Lab Notes for a Scientific Revolution (Physics)

April 13, 2008

Thesis Defense of the Kaluza-Klein, Intrinsic Spin Hypothesis — EARLY DRAFT

I have been engaged in a number of Usenet and private discussions about the paper Intrinsic Spin and the Kaluza-Klein Fifth Dimension, Rev 3.0 which I posted here on this blog on March 30.

A number of critiques have been raised, which you can see if you check out the recent Usenet threads I started related to intrinsic spin under the heading “Query about intrinsic verus [sic] orbital angular momentum,” over at sci.physics.foundations, sci.physics.relativity and sci.physics.research. These are among the “links of interest” provided in the right-hand pane of this weblog.

I believe that these critiques can be overcome, and that this hypothesis relating to Kaluza-Klein and intrinsic spin and the spatial isotropy of the square of the spin will survive and be demonstrated, ultimately, to be in accord with the physical reality of nature.

I have begun a new paper which is linked at: Thesis Defense of the Kaluza-Klein, Intrinsic Spin Hypothesis, Rev 1.0, which will respond thoroughly and systematically to the various critiques.  What is here so far is the introductory groundwork.  But, I would appreciate continued feedback as this development continues.

Note that the links within the PDF file unfortunately do not work, so to get the intrinsic spin paper, you need to go to Intrinsic Spin and the Kaluza-Klein Fifth Dimension, Rev 3.0.  Also, to get Wheeler’s paper which is referenced, go to Wheeler Geometrodynamics.

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March 30, 2008

Revised Paper on Kaluza-Klein and Intrinsic Spin, Based on Spatial Isotropy

I have now prepared an updated revision of a paper demonstrating how the compact fifth dimension of Kaluza-Klein is responsible for the observed intrinsic spin of the charged leptons and their antiparticles.  The more global, underlying view, is that all intrinsic spins originate from motion through the curled up, compact x^5 dimension.  This latest draft is linked at:

Intrinsic Spin and the Kaluza-Klein Fifth Dimension, Rev 3.0

Thanks to some very helpful critique from Daryl M. on a thread at sci.physics.relativity, particularly post #2, I have entirely revamped section 5, which is the heart of the paper wherein we establish the existence of intrinsic spin in x^5 , on the basis of “fitting” oscillations around a 4\pi loop of the compact fifth dimension (this is to maintain not only orientation but entanglement version).  From this approach, quantization of angular momentum in x^5 naturally emerges, it also emerges that the intrinsic x^5 angular momentum in the ground state is given by (1/2) \hbar .

In contrast to my earlier papers where I conjectured that the intrinsic spin in x^5 projects out into the three ordinary space dimensions by virtue of its orthogonal orientation relative to the x^5 plane, which was critiqued in several Usenet posts, I now have a much more direct explanation of how the intrinsic spin projects out of x^5  to where we observe it.

In particular, we recognize that one of the objections sometimes voiced with regards to a compactified fifth dimension is the question: how does one “bias” the vacuum toward one of four space dimensions, over the other three, by making that dimension compact?  This was at the root of some Usenet objections also raised earlier by DRL.

In this draft — and I think this will overcome many issues — we require that at least as regards intrinsic angular momentum, the square of the J^5 = (1/2) \hbar obtained for the intrinsic angular momentum in x^5 , must be isotropically shared by all four space dimensions.  That is, we require that there is to be no “bias” toward any of the four space dimensions insofar as squared intrinsic spin is concerned.  Because J^5 = (1/2) \hbar emanates naturally from the five dimensional geometry, we know immediately that \left(J^{5} \right)^{2} ={\textstyle\frac{1}{4}} \hbar ^{2} , and then, by the isotropic requirement, that \left(J^{1} \right)^{2} =\left(J^{2} \right)^{2} =\left(J^{3} \right)^{2} ={\textstyle\frac{1}{4}} \hbar ^{2} as well.  We then arrive directly at the Casimir operator J^{2} =\left(J^{1} \right)^{2} +\left(J^{2} \right)^{2} +\left(J^{3} \right)^{2} ={\textstyle\frac{3}{4}} \hbar ^{2} in the usual three space dimensions, and from there, continue forward deductively.

For those who have followed this development right along, this means in the simplest terms, that rather than use “orthogonality” to get the intrinsic spin out of x^5 and into ordinary space, I am instead using “isotropy.”

There is also a new section 8 on positrons and Dirac’s equations which has not been posted before, and I have made other editorial changes throughout the rest of the paper.

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 3, 2008

Intrinsic Spin and the Kaluza-Klein Fifth Dimension: Journal Submission

   I mentioned several days ago that I had submitted a Kaluza Klein paper to one of the leading journals.  That lengthy paper was not accepted, and you can read the referee report and some of my comments here at sci.physics.foundations or here, with some other folks’ comments, at sci.physics.relativity.  The report actually was not too bad, concluding that “the author must have worked a considerable amount to learn quite a few thing in gravitation theory, and a number of the equations are correctly written and they do make sense, however those eqs. do not contain anything original.”  I would much rather hear this sort of objection, than be told — as I have been in the past — that I don’t know anything about the subject I am writing about.

   In fact, there is one finding in the above-linked paper which, as I thought about it more and more, is quite original, yet I believe it was lost in the mass of this larger paper.  And, frankly, it took me a few days to catch on to the full import of this finding, and so I downplayed it in the earlier paper.  Namely:  that the compactified fifth dimension of Kaluza-Klein theories is the mainspring of the intrinsic spins which permeate particle physics.

   I have now written and submitted for publication, a new paper which only includes that Kaluza-Klein material which is necessary to fully support this particular original finding.  You may read the submitted paper at Intrinsic Spin and the Kaluza-Klein Fifth Dimension.  I will, of course, let you know what comes from the review of this paper.

   I welcome your comments and feedback.

   Jay.

February 29, 2008

Lab Note 2: Why the Compactified, Cylindrical Fifth Dimension in Kaluza Klein Theory may be the “Intrinsic Spin” Dimension

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,

frac{dx^{5} }{dtau } equiv -frac{1}{b} frac{sqrt{hbar calpha } }{sqrt{G} m} =-frac{1}{b} frac{1}{sqrt{4pi G} } frac{q}{m} . (3.2)

which is used to connect the q/m ratio from the Lorentz law to geodesic motion in five dimensions, and b is a numeric constant of proportionality. Section 4 below picks up from this.

Excerpt from Section 4:

Transforming into an “at rest” frame, dx^{1} =dx^{2} =dx^{3} =0, the spacetime metric equation d/tau ^{2} =g_{/mu /nu } dx^{/mu } dx^{/nu } reduces to dtau =pm sqrt{g_{00} } dx^{0} , and (3.2) becomes:

frac{dx^{5} }{dx^{0} } =pm frac{1}{b} sqrt{frac{g_{00} }{4pi G} } frac{q}{m} . (4.1)

For a timelike fifth dimension, x^{5} may be drawn as a second axis orthogonal to x^{0} , and the physics ratio q/m (which, by the way, results in the q/m material body in an electromagnetic field actually “feeling” a Newtonian force in the sense of F=ma due to the inequivalence of electrical and inertial mass) measures the “angle” at which the material body moves through the x^{5} ,x^{0} “time plane.”

For a spacelikefifth dimension, where one may wish to employ a compactified, hyper-cylindrical x^{5} equiv Rphi (see [Sundrum, R., TASI 2004 Lectures: To the Fifth Dimension and Back, http://arxiv.org/abs/hep-th/0508134 (2005).], Figure 1) and R is a constant radius (distinguish from the Ricci scalar by context), dx^{5} equiv Rdphi . Substituting this into (3.2), leaving in the pm ratio obtained in (4.1), and inserting c into the first term to maintain a dimensionless equation, then yields:

frac{Rdphi }{cdtau } =pm frac{1}{b} frac{sqrt{hbar calpha } }{sqrt{G} m} =pm frac{1}{b} frac{1}{sqrt{4pi G} } frac{q}{m} . (4.2)

We see that here, the physics ratio q/m 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 pm 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 R, in the following manner:

Assume that x^{5} is spacelike, casting one’s lot with the preponderance of those who study Kaluza-Klein theory. In (4.2), move the c away from the first term and move the m over to the first term. Then, multiply all terms by another R. Everything is now dimensioned as an angular momentum mcdot vcdot R, which we have just ascertained is constant irrespective of mass. So, set this all to pm {textstylefrac{1}{2}} nhbar , which for n=1, represents intrinsic spin. The result is as follows:

mfrac{Rdphi }{dtau } R=pm frac{1}{b} frac{sqrt{hbar c^{3} alpha } }{sqrt{G} } R=pm frac{1}{b} frac{c}{sqrt{4pi G} } qR=pm frac{1}{2} nhbar . (4.3)

Now, take the second and fourth terms, and solve for R with n=1, to yield:

R=frac{b}{2sqrt{alpha } } sqrt{frac{Ghbar }{c^{3} } } =frac{b}{2sqrt{alpha } } L_{P} , (4.4)

where L_{P} =sqrt{Ghbar /c^{3} } is the Planck length. This gives a definitive size for the compactification radius, and it is very close to the Planck length. (more…)

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 16, 2008

Lab Note 2 Intermezzo: Change of View to a Spacelike Fifth Dimension, as the Geometric Foundation of Intrinsic Spin

(A PDF version of this Lab Note may be downloaded from Lab Note 2 Intermezzo.) 

 Those who have followed the development of this lab note know that I have been working with a Kaluza-Klein theory which regards the fifth dimension as timelike, rather than spacelike.  After reviewing some key literature in the field including a Sundrum Lecture recommended by Martin Bauer and several articles by Paul Wesson linked over at The 5-D Space-Time-Matter Consortium, I have undergone a conversion to the view that the fifth dimension needs to be spacelike – not timelike – and specifically, that it needs to be a compact, spacelike hypercylinder.  In this conversion, I am motivated by the following reasoning, which gives a geometric foundation to intrinsic spin: 
(more…)

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)
(more…)

February 6, 2008

Lab Note 2, Part 2: Gravitational and Inertial Mass, and Electrodynamics as Geometry, in 5-Dimensional Spacetime

(You may obtain a PDF version of this lab note at Electrodynamic Geodesics) Note: See also Part 3 of this Lab Note, Gravitational and Electrodynamic Potentials, the Electro-Gravitational Lagrangian, and a Possible Approach to Quantum Gravitation, which contains further development.

1.  Introduction  It has been understood at least since Galileo’s refutation of Aristotle which legend situates at the Leaning Tower of Pisa, that heavier masses and lighter masses similarly-disposed in a gravitational field will accelerate at the same rate and reach the ground after identical times have elapsed.  Physicists have come to describe this with the principle that the “gravitational mass” and the “inertial mass” of any material body are “equivalent.”  As a material body becomes more massive and so more-susceptible to the pull of a gravitational field (back when gravitation was viewed as action at a distance), so too this increase in massiveness causes the material body in equal measure to resist the gravitational pull.  By this equivalence, the result is a “wash,” and so with the neglect of any air resistance, all the bodies accelerate and fall at the same rate.  (The other consequence of Galileo’s escapade, is that it strengthened the role of experimental testing, in relation to the “pure thought” upon which Aristotle had relied to make the “obvious” but untested and in fact false argument that heavy objects should fall faster.  In this way, it spawned the essence of what we today know as the scientific method which remains a dynamic blend of thought and creativity, with experience and cold, hard numbers derived from measurement of masses, lengths, and times.)

  Along his path to developing the General Theory of Relativity (GTR), Albert Einstein made a brief stop in 1911 in an imaginary elevator, to conduct a gedanken in which he concluded that the physical experience of an observer falling freely in a gravitational field before terminally hitting the ground is no different from what was commonly thought of as Newton’s inertial motion in which a body in motion remained in motion unless acted upon by a “force.”  (GTR later showed that this was not quite true, the “asterisk” to this insight arising from the so-called tidal forces.)  And, he concluded that the force one feels standing on the floor of an elevator in free fall to which a constant force is then applied, is no different from the force one feels when standing on the surface of the earth.

  The General Theory of Relativity, in the end, captured inertial motion and its close cousin of free-fall motion in a gravitational field, in the most elegant way, as simple geodesic motion in a curved geometry along geodesic paths which coincide precisely with the paths one observes for bodies moving under gravitational influences.  This was a triumph of the highest order, as it placed gravitational theory on the completely-solid footing of Riemannian geometry, and became the “gold standard” against which all other physical theories are invariably measured, even to this day.  (“Marble and wood” is another oft-employed analogy.)

  However, the question of “absolute acceleration,” that is, of an acceleration which is not simply a geodesic phenomenon of unimpeded free fall through a swathe carved out by geometry, but rather one in which an observer actually “feels” a “force” which can be measured by a “weight scale” in physical contact between the observer and that body which applies the force, is in fact not resolved by GTR.  To this day, it is hotly-debated whether or not there is such a thing as “absolute acceleration.”  Surely, the forces we feel on our bodies in elevators and cars and standing on the ground are real enough, but the question is whether there is some way to understand these forces — which are impediments to what would otherwise be our own geodesic free fall motion in spacetime under the influence of gravity and nothing more — as geodesic forces in their own right, simply of a different, supplemental, and perhaps more-subtle character than the geodesics of gravitation.  That is the central question to be examined in this lab note. (more…)

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