- #1

- 7,385

- 6,468

OK - I follow almost all of that. Stating it in painful detail:mathpages said:We can restrict the range of possible metrics based on symmetry considerations. We seek a stationary coordinate system, so the metric coefficients must be independent of ##t##. In addition, the field is clearly independent of the transverse coordinates ##y## and ##z##, so we consider a diagonal metric of the form$$d\tau^2=g_{tt}(x)dt^2-g_{xx}(x)dx^2-g_{yy}(x)dy^2-g_{zz}(x)dz^2$$where ##g_{yy}(x)=g_{zz}(x)##

- We choose the wall to be at rest and pick constant spatial coordinates to be at rest wrt the wall. The wall isn't moving or changing in these coordinates, so the metric can't depend on the time coordinate.
- We pick two spatial directions to lie in the planes at constant distance from the wall. The wall is symmetric under translation and rotation in this plane, so the metric can't depend on these coordinates.
- That leaves perpendicular to the plane of the wall as our remaining direction, and the only possible coordinate the metric can depend on.

There's clearly something basic I'm missing here, but hopefully I won't have to kick myself too hard when I get an answer...