# Maxwellian puzzle or paradox?

Satie
Hi!
Following Landau&Lifschitz (L&L) (I´m not a researcher in GTR) it is possible to arrive to a 3D version of Maxwell equations in a GTR-correct form. These equations resembling the ones in a dispersive electric and magnetic material, suggest, at first glance, that there should be a dispersion relation for EM waves propagating in a gravitational field. In fact, after a naïve algebra-struggling, it is possible to get to equations with a dispersion-like form.
However, if this was true, the radiation of an explosive event would arrive at different times for different wavelenghts, in a classical GTR formulation which is not correct (as far as I know) since then there should be local effects that could tell an observer about absolute positions and velocities with respect to the source.
I know there is some Loop Quantum Gravity results that suggest it is possible, but in Quantum Gravity scenarios, to have this effect.
The point is that I cannot find the L&L-inspired formulation wrong, nor can I accept it.
Can somebody shed some light on this ignorant-fellow?

## Answers and Replies

Can you state which section (in "Classical Theory of Fields", I presume?) you are looking at?

If you think you have found a dispersion relation for EM waves propagating in a vacuum in gtr, you must have goofed (unless I misunderstand what you have in mind). However, this sounds like the kind of mistake which could result from making an inappropriate approximation.

See http://en.wikipedia.org/w/index.php?title=Monochromatic_electromagnetic_plane_wave&oldid=41212306 for a detailed discussion of the simple exact solution (given in closed form) which models, in gtr, an EM plane wave propagating through "empty space"; this wave has a specific frequency and it is a solution to the fully nonlinear field equation. Strangely enough, parallel plane waves are not hard to superimpose (despite the nonlinearity of the EFE), so you should be able to modify this so that you can see two frequency components, and then you will see that the wavefronts move in lockstep: no dispersion.

Satie
Hi! And Thanks! The Section of L&L I am referring to is the Problem Part of Section 90 of The Classical Theory of Fields L&L. They derive the EM field equations in 3D form, using "gamma" determinant and "g" vector. They state that equations 4, 5 and 6 are the 3D versions and they notice the analogy ("purely formal") with Maxwell´s Eq in material bodies. Probably, my mistake was to push this analogy forward and try to eliminate H to get the E equation (just to have a glance to how would it look like). In an isochronous system (vector "g"=0) they are wave equations but with a term proportional to the time-derivative of the field E. If g is not equal zero, there are terms proportional to the curl of E. In the first, the derivative indicates dispersion and in the second dichroism. Of course I am not saying that this is correct.
I know that this cannot be. But all the same, since in Quantum gravity analysis there is some evidence (I am not the one that can tell if this is right ot wrong) that both effects are possible, then I asked where the whole reasoning leads to such unreasonable result in the classical field.
I did not pretend to find flaws in the GTR, nor to have found a "new effect", mine is just plain doubt about the use of such form of Maxwell´s eq...
Thanks again for your next comment! Truly yours
By the way, an e-mail of yours came yesterday but somehow I cannot find it in the server today, so I cannot answer that... sorry!