# Poynting vector and differential forms

It is well known that electromagnetic field is a 2-form and Maxwell's equation can be reformulated in language of differential forms. What is the Poynting vector in this language?

• Seems like this would be a better question on Physics SE. Dec 9 '20 at 17:06
• It seems on-topic here too.
– YCor
Dec 10 '20 at 10:35

It is worth mentioning the controversy that arose about the expression of the Poynting vector $$P$$, whether the Abraham form $$E\times H$$ or the Minkowski form $$D\times B$$ is valid. The notations are that the electro-magnetic field, a closed $$2$$-form in Minkowski space, has coordinates $$(E,B)$$. The vectors $$(D,H)$$ are given by $$D=\frac{\partial L}{\partial E},\qquad H=-\frac{\partial L}{\partial B}$$ where $$L(E,B)$$ is the density of the Lagrangian from which the Maxwell's equations derive. For instance, the standard equations follow from the choice $$L=\frac12(|E|^2-|B|^2)$$ (here the light speed is set to $$1$$). Other choices are possible, like that of Born-Infeld, which yield non-linear models.
The amazing resolution of the controversy is that both expressions equal each other if, and only if $$L$$ is Lorentz invariant. In mathematical terms, this means that $$L$$ is a function of the quantities $$\frac12(|E|^2-|B|^2),\qquad E\cdot B$$ only.
Going further, the Maxwell's equations imply that the following tensor is (row-wise) divergence free $$T=\begin{pmatrix} W & E\times H \\ D\times B & -W_B\otimes B-W_D\otimes D+\sigma I_3 \end{pmatrix}$$ where $$\sigma:=B\cdot W_B+D\cdot W_D-W.$$ Hereabove $$W:=D\cdot E-L(E,B)$$ is the energy density (a partial Legendre transform of $$L$$), and subscripts are differentials. That $${\rm Div}_{t,x}T=0$$ expresses the conservation laws of the energy and of the Poynting vector. The important point is that $$T$$ is a symmetric tensor, and this is ensured by the Lorentz invariant.
Poynting vector is a 3d vector which can be thought of as a part of 4d stress-energy tensor $$T$$. In invariant language, you fix your reference frame by choosing a time-like 4-velocity $$u$$. Consider $$T(u)$$: it gives you a flow of energy-momentum. $$T(u,u)$$ is the flow of energy, and the vector $$P=T(u)+T(u,u)u$$ is the flow of momentum. By construction, $$P(u)=0$$ so you can think of it as 3d space-like vector. So far, this applies to any stress-energy $$T$$. For Maxwell $$T$$, express it in terms of $$F$$: $$T = F.F - 1/4 g (F,F)$$, where $$g$$ is metric and $$F.F$$ is a partial scalar product of $$F$$ with itself, with respect to the first index: this $$T$$ is traceless. You can use this to express $$T(u)$$ and $$T(u,u)$$ and $$P$$ in terms of $$F(u)$$ (aka the electric field) and $$(^*\!F)(u)$$ (aka the magnetic field). In the end you should find something like $$P=u\wedge F(u)\wedge (^*\!F)(u)$$.
• Indeed, $P=u\wedge F(u)\wedge (*F)(u)$ is an amazing expression through exterior calculus operations only! The particular case when $u=(1,0,0,0)$, that is, $P=-(1,0,0,0)\wedge(0,E_x,E_y,E_z)\wedge(0,B_x,B_y,B_z)$ is even clearer. Correcting a possible typo: $T(u,u)$ is energy density, and $T(u)+T(u,u)u$ is energy flux, while the momentum flux is a tensor rather than a vector (see en.wikipedia.org/wiki/Stress%E2%80%93energy_tensor for the particular case $u=(1,0,0,0)$). Dec 14 '20 at 20:04