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I'm quite familiar with Maxwell's equations in the context of real Lorentzian manifolds in $1+2$ and $1+3$ dimensions. But then is there an analogue of Maxwell's equations in $2+2$ dimensions? How about for real pseudo-Riemannian manifolds of dimension $p+q$, i.e. of signature $(p,q)$? Somehow I think the answer is no because maybe Lorentz invariance gets destroyed, though I'm not certain.

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    $\begingroup$ The Maxwell equation, formulated in the language of differential geometry, is completely generalizable to an arbitrary signature. The hard part is having unique solutions then. $\endgroup$
    – Slereah
    Jul 25, 2017 at 22:12

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The Maxwell equation can be fairly simply written as

$$\nabla_{\mu} F^{\mu\nu} = 0$$

This expression does not depend in any way on the metric signature, and as such can be used with any metric signature. In particular, the gauge fixing of the equation does not depend on the signature, as far as I know, so that we can rewrite it as

$$\Box A_\mu - {R^\nu}_{\rho} A^\rho = 0$$

Of course, Lorentz invariance will be lost, and replaced with invariance under the group $O(p,q)$.

Let's see what this implies :

$p = 1$, $q = n-1$

This is the classical case of the Lorentzian manifold. In this case, the principal part of the PDE is a hyperbolic system. This can be generally shown to have a single solution for initial $A_\mu$, $\partial_\nu A_\mu$, if the spacetime is globally hyperbolic and everything is regular enough.

$p = n - 1$, $q = 1$

The opposite case, for which we can show easily that it's equal to

$$- \Box A_\mu + {R^\nu}_{\rho} A^\rho = 0$$

with the same solutions.

$p = 0$, $q = n$

The Riemannian case, in which case we just end up with the Laplace equation.

$$\Delta_g A_\mu - {R^\nu}_{\rho} A^\rho = 0$$

with $\Delta_g$ the Laplace-Beltrami operator. It is fairly well known that Cauchy-type initial conditions are generally too constraining to solve elliptic equations, hence they do not really correspond to what we typically expect of physics. An interesting treatment of physics in the Riemannian case can be found here :

https://web.archive.org/web/20170318151343/http://www.gregegan.net/ORTHOGONAL/ORTHOGONAL.html

Note that in general, it's always possible to have closed timelike curves in a Riemannian space (you can just turn around), as can be shown by the fact that it is invariant under the rotation group $O(n)$. Hence we can "boost" to the frame $(x,t) \to (-x, -t)$. This makes for pretty bad things.

The same is true for the case $p = n$, $q = 0$, simply by taking the opposite sign.

$p > 1$, $q > 1$

This is the ultrahyperbolic case, with more than one timelike dimension. Much like in the Riemannian case, since any timelike plane forms a Riemannian submanifold, there are always closed timelike curves in it. The ultrahyperbolic wave equation tends to either have no solution or non-unique solutions, and they are in general unstable. A good review on the topic can be found here :

http://rspa.royalsocietypublishing.org/content/465/2110/3023

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  • $\begingroup$ Ah, another Egan fan (although Orthogonal's the only of his I have read, but liked it immensely). Nice summary of how the different signatures sit with physics. $\endgroup$ Jul 26, 2017 at 0:37
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Write it in coordinate free formulation first.

$$ dF =0\\ d *_g F =0 $$

Unwrap that universal result for whatever metric. Will expand it if necessary.

Edit:

Make an antisymmetric $F$ below which stands for $\sum F_{\mu \nu} dx^\mu \wedge dx^\nu$

$$ F = \begin{pmatrix} 0 & F_{01} & F_{02} & F_{03}\\ -F_{01} & 0 & F_{12} & F_{13}\\ -F_{02} & -F_{12} & 0 & F_{23}\\ -F_{03} & -F_{13} & -F_{23} & 0\\ \end{pmatrix} $$

$d$ is the exterior derivation so in components

$$ d F_{\mu \nu} = \sum \frac{dF}{dx_\lambda} dx^\mu \wedge dx^\nu \wedge dx^\lambda\\ $$

$*_g$ is the Hodge star for the particular manifold in question. It's the thing that interchanges electric and magnetic fields in usual source free electromagnetism in Minkowski. Like taking $dx \wedge dy \to dz \wedge dt$, whatever 2 were missing. The exact coefficient depends on the metric you choose.

Follow the Wikipedia hole from here

Now you have all the ingredients. So what you have to do is expand out all the terms for that particular $g$. They will turn into some system of first order PDEs replacing usual Maxwell equations. Then give names to the $F_{ij}$ appropriately. Usually 3 of them become E and the other 3 become B.

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  • $\begingroup$ Are you saying that it is possible to have Maxwell's equations in any arbitrary number of temporal and space dimensions? Please expand or elaborate if possible. $\endgroup$
    – Libertron
    Jul 25, 2017 at 22:08
  • $\begingroup$ @Libertron He/she sure is! We're talking here about generalizing the statement that a closed two form has a dual whose flux (exterior derivative) is the current three form. So it's simply a generalization of a really simple statement of a kind of symmetry that's in keeping with charge conservation (because the current three form is an exact three form, so if you hit it again with the exterior derivative, it vanishes). Whether or not this corresponds to possible physics is another matter altogether. Slereah's answer gives a good summary of why different signatures are "more possible" than ... $\endgroup$ Jul 26, 2017 at 0:46
  • $\begingroup$ ... others, in particular yours has a problem at the outset, in that the symmetry group $O(2,\,2)$ for the analogue of Lorentz invariance can simply turn time around. So CTCs and time machines abound, which are amongst Slereah's interests. $\endgroup$ Jul 26, 2017 at 0:49
  • $\begingroup$ Good point. I left it current-less. Put in a $d \star_g F = J$ instead with $J$ being an $n-1$ form. You can use the metric again to turn that into a vector like the usual 4-vector for charge and current density. $\endgroup$
    – AHusain
    Jul 26, 2017 at 1:31

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