Covariant equation with a preferred timelike vector ($({\eta }^{\mu \nu }-\alpha n^{\mu }{n}^{\nu }){\partial }_{\mu }{\partial }_{\nu }\varphi =0$)

In units $c=1$ and metric signature $(-,+,+,+)$, take a fixed unit timelike vector field $n$ and $\alpha >0$. The displayed equation for a scalar field is tensorial; in arbitrary coordinates use the flat Levi-Civita connection in place of ordinary second derivatives. In the rest inertial frame of $n$ it becomes $-(1+\alpha ){\varphi }_{tt}+{\nabla }^2\varphi =0$, with characteristic speed $1/\sqrt{1+\alpha }$. Its fixed $n$ selects a preferred frame. A boost that does not preserve $n$ is not a symmetry with this background held fixed, although transforming the components of every quantity gives an equivalent coordinate description. This separates general covariance from a substantive relativity principle.