Abstract definition of electromagnetic fields on manifolds

[ZuperPosition]

Hello,
In the sources I have looked into (textbooks and articles on differential geometry), I have not found any abstract definition of the electromagnetic fields. It seems that at most the electric field is defined as
$$\bf{E}(t,\bf{x}) = \frac{1}{4\pi \epsilon_0} \int \rho(t,\bf{x}') \frac{\bf{x}-\bf{x}'}{|\bf{x}-\bf{x}'|^3} d^3 x'$$
as seen in e.g. (John David Jackson, Classical electrodynamics, John Wiley & Sons, 2012). I want to make a similar definition of the Electric field for an orientable Lorentz manifold. There are some issues that come to mind. The most obvious is that position is not well-defined on an abstract manifold. Secondly, if we for example model space-time as $\mathbb{R}\times S^2$, with the first factor representing time and $S^2$ is the spatial part, the nominator $\bf{x}-\bf{x}'$ in the integral should be projected to the part that is tangent to $S^2$. This is the case as otherwise the electric field would not be a vector field. Though, this is not really representative of manifolds in full generality, the manifold as a set need not be a subset of a vector field. I am aware that the Whitney embedding theorem might be applicable here which could prove useful. But for various reasons I can not use the Whitney embedding theorem, sadly.

My goal is to later define the electromagnetic field tensor on (a kind of generalised) manifolds. How to define the charge density, and volume form is obvious, but I don't see how the rest could be done in such an abstract manner. Has it even ever been done? If so, could someone please enlighten me?

Regards

 

[Orodruin]

It seems that at most the electric field is defined as

This is not the definition of the electric field. It is an expression for the electric field given some source density $\rho$. It seems more to me as if you want to solve Maxwell's equations in a general manifold. The expression for this is going to be dependent on the particular manifold, but in general the field should depend on the charge and current densities on the past light-cone. Also, the electric field is just a part of the electromagnetic field tensor $F^{\mu\nu}$, what you really want to do is to find that. Its definition is exactly the same as that in special relativity, $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$. Note that you do not need to replace the partial derivatives here with covariant derivatives.

How to define the charge density, and volume form is obvious

It should not be. Charge density is a component of a 4-vector and depends on the coordinate system. What "volume" means also depends on the coordinate system, i.e., which spatial surface you integrate over. The "obvious" things are the 4-current density and the spacetime volume form.

Edit: Also, your proposed spacetime $\mathbb R \times S^2$ only has 3 dimensions. Your electric field with have two components and your magnetic field will have one.

 

[ZuperPosition]

This is not the definition of the electric field. It is an expression for the electric field given some source density $\rho$. It seems more to me as if you want to solve Maxwell's equations in a general manifold. The expression for this is going to be dependent on the particular manifold, but in general the field should depend on the charge and current densities on the past light-cone. Also, the electric field is just a part of the electromagnetic field tensor $F^{\mu\nu}$, what you really want to do is to find that. Its definition is exactly the same as that in special relativity, $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$. Note that you do not need to replace the partial derivatives here with covariant derivatives.It should not be. Charge density is a component of a 4-vector and depends on the coordinate system. What "volume" means also depends on the coordinate system, i.e., which spatial surface you integrate over. The "obvious" things are the 4-current density and the spacetime volume form.

Edit: Also, your proposed spacetime $\mathbb R \times S^2$ only has 3 dimensions. Your electric field with have two components and your magnetic field will have one.

Ah yes, you are right it might be much more fruitful to define the electromagnetic field tensor using the vector potential. I got a bit distracted as some of the literature seems to want to define the electromagnetic fields on Minkowski space before defining the electromagnetic field tensor. In general, each component of $A_\mu$ is usually computed with a similar integral as above, right? Is there a way to express the 4-potential on a Lorentz manifold in general or is any such endeavour not really feasible?

You are also correct about the charge density, my bad.

Yes, having $\mathbb{R} \times S^2$ as spacetime is a bit weird dimensionally, but it is intentional. :)

Thank you!