I am currently writing a Bachelor's thesis in theoretical physics, and since I like the interplay between mathematics and theoretical physics, I am writing about Maxwell's law in terms of differential forms. Maxwell's equations read
$$dF=0 $$
$$
d\star F=\star J
$$
for the $F$ the field tensor, $d$ exterior derivative and $\star$ the Hodge star operation. Does anyone know any interesting applications/advantages of writing Maxwell's equations like this, apart from their stunning beauty?
[Physics] Electromagnetism and differential forms
differential-geometryelectromagnetismmaxwell-equations
Related Solutions
As it is, differential forms don't tell you the whole story--strictly speaking, differential forms only deals with covectors and wedge products of covectors and then uses the hammer of the Hodge star to be able to clumsily do inner products. To me, it is too far removed from the vector calculus you may already know.
Instead, I strongly urge you to look into geometric algebra. All of the results of differential forms apply to geometric algebra as well--the former is strictly contained in the latter--but the notation is much more familiar and the emphasis is on geometric interpretation instead of abstract symbol pushing. David Hestenes has several books on the subject. Probably the authoritative piece as far as using geometric algebra to solve physical problems is Geometric Algebra for Physicists by Doran and Lasenby. You can also read some things quickly in this website, written by Gull, Doran, and Lasenby.
I'll give a quick overview. Geometric algebra has a wedge product like differential forms does, but it also lets you directly use a dot product as well. In fact, it combines the two in a useful operation called the geometric product, defined as follows. For two vectors $a, b$, the geometric product $ab$ is
$$ab = a \cdot b + a \wedge b$$
The geometric product is associative (even though the dot product is not!). This makes it very useful. It is also invertible in Euclidean space, as a consequence of that associativity. This makes possible the formula
$$a = abb^{-1} = (a \cdot b) b^{-1} + (a \wedge b) \cdot b^{-1}$$
Geometrically, this decomposes $a$ into $a_{\parallel, b}$ and $a_{\perp, b}$. We emphasize that $a \wedge b$ denotes an oriented plane, and further wedge products yield oriented volumes and more.
Some applications immediate to physics are as follows:
- Angular momentum as a bivector. This is one of the first times you "need" a cross product, and using the wedge product instead yields a cleaner interpretation. The angular momentum bivector is exactly the plane in which two objects move in relation to one another. This also generalizes to beyond 3d, so it makes sense to talk about angular momentum bivectors in relativity also.
- Unification of integral theorems (the fundamental theorem of calculus). Geometric calculus (like differential forms) makes possible the unification of the divergence theorem, Stokes' theorem, and so on as one basic concept: that the integral of a function over a boundary is equal to the integral of the derivative over the region bounded by that boundary. I think this is a significant quality of life issue; having to remember only one concept is much easier, in my opinion, than remembering many separate integral theorems.
- Relativity without indices or classic tensor calculus. Geometric algebra's combination of the dot and wedge products makes possible all the usual operations one usually needs tensor calculus and index notation for. Relativity can be presented using a modest extension of the methods used in 3d electromagnetism. The geometric product makes it possible to boil down Maxwell's equation in vacuum to one equation (instead of two for differential forms): $\nabla F = J$. This emphasizes the interpretation of the EM field $F$ as a bivector field, a field of oriented planes throughout spacetime.
- Geometric interpretation of quantum mechanics. A lot of the mathematics in quantum is presented as mystical or special to QM, but most of it is actually inherent to the geometric structure of space and time. Geometric algebra allows one to treat the Pauli and Dirac algebras as the algebras of basis vectors in 3d and 3+1d space. This makes the interpretation of spin and spin operators inherently geometrical.
- Construction of spinors. Spinors are things we often deal with in quantum, perhaps with only the understanding that they must be rotated through $4\pi$ instead of $2\pi$ to return to their original form. Geometric algebra shows that spinors underlie all rotations--even those of plain old 3d space. In fact, the spinors of 3d space are quaternions, and the spinors of 2d space are complex numbers. GA gives a framework for constructing spinors and manipulating them like other objects.
Differential forms can do some of these things as well, some of them not (it absolutely can't boil down Maxwell's equations to one expression). Either formalism is a great improvement over traditional methods, however.
In the language of differential forms, the Maxwell-Lorentz equations are simply $$\begin{eqnarray*}\mathrm{d}\!\star\!F = J/\lambda_0 &\text{,}\quad&\mathrm{d}F = 0\text{,}\end{eqnarray*}$$ where $1/\lambda_0$ is the characteristic impedance of free space, and can be fixed to $1$ in appropriate units.
From that point of view, all you need to "move" to curved spacetime is realize that the only place where the metric comes up is the in the Hodge dual operator $\star$.
This was what I expected initially, but I got scared by the existence of the exterior covariant derivative -- I can't figure out why it is needed if the exterior derivative already behaves geometrically.
As already said, the differential structure does not care about either the metric or the manifold-connection, and hence not the curvature. That's simply a matter of how they are defined, and therefore curved spacetime wouldn't care about the exterior derivative, but only have a different Hodge dual.
However, there is an informal way to motivate this by analogy, which is what I gather you're asking for (this is not clear, so I'm guessing). Recall that the notion of covariant derivative for scalar fields is particularly trivial and independent of curvature: $$\nabla_u \phi = u^\mu\phi_{,\mu}\text{.}$$ The same kind of thing happens with the covariant exterior derivative. Recall the $k$-forms are certain types of maps in the form $$\omega_p: (T_pM)^k\to\mathbb{R}\text{,}$$ and are therefore scalar-valued.
On the other hand, one can consider a kind of generalized form that takes values in some vector bundle over over the manifold, and try to have an "exterior derivative" with $\mathrm{d}$ acting on each component in an arbitrary basis. This generally does not work without a connection on that vector bundle, with a notable exception where the values taken are are actually scalars, since then all possible bases for this $1$-dimensional vector space are just proportional to one another.
So the short of it is that the exterior derivative makes sense regardless of curvature and the covariant exterior derivative doesn't do anything interesting for ordinary scalar-valued forms.
Best Answer
By themselves, these equations are just a (indeed beautiful) reformulation of the standard Maxwell equations, so they don't contain new physics. However, they are useful in the sense that they stress the geometric and topological aspects, and they generalise nicely:
Some of these aspects are quite advanced. If you are interested, the book "Geometry, Topology and Physics" by Nakahra should provide a reasonably accessible introduction.