Interpreting the cohomology class of the Maxwell tensor.

Question

In the introduction to Bott and Tu, “Differential forms in algebraic topology” there is the motivating example of a stationary point charge in $3$-space. The electromagnetic field $\omega$ is a $2$-form on $X=\mathbb{R}^4\setminus \mathbb{R}$, and the Hodge star of this generates the de Rham cohomology $H^2(X)=\langle \star\omega \rangle$. As a passing remark, it is claimed that the class $[\star\omega]$ can thus be interpreted as the charge of the source.

Assuming little background in electromagnetism, I’d be grateful for an explanation. In particular what is the physical significance of $\star\omega$ and why is the physical quantity of charge the same as a cohomology class of space-time?

Answer 1

The electromagnetic field of a point charge is

$$\omega=\frac{q}{r^3}\, dt\wedge\left(x\, dx+y\, dy+z\, dz\right)$$

where $r=\sqrt{x^2+y^2+z^2}$, where $q\in \mathbb{R}$ is its charge. With the Minkowski metric, $g=-dt^2+dx^2+dy^2+dz^2$, one has

$$\star\omega=-\frac{q}{r^3}\left(x\, dy\wedge dz+y\, dz\wedge dx+z\, dx\wedge dy\right)$$

Integration on the hypersurface with $x^2+y^2+z^2=1$, i.e. $S^2\times \mathbb{R}$, provides a linear map $\Omega^2(\mathbb{R}^4\setminus \mathbb{R})\to \mathbb{R}$ which extends to an isomorphism on cohomology

$$H^2_{\textrm{dR}}(\mathbb{R}^4\setminus\mathbb{R})\xrightarrow{\sim} \mathbb{R}$$

One can see that this integral is nonzero on $\star \omega$ and proportional to $q$, and thus via this isomorphism, the hodge dual of $\omega$ is sent to the charge $q$.

Answer 2

The definiton of the induction field is when a point charge on charged pointlike metal ball at the end of a stick is placed near a neutral metallic sphere.

By the axiom, that there are no currents and no electric fields in an isolated conductor in static equilibrium , the dynamic electric field at the surface in vacuum in equlibrium is pure radial.

The tangent component has to be continuous through the surface. Any discontinuity means local charge density jump.

Measurable quantities are charge by charge q= current * time in A s, force between charges by by q1^2 q2^2/r^2.

The surface charge density, measured in A s/m^2 was called D and by modern mathematics terminology is a surface density 2-form, to be inteegrated over the surface, giving the charge. Gauss found his famous law, extended by Poisson, that the D-field outside the sphere has $\text{div} D=0 $ in vacuum and the its integral as 2-form over any surface enclosing the charged sphere yields the same charge.

So by the surface integral over a large sphere, fields fall in classes of total charge in a compactum.

The electric force field, on the other hand, the dynamics of Newton side, is direct proprotional to the assumed 2-form D in vacuum.

But its not a surface 2-form, its a vector. This forc field could declared in euclidean $R^3$ as the component of a 1-form $\sum_i K_i dx^i$, the integrand yielding mechanical work done by moving the charge in the electric D field of a static charge distribution along the local tangent vector of the curve.

This incomplete understanding was resolved in the times of Lorentz, Einstein and Poincaré, after Minkwoswki introduced time as the fourth coordinate.

Today we have the mathematical trival equation $$ D = *E $$

with the Hodge star operation in Minkowki space with Lorentz scalar product. $$ D = D_{ik} dx^i\wedge dx^k , E=*D = \left<dx^0 \wedge \dots \wedge dx^3, D\right>$$