Physics | Investigations in Forms

Given a Riemannian manifold $(M^3, g)$ , for a vector field $X$ over $M$ we can define the generalized curl $\nabla \times X = \star (dX)$ where $dX$ is the exterior derivative of (the one-form corresponding to) $X$ and $\star$ is the Hodge star operator with regards to the Riemannian volume form. So for example the vector calculus identity $\nabla \times \nabla f = 0$ in this language becomes $\star d(df)=0$ which follows from $d^2 = 0$ . Similarly we can define the divergence $\nabla \cdot X = \star d( \star X)$ and again the identity “divergence of curl is zero” translates to the equality $\star \ d \star (\star dX) =0$ , which again holds because here we have $\star \star = \mathrm{Id}$ .

In the physical setting, we may have a vector field $E$ over $M$ and a Killing field $\eta$ (i.e. a field with $\mathcal{L}_{\eta} g = 0$ , which will be a Lorentz transformation), and we ask whether there exists a scalar potential for it, meaning that a function $f \in C^\infty(M)$ with $\iota_{\eta}(\star E) = df$ . And what physicists do to test for existence of such an $f$ is to take the curl of the vector field $\iota_{\eta}(\star E)$ and check if it’s zero. But we now see how such a check is justifiable, because $\iota_{\eta}(\star E)$ being curl-free corresponds to it being closed, and in $\mathbb{R}^4$ spacetime closed forms are exact, i.e. potential fields will exist.

Investigations in Forms

Category Archives: Physics

Scalar potential and differential forms