A pure-gauge background in the ramp

A smooth ramp changing from zero to one has ${\sigma }_{\epsilon }^{\prime \prime }\not\equiv 0$ on $(0,\epsilon )$. Indeed, otherwise it would be affine there, and endpoint smoothness would force zero slope and hence a constant.

Choose a nonnegative $\zeta \in {C_c}^{\infty }((0,\epsilon ))$ supported where ${\sigma }_{\epsilon }^{\prime \prime }\ne 0$, with $\zeta \not\equiv 0$, and put

Choose

Set

This is a permitted classical background. The vector field $\xi$ here is not asserted to belong to the plateau algebra: it parametrizes the chosen background, not an admitted plateau transformation.

Let $e_{00}$ be the tensor whose only nonzero covariant component is $(e_{00}{)}_{00}=1$, and take

Both satisfy $P_{\mathrm{gr}}u=P_{\mathrm{gr}}v=0$.