Solar system tests and interpretation of gauge field and Newtonian prepotential in general covariant Ho\v{r}ava-Lifshitz gravity

We study spherically symmetric, stationary vacuum configurations in general covariant theory (U(1) extension) of Ho\v{r}ava-Lifshitz gravity with the projectability condition and an arbitrary coupling constant $\lambda$, and obtain all the solutions in closed forms. If the gauge field $A$ and the Newtonian prepotential $\phi$ do not directly couple to matter fields, the theory is inconsistent with solar system tests for $\lambda ≠1$, no matter how small $\mathrm{\mid }\lambda -1\mathrm{\mid }$ is. This is shown to be true also with the most general ansatz of spherical (but not necessarily stationary) configurations. Therefore, to be consistent with observations, one needs either to find a mechanism to restrict $\lambda$ precisely to ${\lambda }_{GR}=1$, or to consider $A$ and/or $\phi$ as parts of the 4-dimensional metric on which matter fields propagate. In the latter, requiring that the line element be invariant not only under the foliation-preserving diffeomorphism but also under the local U(1) transformations, we propose the replacements, $N \rightarrow N - \upsilon(A - {\cal{A}})/c^2$and$N^i \rightarrow N^i+N\nabla^{i}\varphi$,where$\upsilon$isadimensionlesscouplingconstanttobeconstrainedbyobservations,$N$and$N^i$are,respectively,thelapsefunctionandshiftvector,and${\cal{A}} \equiv - \dot{\varphi} + N^i\nabla_{i}\varphi + N(\nabla_{i}\varphi)^2/2$. With this prescription, we show explicitly that the aforementioned solutions are consistent with solar system tests for both $\lambda =1$ and $\lambda ≠1$, provided that $\mathrm{\mid }\upsilon -1\mathrm{\mid }<{10}^{-5}$. From this result, the physical and geometrical interpretations of the fields $A$ and $\phi$ become clear. However, it still remains to be understood how to obtain such a prescription from the action principle.