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Initial Value Formulation on curved space-time: Maxwell's equations

  1. Apr 24, 2013 #1

    WannabeNewton

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    Hello there Ladies and Gents! This question is (mostly) related to problem 10.2 in Wald which is to show that the source-free Maxwell's equations have a well posed initial value formulation in curved space-times. We start off with a globally hyperbolic space-time ##(M,g_{ab})## and a spacelike cauchy surface ##\Sigma ##, in this space-time, which will soon become our initial data surface. The first part of the problem was to show given the electric and magnetic fields ##E_{a} = F_{ab}n^{b}, B_{a} = -\frac{1}{2}\epsilon_{ab}{}{}^{cd}F_{cd}n^{b}## on ##\Sigma##, where ##n^{a}## is the unit normal to ##\Sigma##, that Maxwell's equations ##\nabla^{a}F_{ab} = -4\pi j_{b}, \nabla_{[a}F_{bc]} = 0## implied ##D_{a}E^{a} = -4\pi j_{a}n^{a}, D_{a}B^{a} = 0## where ##D_{a}## is the derivative operator associated with the induced metric ##h_{ab}## on ##\Sigma##. I will spare you the details of the calculations involved in showing these two relations hold; you can, for now, take my word that I have indeed shown them to be true.

    The next part of the problem, which is the one of relevance here, was to then show that the source-free Maxwell's equations, i.e. ##j^{a} = 0##, have a well posed initial value formulation in the sense that given ##E^{a},B^{a}## on ##\Sigma## subject to the above constraints ##D_{a}E^{a} = D_{a}B^{a} = 0##, there exists a unique solution ##F_{ab}## of Maxwell's equations throughout ##M## with the given initial data and that the solution had the appropriate continuity of initial data to solution map and domain of dependence (causality). We are told to assume global existence of a vector potential ##A_{a}##.

    The key result presented in the text that will be of critical use here is theorem 10.1.2 which states: Let ##(M,g_{ab})## be a globally hyperbolic space-time and let ##\nabla_{a}## be any derivative operator and let ##\Sigma## be a smooth, space-like Cauchy surface. Consider the system of ##n## linear equations for ##n## unknown functions ##\phi_1,...,\phi_n## of the form ##g^{ab}\nabla_{a}\nabla_{b}\phi_{i} + \sum _{j}(A_{ij})^{a}\nabla_{a}\phi_{j} + \sum_{j}B_{ij}\phi_{j}+ C_{i} = 0##, where ##(A_{ij})^{a}## are of course vector fields and ##B_{ij}## smooth scalar fields. Then, given arbitrary smooth initial data ##(\phi_{i}, n^{a}\nabla_{a}\phi_{i})## for ##i = 1,...,n## on ##\Sigma##, there exists a unique solution of the above equation throughout ##M## that has the appropriate continuity and domain of dependence properties.

    Note that given arbitrary initial data ##(A_{a}, n^{b}\nabla_{b}A_{a})## on ##\Sigma## where ##A_{a}## is the 4-potential corresponding to the arbitrary given initial electric and magnetic field on ##\Sigma##, Maxwell's equations ##\nabla^{b}(\nabla_{a}A_{b} - \nabla_{b}A_{a}) = 0 ## are not in the form required by theorem 10.1.2 above in order for a well posed initial value formulation to be guaranteed. However, note that if the Lorentz gauge ##\nabla^{a}A_{a} = 0## is satisfied throughout ##M##, then we can always fix this gauge under which Maxwell's equations become ##\nabla^{a}\nabla_{a}A_{b} - R_{b}{}{}^{d}A_{d} = 0##. Now this does have the form required for a well posed initial value formulation to be guaranteed and hence this equations is satisfied throughout ##M## by ##A_{a}## for the new gauge transformed initial 4-potential and will give us a unique ##F_{ab}## with all the appropriate continuity and causality conditions. Therefore, all we need to show is that ##\nabla^{a}A_{a} = 0## will always hold throughout ##M## given the original arbitrary initial data ##(A_{a}, n^{b}\nabla_{b}A_{a})## associated with the initial ##E^{a},B^{a}## on ##\Sigma## satisfying the constraints ##D_{a}E^{a} = D_{a}B^{a} = 0## on ##\Sigma##.

    ##\nabla^{a}\nabla_{a}A_{b} - R_{b}{}{}^{d}A_{d} = 0## is always satisfied so we can work with this to start off. We have ##\nabla^{b}\nabla^{a}\nabla_{a}A_{b} - \nabla^{b}(R_{b}{}{}^{d}A_{d}) = 0##. ##\nabla^{b}\nabla^{a}\nabla_{a}A_{b} - \nabla^{a}\nabla^{b}\nabla_{a}A_{b} = R^{ba}{}{}_{a}{}{}^{e}\nabla_{e}A_{b} + R^{ba}{}{}_{b}{}{}^{e}\nabla_{a}A_{e} = -R^{be}\nabla_{e}A_{b} + R^{ae}\nabla_{a}A_{e} = -R^{eb}\nabla_{b}A_{e} + R^{ae}\nabla_{a}A_{e} = 0## and ## \nabla^{b}\nabla_{a}A_{b} - \nabla_{a}\nabla^{b}A_{b} = R_{a}{}{}^{d}A_{d}## therefore ##\nabla^{a}\nabla_{a}\nabla^{b}A_{b} + \nabla^{a}(R_{a}{}{}^{d}A_{d}) - \nabla^{b}(R_{b}{}{}^{d}A_{d}) = \nabla^{a}\nabla_{a}\nabla^{b}A_{b} = 0##. As you probably noticed, this equation also has the form required for theorem 10.1.2 to apply thus we will have, by uniqueness of the solution, ##\nabla^{a}A_{a} = 0## throughout ##M## given that we can always arrange for initial conditions ##(\nabla^{a}A_{a}, n^{b}\nabla_{b}\nabla^{a}A_{a})## on ##\Sigma## such that ##\nabla^{a}A_{a} = n^{b}\nabla_{b}\nabla^{a}A_{a} = 0## on ##\Sigma##. Now, we are already given some initial ##A_{a}## on ##\Sigma## itself so we can always make a gauge transformation so that ##\nabla^{a}A_{a} = 0## on ##\Sigma##. All that is left to show is the latter, which will come out of the constraints on the initial electric and magnetic field.

    We first see that on ##\Sigma##, ##D_{a}E^{a} = h_{a}{}{}^{b}h^{a}{}{}_{c}\nabla_{b}E^{c} = \delta^{b}{}{}_{c}\nabla_{c}(F^{cd}n_{d}) = n_{d}R^{db}A_{b} - n_{d}\nabla_{c}\nabla^{d}A^{c} + F^{cd}\nabla_{c}n_{d} = 0##. Now ##\nabla_{c}\nabla^{d}A^{c} - \nabla^{d}\nabla_{c}A^{c} = R^{db}A_{b}## hence the first constraint reduces to ##n^{d}\nabla_{d}\nabla^{c}A_{c} - F^{cd}\nabla_{c}n_{d} = 0##. Similarly we have ##D_{a}B^{a} = -\frac{1}{2}\epsilon^{cdef}\delta^{b}{}{}_{c}\nabla_{b}(F_{ef}n_{d}) = n_{d}\epsilon^{bdef}\nabla_{b}F_{ef} + \epsilon^{bdef}F_{ef}\nabla_{b}n_{d} = 0##. Note that ##n_{d}\epsilon^{bdef}\nabla_{b}F_{ef} = n_{d}\epsilon^{edbf}\nabla_{e}F_{bf} = n_{d}\epsilon^{fdbe}\nabla_{f}F_{be}## therefore ##3n_{d}\epsilon^{bdef}\nabla_{b}F_{ef} = n_{d}\epsilon^{bdef}\nabla_{[b}F_{ef]} = 0## by virtue of Maxwell's equations, leaving us with ##\epsilon^{bdef}F_{ef}\nabla_{b}n_{d} = 0##. Multiplying both sides by ##\epsilon_{ijkl}## we find that ##\delta^{[b}_{i}\delta^{d}_{j}\delta^{e}_{k}\delta^{f]}_{l}F_{ef}\nabla_{b}n_{d} = F_{kl}\nabla_{[i}n_{j]} = 0## giving us ##F^{cd}\nabla_{[c}n_{d]} = F^{cd}\nabla_{c}n_{d} - F^{cd}\nabla_{d}n_{c} = F^{cd}\nabla_{c}n_{d} - F^{dc}\nabla_{c}n_{d} = 2F^{cd}\nabla_{c}n_{d} = 0## which finally implies that ##n^{d}\nabla_{d}\nabla^{c}A_{c} = 0## on ##\Sigma## as desired therefore the source-free Maxwell's equations have a well posed initial value formulation.

    Hopefully that wasn't too long. I posted my work in part so that anyone wanting to check it and/or look over it for themselves could do so but also in part to motivate my concluding question. Now, as you can see, this proof holds for the source free Maxwell's equations but what if ##j^{a}\neq 0##? What if it is some highly non-trivial current density? Wald never discusses in detail what happens to the initial value formulation of Maxwell's equations in such a case, not on curved space-time nor on flat space-time. At least for physically relevant non-vanishing current densities, we should expect a well posed initial value formulation shouldn't we? How do the physics and math work out in special cases of non-vanishing current densities and possibly for general non-vanishing ones with regards to there being a well posed initial value formulation? Thank you very much in advance!
     
  2. jcsd
  3. Apr 24, 2013 #2
    It was.
     
  4. Apr 24, 2013 #3

    Jano L.

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    I did not read the whole post and I do not know about the curved case, but in the simple case of flat space, if the sources are prescribed consistently with the equation of continuity

    $$
    \partial_t \rho + \nabla \cdot \mathbf j = 0,
    $$
    and if the initial fields are consistent with the time-independent Maxwell equations

    $$
    \nabla \cdot \mathbf E = \rho,
    $$
    $$
    \nabla \cdot \mathbf B = 0,
    $$

    then I would expect there is always unique solution of the remaining equations, even with delta sources. I cannot prove it, so if you are looking for more mathematical discussions, perhaps this book can be useful:

    Alain Bossavit: Computational electromagnetism


    http://butler.cc.tut.fi/~bossavit/

    (scroll down to the section "Books", the line "This one, out of print, ... "
     
  5. Apr 24, 2013 #4

    WannabeNewton

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    Thank you very much Jano, I'll download the relevant chapters. If you know other books regarding the issue in flat space-time then I would also be much obliged if you could recommend those as well, even if they aren't available free online. However if the issue for non-vanishing current densities is highly non-trivial even for the flat case then that doesn't bode very well for what I could potentially expect from the curved case :frown:
     
  6. Apr 24, 2013 #5
    Wannabe:

    What kind of math is that?

    Anyway, I think the answer is 17. (I'm kidding, I'm kidding:)
     
  7. Apr 24, 2013 #6

    robphy

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  8. Apr 24, 2013 #7

    WannabeNewton

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    Ah that comment does look like a fell blow. Nevertheless, it looks like this book will be a very very fun read. Thank you as always for the great reference robphy!

    EDIT: Just a quick clarification robphy, when Friedlander refers to the Einstein Maxwell equations (when saying they don't have the form 3.2.2) is he including a non-vanishing current density in the Maxwell equations? It seems like he is but I want to make sure because Wald mentions at the end of ch10 that the Einstein Maxwell equations do have a well posed initial value formulation but he still only considers the vanishing current density case so that the coupled Einstein and Maxwell equations still have the desired form.
     
    Last edited: Apr 24, 2013
  9. Apr 24, 2013 #8

    WannabeNewton

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    Hey VC, I would say that the math involved in this exercise (10.2 in Wald) is mostly if not all tensor calculus. If you end up working through Wald at some point you will notice that many of his problems are heavy on tensor calculus.
     
  10. Apr 25, 2013 #9
    Howdy WN, the way I see it Wald in page 252 makes clear the constraints and gauge choices under wich both GR's EFE and Maxwell equations can be considered as well posed initial value problems. In the Maxwell's eq. case clearly being source-free seems to be the constraint needed to show well posedness, so why should we expect anything different?
     
  11. Apr 25, 2013 #10

    WannabeNewton

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    But he writes down the constraints explicitly for the source-free case alone; I don't see him say anywhere that Maxwell's equations have a well posed IVF if and only if they are source-free.
     
  12. Apr 25, 2013 #11
    True, he doesn't say it. But I figured it was implicit in that in order to formulate ME as hyperbolic equations you need div E=0, if you also want the speed of propagation of the potentials to be c.
     
    Last edited: Apr 25, 2013
  13. Apr 26, 2013 #12

    Jano L.

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    Gerald Holland gives these references for the discussions of the initial value formulation in his book "Quantum Field Theory - A tourist guide":

    "
    The mathematical theory of such systems is well understood. In particular, the
    Cauchy problem is well posed: there is a family of theorems that say that if j is in
    some nice space X of functions or distributions on R4 and the initial data E(to, •)
    and B(£0, •) are in some related space ^ of functions or distributions on R3, there
    is a unique solution (E, B) of this system in another related space Z. Versions of
    this result can be found, for example, in Taylor [115], §6.5, and Treves [120], §15.
    "

    M. E. Taylor, Partial Differential Equations I, Springer, New York, 1996, sec. 6.5
    F. Treves, Basic Linear Partial Differential Equations, Academic Press, New York, 1975., sec. 15

    Perhaps they will be useful.
     
  14. Apr 26, 2013 #13
    Additionally to Friedlander's comment, here is a discussion of why the EM wave eq. with sources doesn't have a physically viable IVF as defined in Wald's page 244 (second property requirement). Yes, it's the old advanced and retarded potentials issue.

    It would be interesting to also analyze the constraints and coordinate choices needed for GR to have a well posed IVF.
     
  15. Apr 26, 2013 #14
    The systems that the quote is referring to are not classical but quantum, right?
     
  16. Apr 26, 2013 #15
    Einstein-Maxwell eq. are in "free-space" with vanishing current density.
     
  17. Apr 26, 2013 #16

    Jano L.

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    I do not have the books, but I think it is about the classical Maxwell equations.
     
  18. Apr 26, 2013 #17

    WannabeNewton

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    Yes that would present causality issues in general but perhaps one of Jano's latest references give sufficient conditions for when the causality condition is satisfied by certain 4-current densities.

    Well Wald does the case in vacuum to show that GR does have a well posed IVF in that case. In the presence of matter sources he says it depends entirely on the dynamical equations satisfied by the matter fields.
     
    Last edited: Apr 26, 2013
  19. Apr 26, 2013 #18

    WannabeNewton

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    But in robphy's link, the author specifically writes down the equations with non-vanishing current density (page 78).
     
  20. Apr 26, 2013 #19

    WannabeNewton

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    I got access to the Taylor one and they are indeed the classical Maxwell equations. I'll read it and see what I can garner from it. From a first glance, it seems he only shows existence of solutions, specifically for the case of electromagnetic waves in vacuum, but I'll dig around the book to see if he goes beyond that.
     
    Last edited: Apr 26, 2013
  21. Apr 26, 2013 #20
    It is ambiguous if he is referring to the Einstein-Maxwell eq. proper or just to the Maxwell covariant equations of the EM field tensor Fab. Actually according to the wikipedia article on the EFE the Einstein-Maxwell are always in free-space so they would automatically conform to the eq. 3.2.2 form. so I can only make sense of what the author says with the second interpretation.
     
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