# The electromagnetic field tensor in curvilinear coordinate systems

I've been researching all things tensor so I can start studying general relativity. One of the main reasons to use tensors is because of their ability to describe physical laws in various coordinate systems. During my research, I found out about the electromagnetic field tensor, $F^{\mu\nu}$, and how it can be used to sum up Maxwell's equations in two equations, these: $$\partial_{\mu}F^{\mu\nu}=\mu_0J^{\nu}$$ $$\partial_{[{\gamma}}F_{\mu\nu]}=0$$ Wikipedia states the form of $F^{\mu\nu}$as this: $$F^{\mu\nu}=\begin{pmatrix}0 & -E_x/c & -E_x/c &-E_x/c\\E_x/c & 0 & -B_z & B_y\\E_y/c & B_z &0 & -B_x\\E_x/c &-B_y & B_x &0\end{pmatrix}$$ However, this definition is in Cartestian coordinates, which defeats the purpose of tensor equations being able to define in all coordinate systems. So, my question is how can I generally define the electromagnetic field tensor for all curvilinear coordinate systems?

NOTE: I have been here, https://www.physicsforums.com/threads/electromagnetic-field-tensor-in-curvilinear-coordinates.618984/ , where this same question was asked. However, I do not understand the way they explained their method, or really the method itself. Also, this is different from the question Electromagnetic tensor in cylindrical coordinates from scratch, because I am asking for a way to find it in all curvilinear coordinate systems, not just cylindical.

• Wouldn't it just be applying the coordinate transformation twice (once for each index)? For example, $F^{\mu\nu}=\Lambda^\mu_{\alpha}\Lambda^\nu_\beta F^{\alpha\beta}$? – Kyle Kanos Feb 21 '17 at 11:22

## 3 Answers

The explicit form for $F^{\mu\nu}$ is just one option to express $F^{\mu\nu}$ in cartesian coordinates using the EM field itself.

The definition over the four potential is coordinate independent and covariant: $$F_{\mu\nu}=\partial_\mu A_\nu -\partial_\nu A_\mu.$$ For an overview the link provided by Leonardo Francisco Cavenaghi in his answer is a good starting point.

For a more detailed but still compact discussion of Maxwells equations and electromagnetism in flat and curved spacetime I can recommend this paper W. C. dos Santos, 2016, Introduction to Einstein-Maxwell equations and the Rainich conditions.

The appropriate definition uses the concept of manifolds and bundles. The electromagnetic tensor is in fact a $2-$form and in particular, the curvature of a particular bundle. You can find more informations here: https://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism

The field-strength tensor $F$ can be described in an entirely coordinate-independent manner using the language of differential forms.

The gauge group of electromagnetism, $U(1)$, is one-dimensional and all structure constants vanish, that is, $f^\gamma_{\alpha\beta} = 0$. The $U(1)$ bundle over four-dimensional Minkowski space $M$ is $\mathbb R^4 \times U(1)$ and is trivial; this can also be seen by the fact $M$ is contractible to a point.

The gauge potential is simply $A = A_\mu dx^\mu$ and $F = dA = 2\partial_{[\mu}A_{\nu]} dx^\mu \wedge dx^\nu$ known as the field-strength corresponds simply to the curvature of the $U(1)$-valued connection or gauge potential.

We can make the identification (up to possibly factors of $i$) that $E_i = F_{i0}$ and $B_i = \frac12 \epsilon_{ijk}F_{jk}$ which makes no explicit reference to the coordinate system we must use.

As an aside, Maxwell's equations in vacuum remarkably reduce to $dF = d\star F = 0$; unpacking these gives you the equivalent of the Bianchi identity and the fact that $F$ is a closed form due to it being an exact form. These two conditions reduce to the four Maxwell equations.