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Maxwell tarpits

  1. Apr 29, 2007 #1
    I am trying to develope magnetic terms in the Minkowski field tensor, based on an electron model positing circular currents and thus [tex]A_{\phi}[/tex]. It is easy enough to decompose this into [tex]A_{x,y}[/tex] and construct the tensor in a Cartesian basis, but essentially my study requires spherical coordinates because my goal is not just to reproduct Maxwell's equations, but to put this magnetic field into the Reissner-Nordstrom solution in spheric coordinates. No reference I've yet found deals with this: the circular vector potential manifests [tex]B_r [/tex] and [tex]B_\theta [/tex]. I need to be sure of the three expressions of the Minkowski form since you need covariant, contravariant, and mixed to build up the RHS of the Einstein equations, the stress-energy tensor. I have already accomplished this with the inhomogeneous electric field, which though not a point source, worked through the usual radial field solution without such difficulty. This problem is represented with only an [tex]E_r[/tex] so nothing usually is shown about the magnetic off-diagonal terms needed here. It is not so hard to effect a rotation of the orthogonal vector basis, as per spheric coordinates, but the formalism is not easy through the rest of this problem. The spheric coord. transform is like a rotation and a scale matrix with diagonal elements of [tex]<1,r,rsin\theta>[/tex]. This is like the square root of the metric [tex]g_{ab}[/tex] usually spoken of. . . . . . . .Part of this issue revolves around getting the expressions for CURL straight, in the spheric coords.
     
    Last edited: Apr 29, 2007
  2. jcsd
  3. Apr 30, 2007 #2

    Chris Hillman

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    "Magnetized" variants of RN?

    Your post is hard to follow (better paragraphing, more careful spelling, and a more careful choice of words might help), but buried somewhere in there you mention the Reissner-Nordstrom solution, which is an exact electrovacuum solution of the Einstein field equation of general relativity. That is, the RN solution is a Lorentzian manifold equipped with a EM field tensor satisfying the source-free curved spacetime Maxwell equations, such that the contributions to the stress energy tensor from the EM field (divided by [itex]8 \pi[/itex]) exactly matches the Einstein tensor computed from the metric tensor given as part of the structure of our Lorentzian manifold. In the literature such a model is sometimes called an Einstein-Maxwell solution (some authors apply this term to more general models including charged matter).

    Now it seems that your stated goal might be to construct a variant of the RN solution which has a magnetic field instead of (or in addition to) the electric field, possibly subject to additional constraints. By "has a magnetic field", I mean "has intrinsic magnetism", i.e. the principle Lorentz invariants of the desired EM field tensor prevent one from being able to remove the magnetic field at any given event by boosting the original local Lorentz frame there.

    If so, be aware that there is a lot known about exact solutions related to the RN electrovacuum, and also about magnetic monopoles, if that is where you are headed. The local versus global distinction is critical here. I can write down a general class of magnetized static axisymmetric electrovacuums locally, but depending upon what boundary conditions you have in mind, and how exotic a hypothesis you are willing to obtain, global solutions answering to a more precise description than you have yet provided might not exist.
     
  4. Apr 30, 2007 #3
    circular current field

    At the moment I'm not making latex work, so I'll just say I am working with a magnetic dipole field as described in my electron paper at: http://laps.noaa.gov/albers/physics/na . I start with a radial charge distribution and let it circulate at 'c'.
     
    Last edited: May 1, 2007
  5. May 1, 2007 #4
    Consider a nearfield charge distribution of: [tex]\rho(r)=r^{-1}e^{-r}[/tex] . If we allow this to manifest circular current of: [tex]j_\phi=c\rho sin(\theta)[/tex]. This particular form is soluble analytically for the implied vector potential: [tex]A_\phi[/tex], which I will not detail here. I looked up the formulae for curl expressed in spherical coordinates and vector basis to get the right forms for the resulting [tex]B_randB_\theta[/tex]. Herein lies part of my problem, to get the correct terms for curl like: [tex]B_\theta=-r^{-1}d_r(rA_\phi)[/tex]. The "extra" terms are, I think, expressible as Christoffel symbols of the transform. I need to be sufficiently clear here to construct the covariant Minkowski tensor. I think it is OK to use, rather than [tex]H_z,H_y,H_x[/tex], the spheric vector basis [tex]H_\phi,H_\theta,H_r[/tex]. Then hopefully I can construct the contravariant form, and then the entire stress-energy tensor for the RHS of the R-N metric solution, to which this will add a term. . . . . .The other issue I'm trying to master is getting the final nearfield form of my original current solution. I have accomplished this with the electric part of the problem. For this one must deal with the CURL and correct metric manipulations. ( This latex composer has been difficult.) . . . . time passes . . . . Come to think of it, the first step of taking the curl of the vector potential is not complete. It is so for a flat space, but the R-N metric has a form: [tex]<e^\nu,e^\lambda, r^2, r^2sin^2(\theta)>[/tex], and this was not included (the first two).
     
    Last edited: May 1, 2007
  6. May 1, 2007 #5

    Chris Hillman

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    Uh oh!

    Hi, Norman,

    Maybe I have simply misunderstood your post, but I fear that something, or several things, which you said appear to be founded upon misconceptions consistent with the hypothesis that you have experience working with Euclidean vector calculus (e.g. for hydrodynamics?), but are new to working with tensor and exterior calculus on curved manifolds.

    1. "Christoffel symbols of the transform" DMS (doesn't make sense).

    2. "I looked up the formulae for curl expressed in spherical coordinates and vector basis to get the right forms for the resulting" (magnetic vector field components). But you can't use the flat spacetime formulae in a curved spacetime, the RN-spacetime! If you want to treat a small magnetic component as a "test field" added to the electrostatic field of the RN electrovacuum, you should use the connection of the RN electrovacuum to compute the "curl". (Or the exterior derivative applied to the potential one-form; note that the RN electrovacuum is NOT conformal to Minkowski vacuum!) If you want an exact solution, you will need to derive that carefully--- if your idea makes sense, I can probably help with that.

    3. "hopefully I can construct the contravariant form, and then the entire stress-energy tensor for the RHS of the R-N metric solution, to which this will add a term." If you seek an exact Einstein-Maxwell solution, aka electrovacuum solution, which resembles RN electrostatic field but includes an intrinsic magnetic field, you can't "add a term" because the Einstein field equations are nonlinear. In particular, if you assume the only source of the gravitational field is the EM stress-energy tensor, this gives a system of coupled nonlinear PDEs.

    4. I didn't look at your paper but I hope you aren't trying to treat the electron as an "RN object", since this is a very old idea which has been known for a long long time not to work out.
     
  7. May 1, 2007 #6
    Thanks Chris for the observations. Don't we generate connections with the transform from <x,y,z> to <r,theta,phi>? I am just learning to put all this together. In the example mentioned above, the extra term is [tex]1/rA_\phi[/tex], and that reads like the Christoffel symbols I've been reading. They add a multiple of the field element, rather than its derivative. Perhaps I am about to hit a wall but I have come to several useful constructions with inhomogeneous electrodynamics. I expect to soon learn, from my standpoint, about your third point. In the electric field substitution I have accomplished, just as I designed to cancel the orders of 'r' producing infinities of observables, so the inverse r-squared term in the R-N metric is cancelled going inward from the classical electron radius. Thus a "pesky gravitational term" never gets too large before the positive hump in the graph of [tex]g_{oo}[/tex] resumes the -1/r dive.
     
    Last edited: May 1, 2007
  8. May 2, 2007 #7

    Chris Hillman

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    Danger Will Robinson!

    I used a technical term, connection, which you might not have yet encountered, which might explain the confusion. A "connection" is a way to compute covariant derivatives. In Riemannian geometry, we always use a particular connection which is determined by the metric tensor (not by any coordinate transformation!). This special connection, called the Levi-Civita connection, has especially nice properties.

    These aren't precise descriptions, of course-- to see the real definitions you need to study a good textbook.

    You seem to have a vivid imagination, but I'd advise you to firmly reign it in until you have mastered the mathematics of manifolds. Frankly it sounds to me like you have been indulging in wild speculation in which you are tossing around technical buzzwords you appear not to understand. This always makes me cringe because harsh experience has taught me that in forums like PF, wild speculation unfettered by any mathematical reasoning rarely leads to good things.
     
    Last edited: May 2, 2007
  9. May 3, 2007 #8
    I'm talking covariant differentiation and because your attitude is in the way you missed it. If you have not read my paper you do not even know what I have developed. I enjoyed a five-month correspondence with Hal Puthoff because we similarly approach a PV approach to the vacuum and gravitation, with metric differences.
     
    Last edited: May 3, 2007
  10. May 4, 2007 #9

    Chris Hillman

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    I need a macro... for my ignore list!

    Ah... now I understand.
     
  11. May 4, 2007 #10
    helping

    Chris, do you understand sufficiently to help me develop the tools I need? You already have, but I am slow.
     
    Last edited: May 4, 2007
  12. May 13, 2007 #11
    I am moving forward on a difficult problem with no help from the likes of the crappy attitudes expressed here.
     
  13. May 13, 2007 #12
    Albers, I see you post here similar things you post on PhysOrg and you reject honest advice here as you do there.

    Both Chris and myself have told you it might be wise you learn the basic mathematics of vector calculus and differential geometry but you don't seem to want to. When you're unable to do a change of basis from Cartesians to Spherical polars or don't know how to go from [tex]\partial_{\mu}[/tex] to [tex]\partial^{\nu}[/tex] in relativity, you have to be asking yourself if you're biting off more than you can chew.

    Chris is one of the best physicists in this area you're likely to ever converse with and rather than taking the advice of someone whose learnt all the relevent material and has a wealth of knowledge and understanding, you throw your toys out the pram at the first hint of someone not saying "Wow, you're a genius!"

    Differential geometry is hard, even for the best people, once you scratch it's surface. There's a reason people like 'Riemann' have so many results and entities in differential geometry named after him, because he was a genius and it took someone of his calibre to first come up with this stuff. As such, it's both extremely unlikely you'd manage it off the top of your head and simply a waste of time you reinventing the wheel even if you could.

    You're showing rather gaping holes in your knowledge on matters considered 'required understanding' for first courses in vector calculus or tensor mathematics. Why not actually use the vast amount of resources available to you both online and in libraries and learn such material?

    I really do struggle to understand the mentality of some people who actively try to avoid learning material that would obviously benefit them. While you try to use more maths than many of the cranks on PhysOrg, you display just as much loathing for learning as any of them. And a much more bitter attitude to negative comments.
     
  14. May 13, 2007 #13
    I am working hard to further my command of what you are talking about. When I started six years ago I worked out the spherical coordinate form of the Laplacian of a vector field . Have you seen this? I worked out all the necessary matrices with thoughts like, what is the matrix of the change of the local unit basis vector with spheric coordinates? You get cross-terms because of the changing basis rotating one into another, like phi-hat changing in the direction of r-hat. If you do all this accounting you get the gnarly expression to deal with a vector wave equation. Today I finished an important (to me) first page where I show how to keep a contravariant form through the expression of an arbitrary 3-vector, starting from Cartesian space and transforming to spherical expression (vector basis). Can you show me a concise exposition of how to develope in tensor notation the spheric coordinate expressions for CURL? The ingredient I've missed is the square root of the metric bilinear tensor for the spheric transform, namely the diagonal matrix <1,r,rsin(theta)>. What do you call this? It gives distance scale to local coordinate displacements. I know more than you think I do on some fronts, and wish to continue a rapid learning. Get with what I do know. I am working in relativity from the text by Adler, Schiffer, Bazin, Intro. to General Relativity. "Loathing of learning?" This is ridiculous, and malicious. I am looking for Chris Hillman's recommendation of electrovacuum studies and see his excellent bibliography, so I respect his broad knowledge here. I've not yet found an intro to this matter. I have enjoyed sufficient strongly positive reactions to some of my results that I will not tolerate a nonproductive atmosphere. . . . .I have announced what I think are significant results of substituting my electron nearfield in the Reissner-Nordstrom form, successfully for the electric part, on ScienceForums and on PhysOrg, though I will not again share on the latter.
     
    Last edited: May 14, 2007
  15. May 14, 2007 #14

    Chris Hillman

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    Tiny correction

    Wow, thanks, AlphaNumeric! But FYI I was trained as a mathematician. I have no formal background in physics whatever, just picked up some stuff by reading on the side.

    (Whozzit is in my "ignore list", BTW, so I am also blissfully ignorant of whatever he said about me--- I gather from your reply that it wasn't flattering, heh!)
     
  16. May 14, 2007 #15
    I have reached an elegant solution to this spherical tensor representation involving the 3x3 matrix I mentioned. Without it the magnetic vector potential is not a tensor. With it, it is and you can follow this through the formalism to produce curl. Cheers, I won't bother any of you further. Too bad, heh.
     
  17. May 14, 2007 #16
    I started as a mathematician by degree and ended up in a physics department as postgrad. It's the best way to do things :tongue:
    Oh no, we won't have someone to explain vector calculus a first year should know to. How will we stumble from one day to the next :cry:
     
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