This post is a follow-up to the previous post [LINK] in which I tried to make sense of the mathematics of differential geometry, especially in the context of general relativity, and proposed notation that may be less confusing, more consistent, and arguably more powerful than traditional notation given the existence of a metric in general relativity. With similar motivations, this post explores how some of the ideas of electromagnetic (EM) theory may change on curved manifolds. Follow the jump to see more.
Maxwell's equations on a curved manifold
On a curved manifold (which may include curvature along the timelike coordinate for a faraway observer), Maxwell's equations in vacuum involve the metric only in the definition of the EM displacement field [LINK from Wikipedia]. (The situation is more complicated when polarizable media are present.) However, because the EM field contains energy & momentum which are sources for manifold curvature, the metric depends on the EM field. Thus, even in vacuum, Maxwell's equations are nonlinear. This means that it is no longer possible to study solutions to Maxwell's equations with linear changes of basis, like Fourier transforms or multipole expansions. However, even if one did not know about the dependence of Maxwell's equations on the metric & resulting nonlinearity, one could still conclude that solutions to Maxwell's equations cannot be studied with linear changes of basis, like Fourier transforms or multipole expansions. This is for the following reasons that will frequently refer to work from my PhD.
Scalar quantities
On a Riemannian/Lorentzian (flat) manifold, it is customary to take dot products of vector fields with each other and integrate them over space & time to yield scalars (not scalar fields). For example, the work done by an electric field on a polarization field can be written (possibly with the opposite sign) using conventional multivariable calculus as \( W = \int_{\mathbb{R}^{3}} \int_{-\infty}^{\infty} \vec{P}(t, \vec{x}) \cdot \vec{E}(t, \vec{x})~\mathrm{d}t~\mathrm{d}^{3} x \). If there is no explicit time dependence of polarizable media in Maxwell's equations, then it is possible to take the Fourier transform from time \( t \) to angular frequency \( \omega \). Additionally, depending on the polarizable media present, evaluating this integral in practice involves linearly changing from the position basis to some other spectral basis, like the Fourier basis or the multipole basis, or localized basis, like the finite element basis or the Rao-Wilton-Glisson or Schaubert-Wilton-Glisson bases.
It may seem like the fact that vectors on a curved manifold cannot be added to each other precludes generalization of such scalar quantities. However, the dot product integrand is a scalar, which can be integrated even on a curved manifold. In particular, at every manifold point \( p \), the electric field is properly a covector \( \tilde{E}(p) \), the polarization field is properly a vector \( \vec{P}(p) \), and the integration measure can be replaced by the differential 4-form \( \tilde{\mathrm{d}}x^{0}(p) \wedge \tilde{\mathrm{d}}x^{1}(p) \wedge \tilde{\mathrm{d}}x^{2}(p) \wedge \tilde{\mathrm{d}}x^{3}(p) \) (where the superscripts are indices, not exponents). Thus, the generalization of the work is \( W = \int (\vec{P}(p) \cdot \vec{E}(p)) \tilde{\mathrm{d}}x^{0}(p) \wedge \tilde{\mathrm{d}}x^{1}(p) \wedge \tilde{\mathrm{d}}x^{2}(p) \wedge \tilde{\mathrm{d}}x^{3}(p) \), where the integral is taken over all space & time.
Vector or covector quantities
On a Riemannian/Lorentzian (flat) manifold, it is customary to take the dot product of a tensor field with a vector fields integrate the result over space & time to yield another vector field. For example, given the vacuum Green's function tensor \( \leftrightarrow{G}^{\mathrm{vac}}(t, \vec{x}, t', \vec{x}') \), I would often use the equation \( \vec{E}(t, \vec{x}) = \int_{\mathbb{R}^{3}} \int_{-\infty}^{\infty} \leftrightarrow{G}^{\mathrm{vac}}(t, \vec{x}, t', \vec{x}') \cdot \vec{P}(t', \vec{x}')~\mathrm{d}t'~\mathrm{d}^{3} x' \) to obtain the electric field from the polarization field.
It may seem like the fact that vectors on a curved manifold cannot be added to each other precludes generalization of such nonlocal integrals to yield vector fields. However, with some creativity, this generalization is indeed possible. In particular, \( \leftrightarrow{G}^{\mathrm{vac}}(t, \vec{x}, t', \vec{x}') \) can be generalized to a nonlocal tensor of order (0, 2) at manifold points \( (p, p') \) with the definition \( G^{\mathrm{vac}}(p, p') = G^{\mathrm{vac}}_{ij}(p, p') \tilde{\mathrm{d}}x^{i}(p) \otimes \tilde{\mathrm{d}}x^{j}(p') \). This means that the generalization of the dot product can only be done at the manifold point \( p' \), so that \( G^{\mathrm{vac}}(p, p') \cdot \vec{P}(p') = G^{\mathrm{vac}}_{ij}(p, p') P^{j} (p') \tilde{\mathrm{d}}x^{i}(p) \). There is no problem with integrating this, because the components depend on \( p' \) as well as \( p \), but the basis covectors only exist at \( p \), which is not the integration variable. Thus, the generalization becomes \( \tilde{E}(p) = \int (G^{\mathrm{vac}}(p, p') \cdot \vec{P}(p')) \otimes (\tilde{\mathrm{d}}x^{0}(p') \wedge \tilde{\mathrm{d}}x^{1}(p') \wedge \tilde{\mathrm{d}}x^{2}(p') \wedge \tilde{\mathrm{d}}x^{3}(p')) \), which can be evaluated in terms of explicit components as \( \tilde{E}(p) = E_{i}(p) \tilde{\mathrm{d}}x^{i}(p) \) where \( E_{i}(p) = \int G^{\mathrm{vac}}_{ij}(p, p') P^{j} (p') \tilde{\mathrm{d}}x^{0}(p') \wedge \tilde{\mathrm{d}}x^{1}(p') \wedge \tilde{\mathrm{d}}x^{2}(p') \wedge \tilde{\mathrm{d}}x^{3}(p') \) and the integral is taken over all space & time.
Linear changes of basis
Examples of linear changes of basis in the conventional multivariable calculus formulation of EM theory on flat Riemannian/Lorentzian manifolds are Fourier transforms like \( \int_{-\infty}^{\infty} \vec{E}(t, \vec{x})\exp(\mathrm{i}\omega t)~\mathrm{d}t \) or the multiple expansions, with a particular example of the latter being the dipole moment \( \int_{\mathbb{R}^{3}} \vec{P}(t, \vec{x})~\mathrm{d}^{3} x \). Unfortunately, as far as I can tell, these kinds of linear transformations are no longer possible, even if one were to ignore the nonlinearity created by the dependence of the metric on the EM field. This is simply because this is the kind of addition of vectors or covectors at different points on the manifold that is not allowed when the manifold is curved.
