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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!
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!
