Monday, 18 February 2019

Maxwell's equation with three-forms

James Clerk Maxwell was one of the greatest theoretical physicists of all time. He was up there with Newton and Einstein. He published his famous equations in about 1865. Quite a bit was known about electricity and magnetism and the relationships between them at the time but he spotted a missing component and was able to show that electromagnetic waves must exist and moreover travel at the speed of light, which was already known. Therefore light had to be electromagnetic waves. It must have made physicists feel queasy. I remember being astonished when we did Maxwell's equations and proved it at university. Soon after his equations were known, radio waves (1887, Hertz) and  X-rays (1895, Roentgen ) were discovered. Then came Einstein with Special (1905) and General (1915) Relativity.

Carroll writes "The other one of Maxwell's equations, (1.96) can be expressed as an equation between three-forms:
\mathrm{d}\left(*F\right)=*J & \phantom {10000}(1) \\
\end{align}where the current one-form ##J## is the current four-vector with index lowered. Filling in the details is left for you, as good practice converting from differential form notation to ordinary index notation." I need the practice .....

##J## was given as\begin{align}
J^{\mu }=\left(\rho ,\ J^x,J^y,J^z\right) & \phantom {10000}(2) \\
\end{align}where ##\rho ## is the classical charge density and ##J^i## the current.

His equation (1.96) ("the other one of Maxwell's equations") was
{\partial }_{\mu }F^{\nu \mu }=J^{\nu } & \phantom {10000}(3) \\
\end{align}This equation is really four equations one for each value of ##\nu = 0,1,2,3## and on the Left Hand Side we might have four terms for the summation over ##\mu## but one is zero because of the antisymmetry of ##F## (in the next equation). The four equations are written out in (8)-(10).

We have a new form of the electromagnetic field strength tensor:
F^{\mu \nu }=\left( \begin{array}{cccc}
0 & E^1 & E^2 & E^3 \\
-E^1 & 0 & B^3 & -B^2 \\
-E^2 & -B^3 & 0 & B^1 \\
-E^3 & B^2 & -B^1 & 0 \end{array}
\right)=-F^{\nu \mu } & \phantom {10000}(7) \\
I am not certain if the indices on the ##B##'s should be up or down.

We expand (3) and we get the two relevant Maxwell's equations (one in three parts):
{\partial }_1F^{01}+{\partial }_2F^{02}+{\partial }_3F^{03} & =J^0 & \phantom {10000}(8) \\
{\partial }_0F^{10}+{\partial }_2F^{12}+{\partial }_3F^{13} & =J^1 & \phantom {10000}(9) \\
{\partial }_0F^{20}+{\partial }_1F^{21}+{\partial }_3F^{23} & =J^2 & \phantom {10000}(10) \\
{\partial }_0F^{30}+{\partial }_1F^{31}+{\partial }_2F^{32} & =J^3 & \phantom {10000}(11) \\
We now convert (1) into index notation. It becomes
{\widehat{\varepsilon }}^{\ \ \ \ \ }_{\beta \gamma \delta \iota }{\partial }_{\alpha }F^{\delta \iota }+{\widehat{\varepsilon }}^{\ \ \ }_{\gamma \alpha \delta \iota }{\partial }_{\beta }F^{\delta \iota }+{\widehat{\varepsilon }}^{\ \ \ }_{\alpha \beta \delta \iota }{\partial }_{\gamma }F^{\delta \iota }=2{\widehat{\varepsilon }}_{\kappa \alpha \beta \gamma }J^{\kappa } & \phantom {10000}(29) \\
\end{align}On the face of it there the free indices ##\alpha ,\beta ,\gamma ## will give 64 components of the equation to check and the double summation on the LHS gives ##16\times 64=1,024## terms and the summation on the RHS a further 256. In total 1,280 terms which is rather more than the 16 terms in Maxwell's equations. Conceivably we might need to write out every combination. Fortunately the number of terms will be reduced by the antisymmetry of the Levi-Civita symbol ##\widehat{\varepsilon }## and the electromagnetic field tensor ##F##.

We prove that all components of (29) vanish except ones where ##\alpha ,\beta ,\gamma ## all differ. We only have 24 components (or 96 terms) to check! After plodding through all of them, I found six of each of Maxwell's equations (8)-(11) as predicted. The other forty are all 0 = 0. (You can see my notes calculating ##\widehat{\varepsilon }## in the previous post also on the Levi-Civita symbol.)

Having done all that I did discover the easy way to get there without all the plodding tables and paper work. But it was worth it, because as Carroll said, I needed practice converting from differential form notation to ordinary index notation and moreover practice with manipulating indexed equations. The plodding inspired me.

The full six page story is here: Commentary 2.9 Maxwell equation with 3 forms.pdf

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