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I leave for the airport shortly so do not have a lot of time, but I will report that the paper was declined by PRD not for being wrong, but for being merely an "algoritm." The exact rejection is below:

[quote="PRD"]The spacetime metric should not depend on the nature of the test particles moving within the spacetime. Because various types of particles have both different electric charges and different charge to mass ratios, the metric you propose in Eq. (2.1) would have to depend on the particular type of test particle whose geodesic was being determined, and could not be a property of the background spacetime and electromagnetic fields.

If instead you want to view this simply as an algorithm for obtaining the equations of motion of a test particle, then the manuscript should be directed to a journal of mathematical physics.[/quote]
I immediately resubmitted, and replied with the following https://jayryablon.files.wordpress.com/2016/01/prd-reply-1-7-16.pdf. To say that the commonly used canonical gauge prescription [tex]p^\mu\rightarrow p^\mu+eA^\mu[/tex] is merely an "algorithm" and not really physics is really a stretch. I am certain that there have been hundreds of papers published in reputable physics journals that use this "prescription" which one might choose to call an "algorithm." But that does not mean this is not really physics.

Now, the point about the metric is interesting, but I think we have to yield our theoretical preconceptions about what the metric "should be" and look at the empirical evidence as to what a metric "is." The Lorentz motion, which occurs for a test particle that has an interaction mass which is not equal to its inertial mass in contrast to gravitation for which these are equivalent, means that the associated metric will also have a different character than occurs for pure gravitation. In particular, a test particle of a given charge e and mass m [i]will and does affect the metric in its local vicinity[/i], just like, as they say, "a hurricane makes its own weather." If we take the Lorentz force as clear empirical evidence about motion, and if we take the fact that the interaction and inertial masses are different for an electric charge as further clear empirical evidence about the world, and if that motion is to follow a path of least action as it must unless we want to change some very fundamental precepts of physics, then what the Lorentz force tells us by empirical evidence is that a charged particle does indeed change the metric interval [tex]d\sigma[/tex], albeit not the [i]metric tensor[/i], in its local environment.

To the point: The "background spacetime and electromagnetic fields" are [i]not[/i] independent of the test particle in those fields. The test particle itself affects those fields. If you carefully think this through as I will elaborate upon my return home, even in gravitation the mass of the test particle will alter the background spacetime, albeit extremely negligibly for a small mass like a baseball or a person and much more so for a large mass like a planet or a star.

I am actually encouraged by this: Joy suggested recently that I seek peer review to confirm that my calculations are correct. Weinberg has confirmed that they are. Now we are discussing how to interpret and understand the mathematical results.

I will expect to have more to add upon my return home.

Jay

[quote="Ben6993"]Joy holds that the fifth dimension was what Bell overlooked in his important papers. (Though Bellians, in an overwhelming magority, seem to disagree.) But microscopically, that adds nothing to resolve the above. Joy also holds that the fifth dimension can direct macroscopic effects too, see historical threads about 'exploding balls'. (Though Bellians and some anti-Bellians seem to disagree.) Presumably, the logical implication of a macro effect is that in some cases, but not necessarily in all cases, macro formulae could be influenced? Maybe only insofar as spin effects are involved, so not sure if this could affect the above.[/quote]
Ben you are mistaken about this. Bell didn't overlook or needed the fifth dimension for his argument. His hidden variable "lambda" covers [u]everything[/u]. Nor do I need a fifth dimension for my argument against Bell. What Bell overlooked is the [u]topology[/u] of S^3, which is a 3-dimensional space, just like R^3. Only its topology is spherical rather than "cubical." Of course I use S^7 for my general theorem for quantum correlations, but S^3 -- a three-dimension space -- is all that is needed to counter Bell.

But this discussion does not belong in this thread, so let us not clutter Jay's thread and take this discussion elsewhere if you have any further questions.

Hi Jay

Your opinions outweigh mine by 1000 times or more on this issue. However, let me have my uninformed say.

I think you are appealing to 'nature' as a judge and assuming nature to have been 4D over the last century or so?
If a hypothetical fifth dimension is compactified, as occurs in microscopic string dimensions, then this may not need to show up in formulae containing macroscopic dimensions? (Despite infinitesimals of calculus?) And yet may still hypothetically exist despite being hidden?

Joy holds that the fifth dimension was what Bell overlooked in his important papers. (Though Bellians, in an overwhelming magority, seem to disagree.) But microscopically, that adds nothing to resolve the above. Joy also holds that the fifth dimension can direct macroscopic effects too, see historical threads about 'exploding balls'. (Though Bellians and some anti-Bellians seem to disagree.) Presumably, the logical implication of a macro effect is that in some cases, but not necessarily in all cases, macro formulae could be influenced? Maybe only insofar as spin effects are involved, so not sure if this could affect the above.

Your formulae use x, e and m. To my mind, until one knows where these terms emerge from, one cannot rule out a fifth dimension. Maybe such a ruling should wait until e and m are explained, perhaps in a unification of quantum gravity with classical gravity? (In my own 24D model, e is easier to explain than x and m. x depends on a Rasch-like process dependent on hidden variables and is by far the hardest to explain, while m is associated with a wholly attractive macro gravity whereas IMO there can be gravitational repulsion acting on individual particles.)

But assuming your formulae are correct, you could use Occam's Razor to prefer a 4D scenario, pending more ideas as to what are e and m?

[quote="Yablon"]In a sense, Ben is correct that it it may boil down to which approach one prefers to take based on taste, because one can get to the same place -- or at least the Lorentz force -- either way.[/quote]
My momentary concession to "taste," or to Occam's Razor if one prefers, has made me think whether there might be something in what I just posted at https://jayryablon.files.wordpress.com/2016/01/kaluza-klein-mapping.pdf to actually [i]prove[/i] the 4D rather than 5D approach to be the one adopted by nature. Not just taste, but empirical proof. And I think I have it, but bear with me as I think out load and ask for the views of anyone else who has a view:

While the invariant metric linear element [tex]\sigma[/tex] is the [i]mathematically [/i]same in both equations (1) and (4) and via the variation (2) either one leads to the gravitational and Lorentz motion (3), there is one very important [i]structural [/i]difference between [tex]\sigma[/tex] as used in equation (1) versus as used in equation (4). In (1), [tex]d\sigma[/tex] is the differential invariant in 4D spacetime and in (4), although the results (3) are mathematically identical once (8), (9) and (10) are applied, [tex]d\sigma[/tex] is the differential invariant [i]in 5D spacetime-plus-gauge-dimension[/i]. (I call it the gauge dimension because it contains both [i]e[/i] and [tex]A^\mu[/tex] that are part and parcel of the gauge transformations that have a "[tex]+ieA^\mu[/tex]" in them.)

I say this for the following reason: Ponder equation (3) carefully. If the [tex]d\sigma[/tex] appearing is a 5D invariant, then by (4) this is [i]not the same[/i] as the 4D invariant. If one uses the 4D invariant from (4) alone, one only gets the gravitational geodesic equation without the Lorentz motion. When the 5D invariant is used, the gravitational equation is unchanged, expect for the fact that the gravitational motion is supplemented with the Lorentz motion.

Now, a century and more of experiments have proved beyond doubt that the [tex]d\sigma[/tex] of a 4D spacetime is observed as a key invariant everywhere: Lorentz symmetry for relative motion in special relativity, gravitational motion both Newtonian and with the enhancements coming from GR that explain perihelion precession, light bending, and so on. And -- when we observe Lorentz motion -- the same [tex]d\sigma[/tex] that appears in all of the foregoing is the one that appears in the Lorentz "force" motion also. In other words, [i]the Lorentz "force" motion operates and is observed in relation to the same invariant as is everything else that has been observed in relativistic physics since 1905 and 1916[/i]. (I put "force" in quotes, because once established as geodesic motion, the Lorentz motion is no longer really a "force.")

So: here is my argument: given that the Lorentz motion is observed to occur in reference to the very same invariant [tex]d\sigma[/tex] that is found everywhere else throughout relativistic physics, and given that this [tex]d\sigma[/tex] in all these other realms of physics has been firmly established as the invariant of a 4D spacetime and not something of 5D, this means that the [tex]d\sigma[/tex] in the Lorentz motion must be a 4D invariant, not a 5D invariant. As a result, the Lorentz motion must originate following the variation (2), from the [tex]d\sigma[/tex] in equation (1) and not the [tex]d\sigma[/tex] in equation (5).

Consequently, nature proves the 4D linear element of (1) and disproves the 5D element of (5). More generally, even if you develop Kaluza-Klein with some minor variation in relation to what is in https://jayryablon.files.wordpress.com/2016/01/kaluza-klein-mapping.pdf, the Lorentz motion occurs in relation to the 5D invariant and that is different from the 4D invariant that has established itself everywhere else in relativistic physics for over a century.

While I have been a big fan of Kaluza-Klein for 30 years, especially for its explanation of matter, I believe I have just disproved it. :) :(

Any thoughts? Can you help me strengthen and better articulate this? Or, am I missing something?

FYI, I will be in Mexico for the next week and a half with minimal online access. So if I take awhile to reply, or reply briefly and cryptically, that is the reason.

Jay

[quote="Joy Christian"]You can see what I had in mind by simply multiplying out the two terms of your line element (2.1). You then see the four terms explicitly, just like the four terms of the last equation on the page 4 of the paper you have just linked.

Now you can absorb the extra terms into the metric, as usually done in the 5D KK formulation, or you can absorb the extra terms in the differential, as you are doing:

[tex]dx^\mu \rightarrow d\chi^\mu=dx^\mu+ds(e/m)A^\mu.[/tex]

But this is still 4D + 1D = 5D, the fifth dimension now being in the differential rather than in the metric. So it seems to me that what you have got is essentially the same as the usual 5D KK, but done more neatly by using an extended, 5D differential. I don't necessarily mean this as a criticism. I am just thinking out loud.[/quote]
Hi again Joy:

Thanks for motivating me to look at the parallels to Kaluza-Klein. I prepared the attached two-page document which lays this out fully at https://jayryablon.files.wordpress.com/2016/01/kaluza-klein-mapping.pdf. In fact, one can develop these as equivalent formalisms. Mine makes canonical use of gauge theory and is restructed to four spacetime dimensions. Kaluza-Klein, with an equivalent path to the Lorentz force law, uses five spacetime dimensions without the need for a canonical use of gauge theory.

In a sense, Ben is correct that it it may boil down to which approach one prefers to take based on taste, because one can get to the same place -- or at least the Lorentz force -- either way. So let's talk about taste: Kaluza-Klein adds a new dimension that could be deemed superfluous and that must be explained physically in any event (which has been the main reason it is not universally accepted), and one can take the view that the gauge fields and the charge are worked into the five dimensions to fit the Lorentz force law, i.e., by taking the Lorentz force as a given and then constructing a five-dimensional metric that will lead back to that upon variation.

On the other hand, I do not need a fifth dimension so there is no need to explain it, and I do not need to start with the Lorentz force. If I accept gauge theory as part of four-dimensional physics -- and that is universally accepted -- then the correct canonical use of that gauge theory, with no other assumptions whatsoever, will lead to the Lorentz force [i]even if one did not know about the Lorentz force to begin with[/i].

For the moment, I have not added this to the paper, but it could obviously be another section. Since the paper is in review at PRD (transferred from PRL after they said it should go to a specialized journal), I think I will wait for now and see what I hear back. If it gets rejected as just being Klauza Klein in another guise, I will add this to the paper and develop the points I have made here.

Jay

Hi Jay

I wasn't going to comment (too apprehensive as it is way over my head!) but while Joy has put a query so will I.
I was going to ask whether you thought that having your equations in 4D was somehow more 'beautiful' [and therefore better aesthetically] than equations in 5D, But I suppose that depends on whether you feel the need for that extra dimension?

In my model you need the extra dimension, well, it is really 12 extra dimensions, four for each colour brane. Just as positrons are maybe travelling backwards in time, positive charge in my model is associated with travelling backwards in the colour branes' times, ie positive charge equates with anticolour [though that association is lost through confounding at quark level]. So you could loosely equate -charge with crests in this extra dimension and +charge with troughs in the usual analogy.

Recently you alerted me to distinguish between QCD magnetic monopoles and QED magnetic monopoles. In one of your QCD monopole papers a year or two ago (sorry, cannot remember which one) I remember you mentioning trying to find gravity equations was too difficult? Is that something you have now done here or is that still a separate and unresolved issue? That may seem to be a strange question, but in my model there is a QED-like gravity but also a QCD-like gravity [with a long range]. I know there are lots of problems with the latter, like QCD not obeying the inverse square law (but I have ideas to maybe circumvent that). With my model, I am thinking of how my quantum gravitons fit in with classical gravity. Since I can have [but too small to measure] graviton repulsion, it means that there can never be an exact relationship with classical mass which is always attractive. Also, in my model the bulk of the gravity is via QCD-like gravitons which is why I am asking about the 'too difficult' gravity calculations you mentioned in a paper about the QCD magnetic monoploles.

Best wishes

[quote="Yablon"][quote="Joy Christian"]Hi Jay, Your line element (2.1) is just the 5D Kaluza–Klein line element (albeit written differently). What am I missing?[/quote]
Hi Joy. The most important difference is that this line element (2.1) is confined four spacetime dimensions only, and that there is no change whatsoever to the form of the metric tensor. Jay

PS: Could you please point out a reference online that shows the line element you have in mind?

PPS: The link at http://www.weylmann.com/kaluza.pdf is a good case in point. The first (unnumbered) equation in their section 6 is similar in form to (2.1) in my paper multiplied by m/ds, sans the factor of 2, and sans the final term with A^2. Those are different enough to be different, and more importantly, mine has a far simpler root: all you do is apply gauge-covariant derivatives:

[tex]dx^\mu \rightarrow d\chi^\mu=dx^\mu+ds(e/m)A^\mu[/tex]

to the coordinate element [tex]dx^\mu[/tex], and [i]everything else[/i] follows with no heavy lifting or additional suppositions anywhere. The Lorentz force -- and Einstein equations that incorporate Maxwell's equations lock stock and barrel -- flow straight from gauge theory, purely and simply, and nothing more is needed.[/quote]
You can see what I had in mind by simply multiplying out the two terms of your line element (2.1). You then see the four terms explicitly, just like the four terms of the last equation on the page 4 of the paper you have just linked.

Now you can absorb the extra terms into the metric, as usually done in the 5D KK formulation, or you can absorb the extra terms in the differential, as you are doing:

[tex]dx^\mu \rightarrow d\chi^\mu=dx^\mu+ds(e/m)A^\mu.[/tex]

But this is still 4D + 1D = 5D, the fifth dimension now being in the differential rather than in the metric. So it seems to me that what you have got is essentially the same as the usual 5D KK, but done more neatly by using an extended, 5D differential. I don't necessarily mean this as a criticism. I am just thinking out loud.

[quote="Joy Christian"]Hi Jay, Your line element (2.1) is just the 5D Kaluza–Klein line element (albeit written differently). What am I missing?[/quote]
Hi Joy. The most important difference is that this line element (2.1) is confined four spacetime dimensions only, and that there is no change whatsoever to the form of the metric tensor. Jay

PS: Could you please point out a reference online that shows the line element you have in mind?

PPS: The link at http://www.weylmann.com/kaluza.pdf is a good case in point. The first (unnumbered) equation in their section 6 is similar in form to (2.1) in my paper multiplied by m/ds, sans the factor of 2, and sans the final term with A^2. Those are different enough to be different, and more importantly, mine has a far simpler root: all you do is apply gauge-covariant derivatives:

[tex]dx^\mu \rightarrow d\chi^\mu=dx^\mu+ds(e/m)A^\mu[/tex]

to the coordinate element [tex]dx^\mu[/tex], and [i]everything else[/i] follows with no heavy lifting or additional suppositions anywhere. The Lorentz force -- and Einstein equations that incorporate Maxwell's equations lock stock and barrel -- flow straight from gauge theory, purely and simply, and nothing more is needed.

Hi Jay, Your line element (2.1) is just the 5D Kaluza–Klein line element (albeit written differently). What am I missing?

Hello to all:

I wanted in a separate thread to highlight the equations I have found which fully unify classical gravitation and electromagnetism, following my pursuit discussed on this forum over the past few weeks of a geodesic foundation for the Lorentz force law which led me to all of this. The full paper showing these equations is at http://vixra.org/pdf/1512.0489v5.pdf. Equation numbers below reference where you can find these in the paper.

1) The canonically-gauge-extended spacetime metric equation:

[tex]ds^2=g_{\mu\nu}\big(dx^\mu+ds\big(e/m\big)A^\mu\big)\big(dx^\nu+ds\big(e/m\big)A^\nu\big)[/tex] (2.1)

2) The variational equation:

[tex]0=\delta\int_A^Bds[/tex] (2.4)

3) The geodesic equation of motion that results when 2) is applied to 1):

[tex]\frac{d^2x^\beta}{ds^2}=-\Gamma^\beta_{\mu\nu}\frac{dx^\mu}{ds}\frac{dx^\nu}{ds}+\frac{e}{m}F^\beta\!_\sigma\frac{dx^\sigma}{ds}[/tex] (2.17)

4) Definition of the canonically-gauge-extended Riemann tensor (";" designates the gravitationally-covariant derivative, as is customary):

[tex]\Re^\alpha\!_{\beta\mu\nu}V_\alpha\equiv\big[\partial_{;\mu}+ieA_\mu,\partial_{;\nu}+ieA_\nu\big]V_\beta[/tex] (4.2)

5) Maxwell's equation for electric charge:

[tex]ieJ^\nu=\partial_{;\mu}\dig(\Re^{\mu\nu}-(1/2)g^{\mu\nu}\Re\big)\ne 0[/tex] (4.19 first equation)

6) Maxwell's equation for magnetic charge:

[tex]0=\partial_{;\sigma}\Re^{\sigma\beta\mu\nu}}+\partial^{;\mu}\Re^{\beta\nu}+\partial^{;\nu}\Re^{\beta\mu}[/tex] (4.19 second equation)

7) Canonically-extended Einstein equation for gravitation and electromagnetism:

[tex]-\kappa T^\mu\!_\nu=(1/2)\Re^{\mu\beta}\!_{\beta\nu}+(1/2)\Re^\mu\!_\nu-(1/2)\delta^\mu\!_\nu\Re[/tex] (4.19 third equation)

8) Conservation of energy-momentum:

[tex]0=-\kappa \partial_{;\mu} T^\mu\!_\nu=\partial_{;\mu}\Big((1/2)\Re^{\mu\beta}\!_{\beta\nu}+(1/2)\Re^\mu\!_\nu-(1/2)\delta^\mu\!_\nu\Re\Big)[/tex] (4.16)

Keep in mind that a key finding is that [tex]\Re^{\mu\beta}\!_{\beta\nu}\ne\Re^\mu\!_\nu[/tex]. Indeed, the electric charge is a measure of the degree to which one cannot contract from the inner, but only the outer indexes. It took me a few days to pinpoint this; this is why my earlier drafts of the paper were "source-free," but this draft no longer is. This realization led me to see that the non-zero electric sources were also fully covered. Including sources has always been a challenge for geometric theories of E&M. Thus, on this key point:

9) The electric charge as a measure of the difference between the canonically-extended Ricci tensor and the inner-contracted canonically-extended Riemann tensor:

[tex]ieJ_\nu=\partial_{;\sigma}\big((1/2)\Re^\sigma\!\nu-(1/2)\Re^{\sigma\beta}\!_{\beta\nu}\big)[/tex] (4.15)

Jay