Question

Why not use the Weyl gauge?

Answer 1

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:

∇aFabFab=0=∇aAb−∇bAa=∂aAb−∂bAa∇aFab=0Fab=∇aAb−∇bAa=∂aAb−∂bAa

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

□Aa=RabAb◻Aa=RbaAb

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

∂a(−g−−−√gacgbd(∂cAd−∂dAc))=0∂a(−ggacgbd(∂cAd−∂dAc))=0

which, upon identifying the variables,

AabΠi=∂bAa=−g−−−√gtcgid(Adc−Acd)Aab=∂bAaΠi=−ggtcgid(Adc−Acd)

can be nicely decomposed into first order form,

∂tΠi∂tAab∂iΠi=−∂j(−g−−−√gjcgid(Adc−Acd))=∂bAat=0∂tΠi=−∂j(−ggjcgid(Adc−Acd))∂tAab=∂bAat∂iΠi=0

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 ΠiΠi's to get the AitAit's). If one evolves these equations as is, you are forced to solve the elliptic equation to get the AtAt component (which is computationally expensive). If you instead use the Lorenz gauge (∇aAa=0∇aAa=0), you can recover a hyperbolic form of the equations which is relatively simple in general coordinates.

The thing is, adopting the Weyl gauge (At=0At=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..

Answer 2

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.