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

In summary, the conversation discusses the problem 10.2 in Wald which aims to show that the source-free Maxwell's equations have a well posed initial value formulation in curved space-times. The first part of the problem shows that given the electric and magnetic fields on a spacelike cauchy surface, Maxwell's equations imply certain constraints. The second part of the problem shows that with these constraints, there exists a unique solution for Maxwell's equations throughout the space-time. The key result used in the proof is theorem 10.1.2 which guarantees a well posed initial value formulation for certain linear equations. The conversation also brings up the question of what happens when the current density is non-zero, and whether there is still a well posed initial
  • #1
WannabeNewton
Science Advisor
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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!
 
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  • #2
WannabeNewton said:
Hopefully that wasn't too long.
It was.
 
  • #3
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?
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, ... "
 
  • #4
Jano L. said:
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 electromagnetismhttp://butler.cc.tut.fi/~bossavit/

(scroll down to the section "Books", the line "This one, out of print, ... "
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:
 
  • #5
Wannabe:

What kind of math is that?

Anyway, I think the answer is 17. (I'm kidding, I'm kidding:)
 
  • #7
robphy said:
This may be a useful comment from Friedlander's The wave equation on a curved space-time (p 76-)
http://books.google.com/books?id=RDmpajLTw1oC&printsec=frontcover#v=onepage&q=maxwell&f=false
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.
 
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  • #8
VeryConfusedP said:
Wannabe:
What kind of math is that?
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.
 
  • #9
WannabeNewton said:
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!

Howdy WN, the way I see it Wald in page 252 makes clear the constraints and gauge choices under which 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?
 
  • #10
TrickyDicky said:
Howdy WN, the way I see it Wald in page 252 makes clear the constraints and gauge choices under which 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?
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.
 
  • #11
WannabeNewton said:
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.

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.
 
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  • #12
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.
 
  • #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.
 
  • #14
Jano L. said:
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.
The systems that the quote is referring to are not classical but quantum, right?
 
  • #15
WannabeNewton said:
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.

Einstein-Maxwell eq. are in "free-space" with vanishing current density.
 
  • #16
I do not have the books, but I think it is about the classical Maxwell equations.
 
  • #17
TrickyDicky said:
Yes, it's the old advanced and retarded potentials issue.
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.

TrickyDicky said:
It would be interesting to also analyze the constraints and coordinate choices needed for GR to have a well posed IVF.
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.
 
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  • #18
TrickyDicky said:
Einstein-Maxwell eq. are in "free-space" with vanishing current density.
But in robphy's link, the author specifically writes down the equations with non-vanishing current density (page 78).
 
  • #19
Jano L. said:
I do not have the books, but I think it is about the classical Maxwell equations.
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.
 
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  • #20
WannabeNewton said:
But in robphy's link, the author specifically writes down the equations with non-vanishing current density (page 78).

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.
 
  • #21
Yeah Wald, at least at the end, makes it clear that a sufficient (but not necessary) condition for ##G_{ab} = 8\pi T_{ab}## to have a well posed initial value formulation is that the dynamical equations satisfied by the matter fields associated with ##T_{ab}## have the form required for the uniqueness theorems he presents in the text. In particular if ##T_{ab}## is the stress-energy of an electromagnetic field ##F_{ab}## satisfying ##\nabla^{a}F_{ab} = 0, \nabla_{[a}F_{bc]} = 0## then ##G_{ab} = 8\pi T_{ab}## will be well posed given the proof that ##G_{ab} = 0## itself is well posed with the geometrical constraints and gauge constraints that Wald describes. Similarly, if ##T_{ab}## describes the Klein-Gordon matter field ##\phi## satisfying ##\partial^{a}\partial_{a}\phi - m^2\phi = 0## then ##G_{ab} = 8\pi T_{ab}## is again well posed.

As a side note, I love how geometrical the proof is that ##G_{ab} = 0## is well posed. I haven't finished all of Wald's proof regarding it yet though as I still have to finish one of the end of chapter problems, which requires proving a result that he makes critical use of. In particular, I've shown ##\frac{1}{2}\mathcal{L}_{n}h_{ab} = h_{a}{}{}^{c}\nabla_{c}n_{b}## (which is problem 10.3) but problem 10.4, which is to show the Gauss-Codacci equations ##D_{a}K^{a}{}{}_{b} - D_{b}K^{a}{}{}_{a} = R_{cd}n^{d}h^{c}_{}{}_{b}## (where ##K_{ab} = h_{a}{}{}^{c}\nabla_{c}n_{b}## is the extrinsic curvature of the spacelike cauchy surface), is giving me some trouble. Quite a beautiful result though isn't it :D? Anyways, I'm getting needlessly side-tracked haha.
 

What is the Initial Value Formulation on curved space-time?

The Initial Value Formulation is a mathematical framework used to describe the evolution of physical systems on curved space-time, specifically in the context of Einstein's theory of general relativity. It incorporates both the space-time geometry and the matter content of the system to predict how it will change over time.

What are Maxwell's equations?

Maxwell's equations are a set of four partial differential equations that describe the fundamental principles of electricity and magnetism. They show the relationship between electric and magnetic fields, as well as how these fields interact with charged particles.

How do Maxwell's equations apply to curved space-time?

In the Initial Value Formulation, Maxwell's equations are modified to take into account the curvature of space-time. This is necessary because in curved space-time, the laws of physics are different than in flat space-time. The equations are written in a covariant form, meaning they are valid in any coordinate system.

What are the challenges of solving Maxwell's equations on curved space-time?

The main challenge in solving Maxwell's equations on curved space-time is the complexity of the equations and the difficulty in finding solutions. In many cases, numerical methods must be used to approximate the solutions. Additionally, the presence of matter and its effects on the curvature of space-time can make the equations even more complicated.

Why is the Initial Value Formulation important for understanding the behavior of physical systems?

The Initial Value Formulation is important because it allows us to make predictions about the behavior of physical systems on curved space-time, such as the motion of celestial bodies and the propagation of electromagnetic waves. It also helps us understand how matter and energy interact with the curvature of space-time, providing a deeper understanding of the fundamental laws of the universe.

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