs against it.

However, at the point of original derivation in sections 8-11, there is actually a choice: one can construct the variation of the Lagrangian density of matter with respect to g_{\mu \nu}  such that a symmetric tensor will result, or one can choose not to, by creating a symmetric term or not.  This is actually a form of gravitational “symmetry breaking” that occurs in the process of taking the variation of the matter Lagrangian density with respect to the metric tensor.  I think both paths need to be developed, because they lead to on the one hand to a symmetric energy tensor, and on the other to a non-symmetric energy tensor.  In either case, the key term distinguishing this energy tensor from the Maxwell tensor is J^\mu A^\nu .

If we impose symmetry on the energy tensor, then the Maxwell tensor will be the J^u=0  special case of a broader tensor which includes a J^\mu A^\nu + J^\nu A^\mu term and which applies, e.g., to energy flux densities (Poynting components) T^{0k} , k=1,2,3 for “waves” of large numbers of electrons.

Then, the experiment becomes — not a test of the torsion tensor — but a test as between an energy tensor with and without torsion.  That is, the experiment as reformulated, becomes a test of metric-style versus Cartan-style theories of gravitation.

Dealing with the currents J^\mu  is clear.  Regarding how to deal with the potential A^\mu  in doing the experiment, think about a beam of electrons. They of course will all repel, so the beam will emerge conically from the electron gun if nothing is done to force them onto a parallel path. Now, take a circular cross section of electrons from the beam striking an energy flux detector.  One can think of the cross-sectional surface where the electron stream meets the detector as a “disk,” not unlike a charged, flat, frisbee, which is also productively thought about as a dielectric.  I would submit that one can assign a “zero” potential to the center of the cross section, and a varying non-zero potential to the periphery.  That is, if one were to take a circular dielectric disk and fill it with electric charge, then float some positive charge nearby, the positive charge — I believe — would be attracted toward and seat itself at the center of the disk, and so that would be a natural place to define the zero of potential.

This would also mean that regional detections of flux toward the fringes of the detector will be different than toward the center, assuming uniformity of charge distribution, because the energy created by the potentials among the electrons are different in different regions.  So, there is a way to assign potentials even without applying an external voltage, though someone conducting this experiment may want to also apply an external voltage simply to vary the range of experiments.

Now, to the main point: one should do the experiment with random, unpolarized electrons, and then again with spins aligned with and against the direction of propagation, merely to test the symmetric versus non-symmetric energy tensors one to the other.   One will win, the other not.  Metric versus torsion.

I am planning a restructuring of the paper at the above link.  In the near future, I will outline the proposed restruturing — what I would plan to keep and what I plan to change.

Jay.

March 22, 2008

A Possible Kaluza-Klein Experiment

It has been suggested — appropriately so — that I consider whether there might be one or more experiments which can be designed to validate or falsify some of the Klauza-Klein results which I have been posting of late. I believe that one possible experiment resides in the non-symmetric energy tensor of trace matter derived in (11.6) of my latest posted paper. Thus, I have added a new section 15 to this paper, and reposted the entire paper, with this new section 15, at Kaluza-Klein Theory and Lorentz Force Geodesics Rev. 6.0 Because this is of particular interest as it may open some new experimental windows, I have posted section 15 below as well. Please note: the specific discussion of the connection between the compactified fifth dimension, and intrinsic spin, is not updated in this paper, and the latest discusssion I have written up on this topic, is at Intrinsic Spin and the Kaluza-Klein Fifth Dimension.

  Section 15: At this juncture, we have enough information to propose an experiment to validate or falsify some of the results derived thus far.  We turn for this purpose to the stress energy tensor of matter (11.6), which we raise into contravariant notation as follows:

\kappa T^{\nu \mu } =-\kappa \left(F^{\mu \tau } F^{\nu } _{\tau } -{\textstyle\frac{1}{4}} g^{\mu \nu } F^{\sigma \tau } F_{\sigma \tau } \right)+{\textstyle\frac{\sqrt{2} }{2}} \overline{\kappa }g^{5\mu } J^{\nu } =\kappa T^{\mu \nu } _{Maxwell} +{\textstyle\frac{\sqrt{2} }{2}} \overline{\kappa }g^{5\mu } J^{\nu } . (15.1)

The Maxwell tensor T^{\mu \nu } _{Maxwell} =T^{\nu \mu } _{Maxwell} is, of course, a symmetric tensor.  But the added trace matter term g^{5\mu } J^{\nu } is not necessarily symmetric, that is, there is no a priori reason why g^{5\mu } J^{\nu } must be equal to g^{5\nu } J^{\mu } .  The origin of this non-symmetry was discussed earlier in Section 9.

 With an eye toward conducting an experiment, let us now consider (15.1) in the linear approximation of (13.6) where {\rm L}_{QCD} \approx -A^{\beta } J_{\beta } -{\textstyle\frac{1}{4}} F^{\sigma \tau } F_{\sigma \tau } .  In the linear approximation, as used to reach (13.3), (12.11) reduces to g^{5\mu } \approx \overline{\kappa }\left(\frac{\phi ^{5\mu } -{\textstyle\frac{1}{2}} bA^{\mu } }{1+{\textstyle\frac{1}{2}} \overline{\kappa }\phi } \right)\approx -{\textstyle\frac{1}{2}} \overline{\kappa }bA^{\mu } , and (15.1) becomes:

T^{\nu \mu } \approx -\left(F^{\mu \tau } F^{\nu } _{\tau } -{\textstyle\frac{1}{4}} g^{\mu \nu } F^{\sigma \tau } F_{\sigma \tau } \right)-2J^{\nu } A^{\mu } =T^{\mu \nu } _{Maxwell} -2J^{\nu } A^{\mu } , (15.2)

where we have also used b^{2} =8 and 2\kappa =\overline{\kappa }^{2} , and divided out \kappa .  The transpose of this non-symmetric energy tensor is:

T^{\mu \nu } \approx -\left(F^{\mu \tau } F^{\nu } _{\tau } -{\textstyle\frac{1}{4}} g^{\mu \nu } F^{\sigma \tau } F_{\sigma \tau } \right)-2J^{\mu } A^{\nu } =T^{\mu \nu } _{Maxwell} -2J^{\mu } A^{\nu } , (15.3)

 Now, it is known that a non-symmetric energy tensor, physically, is indicative of a non-zero spin density.  In particular, using (15.2) and (15.3), the non-symmetry of the energy tensor is related to a non-zero spin density tensor S^{\mu \nu \alpha } according to: [A good, basic discussion of the spin tensor is at http://en.wikipedia.org/wiki/Spin_tensor.]

S^{\mu \nu \alpha } _{;\alpha } =T^{\mu \nu } -T^{\nu \mu } =-2J^{\mu } A^{\nu } +2J^{\nu } A^{\mu } . (15.4)

For such a non-symmetric tensor, the “energy flux” is not identical to the “momentum density, as these differ by (15.4), for \mu =0, \nu =k=1,2,3 and vice versa.  If the spin density S^{\mu \nu \alpha } =0, then  in this special case, (15.4) yields:

J^{\mu } A^{\nu } =J^{\nu } A^{\mu } . (15.5)

So, for S^{\mu \nu \alpha } =0, (15.3) may be written using (15.5) as the explicitly-symmetric tensor:

T^{\mu \nu } \approx -\left(F^{\mu \tau } F^{\nu } _{\tau } -{\textstyle\frac{1}{4}} g^{\mu \nu } F^{\sigma \tau } F_{\sigma \tau } \right)-J^{\mu } A^{\nu } -J^{\nu } A^{\mu } =T^{\mu \nu } _{Maxwell} -J^{\mu } A^{\nu } -J^{\nu } A^{\mu } . (15.6)

 Now, let’s consider a experiment which is entirely classical.  The T^{0k} “Poynting” components of (15.4), (15.6) represent the energy flux across a two-dimensional area, for a flux of matter which we will take to be a stream of electrons, while the T^{k0} components represent the momentum density.  The proposed experiment, then, will be to fire a stream of a very large number of electrons thereby constituting an electron “wave,” and to detect the aggregate flux of energy across a two-dimensional surface under various spin preparations, in precisely the same manner that one might test the flow of luminous energy across a surface when using light waves rather than electron waves.  Specifically, we propose in test I to fire electrons without doing anything to orient their spins, so that, statistically, the number of electrons flowing through the flux surface with positive helicity is equal to the number with negative helicity and so the spin density is zero, and (15.6) applies.  In test II, we fire electrons, but apply a magnetic field before detecting the flux, to ensure that all of the electrons are aligned to positive helicity.  In this event, the spin density, by design, is non-zero, and one of (15.2) or (15.3) will apply.  In test III, we do the same, but now apply the magnetic field to ensure that all of the electrons have negative helicity, before detecting the flux.

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February 27, 2008

Lab Note 2: Derivation of the Maxwell Stress-Energy Tensor from Five-Dimensional Geometry, using a Four-Dimensional Variation

FOR SOME REASON, THESE EQUATIONS APPEAR CORRUPTED.  I AM CHECKING WITH WORDPRESS TECHNICAL SUPPORT AND HOPE TO HAVE THIS FIXED IN THE NEAR FUTURE. FOR NOW, PLEASE USE THE LINK IN THE FIRST PARAGRAPH, AND GO TO SECTION 10 — JAY.  

As mentioned previously, I have been able to rigorously derive the Maxwell tensor from a five dimensional Kaluza-Klein geometry based on Lorentz force geodesics, using a variational principle over the four spacetime dimensions of our common experience.  At the link: Kaluza-Klein Theory, Lorentz Force Geodesics and the Maxwell Tensor with QED, I have attached a complete version of this paper, which includes connections to quantum theory as well as an extensive summary not included in the version of the paper now being refereed at one of the leading journals.  This is a strategic decision not to overload the referee, but to focus on the mathematical results, the most important of which is this derivation of the Maxwell tensor.

Because this paper is rather large, I have decided on this weblog, to post section 10, where this central derivation occurs.  Mind you, there are nine sections which lay the foundation for this, but with the material below, plus the above link, those who are interested can see how this all fits together.  The key result emerges in equation (10.15) below.  Enjoy!

Excerpt: Section 10 — Derivation of the Maxwell Stress-Energy Tensor, using a Four-Dimensional Variation

 In section 8, we derived the energy tensor based on the variational calculation (8.4), in five dimensions, i.e., by the variation delta g^{{rm M} {rm N} } .  Let us repeat this same calculation, but in a slightly different way. 

 In section 8, we used (8.3) in the form of {rm L}_{Matter} =-{textstylefrac{1}{8kappa }} boverline{kappa }g^{5{rm B} } J_{{rm B} } =-{textstylefrac{1}{8kappa }} boverline{kappa }g^{{rm M} {rm N} } delta ^{5} _{{rm M} } J_{{rm N} } , because that gave us a contravariant g^{{rm M} {rm N} } against which to obtain the five-dimensional variation delta {rm L}_{Matter} /delta g^{{rm M} {rm N} } .  Let us instead, here, use the very last term in (8.3) as {rm L}_{Matter} , writing this as:

{rm L}_{Matter} equiv {textstylefrac{1}{2kappa }} R^{5} _{5} =-{textstylefrac{1}{8kappa }} boverline{kappa }left(g^{5beta } J_{beta } +{textstylefrac{1}{4}} g^{55} boverline{kappa }F^{sigma tau } F_{sigma tau } right)=-{textstylefrac{1}{8kappa }} boverline{kappa }left(g^{mu nu } delta ^{5} _{nu } J_{mu } +{textstylefrac{1}{4}} g^{55} g^{mu nu } boverline{kappa }F_{mu } ^{tau } F_{nu tau } right). (10.1)

It is important to observe that the term g^{5beta } J_{beta } is only summed over four spacetime indexes.  The fifth term, g^{55} J_{5} ={textstylefrac{1}{4}} g^{55} boverline{kappa }F^{sigma tau } F_{sigma tau } , see, e.g., (6.8).  For consistency with the non-symmetric (9.5), we employ g^{5beta } J_{beta } =g^{mu nu } delta ^{5} _{nu } J_{mu } rather than g^{5beta } J_{beta } =g^{mu nu } delta ^{5} _{mu } J_{nu } .  By virtue of this separation, in which we can only introduce g^{mu nu } and not g^{{rm M} {rm N} } as in section 8, we can only take a four-dimensional variation delta {rm L}_{Matter} /delta g^{mu nu } , which, in contrast to (8.4), is now given by:

T_{mu nu } equiv -frac{2}{sqrt{-g} } frac{partial left(sqrt{-g} {rm L}_{Matter} right)}{delta g^{mu nu } } =-2frac{delta {rm L}_{Matter} }{delta g^{mu nu } } +g_{mu nu } {rm L}_{Matter} . (10.2)

Substituting from (10.1) then yields:

T_{mu nu } ={textstylefrac{1}{4kappa }} boverline{kappa }left(delta ^{5} _{nu } J_{mu } +{textstylefrac{1}{4}} g^{55} boverline{kappa }F_{mu } ^{tau } F_{nu tau } right)-{textstylefrac{1}{2}} g_{mu nu } {textstylefrac{1}{4kappa }} boverline{kappa }left(g^{5beta } J_{beta } +{textstylefrac{1}{4}} g^{55} boverline{kappa }F^{sigma tau } F_{sigma tau } right). (10.3)

Now, the non-symmetry of sections 8 and 9 comes into play, and this will yield the Maxwell tensor.  Because delta ^{5} _{nu } =0, the first term drops out and the above reduces to:

kappa T_{mu nu } ={textstylefrac{1}{4}} boverline{kappa }left({textstylefrac{1}{4}} g^{55} boverline{kappa }F_{mu } ^{tau } F_{nu tau } right)-{textstylefrac{1}{2}} g_{mu nu } {textstylefrac{1}{4}} boverline{kappa }left(g^{5beta } J_{beta } +{textstylefrac{1}{4}} g^{55} boverline{kappa }F^{sigma tau } F_{sigma tau } right). (10.4)

Note that this four-dimensional tensor is symmetric, and that we would arrive at an energy tensor which is identical if (10.3) contained a delta ^{5} _{mu } J_{nu } rather than delta ^{5} _{nu } J_{mu } .  One again, the screen factor delta ^{5} _{nu } =0 is at work. 

 In mixed form, starting from (10.3), there are two energy tensors to be found.  If we raise the mu index in (10.3), the first term becomes delta ^{5} _{nu } J^{mu } =0 and we obtain:

-kappa T^{mu } _{nu } =-{textstylefrac{1}{4}} boverline{kappa }left({textstylefrac{1}{4}} g^{55} boverline{kappa }F^{mu tau } F_{nu tau } right)+{textstylefrac{1}{2}} delta ^{mu } _{nu } {textstylefrac{1}{4}} boverline{kappa }left(g^{5beta } J_{beta } +{textstylefrac{1}{4}} g^{55} boverline{kappa }F^{sigma tau } F_{sigma tau } right). (10.5)

with this first term still screened out.  However, if we transpose (10.3) and then raise the mu index, the first term becomes g^{5mu } J_{nu } and this term does not drop out, i.e.,

-kappa T_{nu } ^{mu } =-{textstylefrac{1}{4}} boverline{kappa }left(g^{5mu } J_{nu } +{textstylefrac{1}{4}} g^{55} boverline{kappa }F^{mu tau } F_{nu tau } right)+{textstylefrac{1}{2}} delta ^{mu } _{nu } {textstylefrac{1}{4}} boverline{kappa }left(g^{5beta } J_{beta } +{textstylefrac{1}{4}} g^{55} boverline{kappa }F^{sigma tau } F_{sigma tau } right). (10.6)

So, there are two mixed tensors to consider, and this time, unlike in section 8, these each yield different four-dimensional energy tensors.  Contrasting (10.5) and (10.6), we see that delta ^{5} _{nu } =0 has effectively “broken” a symmetry that is apparent in (10.6), but “hidden” in (10.5).  At this time, we focus on (10.5), because, as we shall now see, this is the Maxwell stress-energy tensor T^{mu } _{nu } =-left(F^{mu tau } F_{nu tau } -{textstylefrac{1}{4}} delta ^{mu } _{nu } F^{sigma tau } F_{sigma tau } right), before reduction into this more-recognizable form.
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February 25, 2008

Lab Note 2 Term Paper: Kaluza-Klein Theory and Lorentz Force Geodesics . . . and the Maxwell Tensor

Dear Friends: 

I have just today completed a paper titled “Kaluza-Klein Theory and Lorentz Force Geodesics,” which I have linked below:

Kaluza-Klein Theory and Lorentz Force Geodesics.

I have also submitted the draft linked above, to one of the leading physics journals for consideration for publication.

One of the things I have been beating my head against the wall over these past few weeks, is to deduce the Maxwell stress-energy tensor from the 5-dimensional geometry using Einstein’s equation including its scalar trace.  I finally got the proof nailed down this morning, and that is section 10 of the paper linked above.

I respectfully submit that the formal derivation of the Maxwell stress-energy tensor in section 10, provides firm support for the Spacetime-Matter (STM) viewpoint that our physical universe is a five-dimensional Kaluza-Klein geometry in which the phenomenon we observe in four dimensions are “induced” out of the fifth dimension, and that it supports the correctness of the complete line of development in this paper.  Section 10 — as the saying goes — is the “clincher.”

As is apparent to those who have followed the development of this particular “Lab Note,” my approach is to postulate the Lorentz force, and require that this be geodesic motion in 5-dimensions.  Everything else follows from there.  The final push to the Maxwell tensor in section 10, rests on adopting and implementing the STM viewpoint, and applying a 4-dimensional variational principle in a five-dimensional geometry.  If you have a serious interest in this subject, in addition to my paper, please take a look at The 5D Space-Time-Matter Consortium.

Best to all,

Jay.

February 19, 2008

Lab Note 2 Progress Report: Draft Paper on Kaluza-Klein Theory and Lorentz Force Geodesics

Hello to all my readers and contributors:

I have been very busy these past several days preparing my research on Kaluza-Klein five-dimensional theory into a formal paper.  I now have a draft paper sufficiently advanced, that I would like to share it with my readers and contributors for their comment.

I am not going to reproduce this directly on the blog as there are dozens of equations and the paper itself is already 25 pages.  However, I have linked a PDF copy of the latest draft below, for your perusal and comment.

Kaluza-Klein Theory and Lorentz Force Geodesics — 2-19-08 Draft

I know that there are literally dozens if not hundreds of Kaluza-Klein papers already out there in the world.  This one, I believe, is the one that actually describes how nature works, and how classical gravitation and electrodynamics actually do become united in nature.

Looking forward to your thoughts.

Jay. 

February 14, 2008

Lab Note 2, Part 3: Gravitational and Electrodynamic Potentials, the Electro-Gravitational Lagrangian, and a Possible Approach to Quantum Gravitation

Note:  You may obtain a PDF version of Lab Note 2, with parts 2 and 3 combined, at Lab Note 2, with parts 2 and 3.

Also Note: This Lab Note picks up where Lab Note 2, Part 2, left off, following section 7 thereof.  Equation numbers here, reference this earlier Lab Note.

8.  The Electrodynamic Potential as the Axial Component of the Gravitational Potential

 Working from the relationship F^{{\rm M} } _{{\rm T} } \propto 2\Gamma ^{{\rm M} } _{{\rm T} 5} which generalizes (5.4) to five dimensions, and recognizing that the field strength tensor F^{\mu \nu } is related to the four-vector potential A^{\mu } \equiv \left(\phi ,A_{1} ,A_{2} ,A_{3} \right) according to F^{\mu \nu } =A^{\mu ;\nu } -A^{\nu ;\mu } , let us now examine the relationship between A^{\mu } and the metric tensor g_{{\rm M} {\rm N} } .  This is important for several reasons, one of which is that these are both fields and so should be compatible in some manner at the same differential order, and not the least of which is that the vector potential A^{\mu } is necessary to establish the QED Lagrangian, and to thereby treat electromagnetism quantum-mechanically.  (See, e.g., Witten, E., Duality, Spacetime and Quantum Mechanics, Physics Today, May 1997, pg. 28.)

 Starting with {\tfrac{1}{2}} F^{{\rm M} } _{{\rm T} } \propto \Gamma ^{{\rm M} } _{{\rm T} 5} , expanding the Christoffel connections \Gamma ^{{\rm A} } _{{\rm B} {\rm N} } ={\tfrac{1}{2}} g^{{\rm A} \Sigma } \left(g_{\Sigma {\rm B} ,{\rm N} } +g_{{\rm N} \Sigma ,{\rm B} } -g_{{\rm B} {\rm N} ,\Sigma } \right), making use of g^{{\rm M} {\rm N} } _{,5} =0 which as shown in (6.5) is equivalent to F^{{\rm M} {\rm N} } =-F^{{\rm N} {\rm M} } , and using the symmetry of the metric tensor, we may write:

{\tfrac{1}{2}} F^{{\rm M} } _{{\rm T} } \propto \Gamma ^{{\rm M} } _{{\rm T} 5} ={\tfrac{1}{2}} g^{{\rm M} \Sigma } \left(g_{\Sigma {\rm T} ,5} +g_{5\Sigma ,{\rm T} } -g_{{\rm T} 5,\Sigma } \right)={\tfrac{1}{2}} g^{{\rm M} \Sigma } \left(g_{5\Sigma ,{\rm T} } -g_{5{\rm T} ,\Sigma } \right).  (8.1)

It is helpful to lower the indexes in field strength tensor and connect this to the covariant potentials A_{\mu } , generalized into 5-dimensions as A_{{\rm M} } , using F_{\Sigma {\rm T} } \equiv A_{\Sigma ;{\rm T} } -A_{{\rm T} ;\Sigma } , as such:

A_{\Sigma ;{\rm T} } -A_{{\rm T} ;\Sigma } \equiv F_{\Sigma {\rm T} } =g_{\Sigma {\rm M} } F^{{\rm M} } _{{\rm T} } \propto g_{\Sigma {\rm M} } g^{{\rm M} {\rm A} } \left(g_{5{\rm A} ,{\rm T} } -g_{5{\rm T} ,{\rm A} } \right)=\left(g_{5\Sigma ,{\rm T} } -g_{5{\rm T} ,\Sigma } \right). (8.2)

The relationship F_{\Sigma {\rm T} } \propto \left(g_{5\Sigma ,{\rm T} } -g_{5{\rm T} ,\Sigma } \right) expresses clearly, the antisymmetry of F_{\Sigma {\rm T} } in terms of the remaining connection terms involving the gravitational potential.  Of particular interest, is that we may deduce from (8.2), the proportionality

A_{\Sigma ;{\rm T} } \propto g_{5\Sigma ,{\rm T} } . (8.3)

(If one forms A_{\Sigma ;{\rm T} } -A_{{\rm T} ;\Sigma } from (8.3) and then renames indexes and uses g_{{\rm M} {\rm N} } =g_{{\rm N} {\rm M} } , one arrives back at (8.2).)  Further, we well know that F_{\Sigma {\rm T} } =A_{\Sigma ;{\rm T} } -A_{{\rm T} ;\Sigma } =A_{\Sigma ,{\rm T} } -A_{{\rm T} ,\Sigma } , i.e., that the covariant derivatives of the potentials cancel out so as to become ordinary derivatives when specifying F_{\Sigma {\rm T} } , i.e., that F_{\Sigma {\rm T} } is invariant under the transformation A_{\Sigma ;{\rm T} } \to A_{\Sigma ,{\rm T} } .  Additionally, the Maxwell components (7.10) of the Einstein equation, are also invariant under A_{\Sigma ;{\rm T} } \to A_{\Sigma ,{\rm T} } , because (7.10) also employs only the field strength F^{\sigma \mu } .  Therefore, let is transform A_{\Sigma ;{\rm T} } \to A_{\Sigma ,{\rm T} } in the above, then perform an ordinary integration and index renaming, to write:

A_{{\rm M} } \propto g_{5{\rm M} } . (8.4)

In the four spacetime dimensions, this means that the axial portion of the metric tensor is proportional to the vector potential, g_{5\mu } \propto A_{\mu } , and that the field strength tensor F_{\Sigma {\rm T} } and the gravitational field equations -\kappa T^{{\rm M} } _{{\rm N} } =R^{{\rm M} } _{{\rm N} } -{\tfrac{1}{2}} \delta ^{{\rm M} } _{{\rm N} } R are invariant under the transformation A_{\Sigma ;{\rm T} } \to A_{\Sigma ,{\rm T} } used to arrive at (8.4).  We choose to set A_{\Sigma ;{\rm T} } \to A_{\Sigma ,{\rm T} } , and can thereby employ the integrated relationship (8.4) in lieu of the differential equation (8.3), with no impact at all on the electromagnetic field strength or the gravitational field equations, which are invariant with respect to this choice.

9.  Unification of the Gravitational and QED Lagrangians

 The Lagrangian density for a gravitational field in vacuo is {\rm L}_{gravitation} =\sqrt{-g} R, where g is the metric tensor determinant and R=g^{\mu \nu } R_{\mu \nu } is the Ricci tensor.  Let us now examine a Lagrangian based upon the 5-dimensional Ricci scalar, which we specify by:

{\rm R} \equiv R^{\Sigma } _{\Sigma } =R^{\sigma } _{\sigma } +R^{5} _{5} =R+R^{5} _{5} . (9.1)
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