- #1

ZuperPosition

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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