# Why not use the Weyl gauge?

In E&M in Minkowski space, the Lorenz and Coulomb gauges are typically used since they make things vastly simpler. On a curved background, Maxwell's equations (without sources) can be written as:

\begin{align} \nabla_a F^{a b} &= 0 \\ F_{a b} &= \nabla_a A_b - \nabla_b A_a = \partial_a A_b - \partial_b A_a \end{align}

Assuming the Lorenz condition, this can be written in the visually pleasing form:

\begin{align} \Box A^a = R^a_bA^b \end{align}

Unfortunately, the coordinate expression of this equation in general coordinates is not very pretty. Conversely, using the fact that $F_{ab}$ is antisymmetric, we can write Maxwell's equations (in any gauge) as,

\begin{align} \partial_a\left(\sqrt{-g}g^{a c}g^{b d}\left(\partial_c A_d - \partial_d A_c\right)\right) = 0 \end{align}

which, upon identifying the variables,

\begin{align} A_{ab} &= \partial_b A_a \\ \Pi^i &= \sqrt{-g}g^{t c}g^{i d}\left(A_{d c} - A_{c d}\right) \\ \end{align}

can be nicely decomposed into first order form,

\begin{align} \partial_t \Pi^i &= -\partial_j \left(\sqrt{-g}g^{j c}g^{i d}\left(A_{d c} - A_{c d}\right)\right) \\ \partial_t A_{a b} &= \partial_b A_{at} \\ \partial_i \Pi^i &= 0 \end{align}

I may have gotten these slightly wrong as I am going mostly from memory and it is pretty late, but the general idea still stands (You invert the definition of the $\Pi^i$'s to get the $A_{it}$'s). If one evolves these equations as is, you are forced to solve the elliptic equation to get the $A_t$ component (which is computationally expensive). If you instead use the Lorenz gauge ($\nabla_a A^a = 0$), you can recover a hyperbolic form of the equations which is relatively simple in general coordinates.

The thing is, adopting the Weyl gauge ($A_t=0$) makes the equations much simpler than the Lorenz gauge and is still hyperbolic. Is there some property of the Weyl gauge that makes it unsuitable for this kind of calculation which would explain why it is not more commonly used?

• The Weyl (or "temporal") gauge is sometimes used, but people are reluctant to use it more generally because it's not a relativistically covariant gauge condition. However, as with all gauge choices, none is "better" than any other, certain ones are just more convenient for certain situations where "convenient" is a matter of personal opinion. Therefore, I think this question is primarily opinion-based. – ACuriousMind Sep 8 '16 at 13:56
• I am just wondering if I am missing something obvious. The Coulomb gauge isn't relativistic covariant either, but it and it's generalisations are commonly used. I am just looking for a reason; perhaps when implemented computationally it is in practice difficult to maintain the elliptic constraint? The point is that I don't know and I am just trying to find others thoughts on the matter. – Graham Reid Sep 8 '16 at 15:31
• In general, I am looking for a gauge, such that Maxwell's equations in an arbitrary background can be written as a system of hyperbolic PDEs with an elliptic constraint on the hypersurface where the initial data is defined. I would also like this system to be as simple as possible, so I would like suggestions for good gauge choices. – Graham Reid Sep 8 '16 at 15:35