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

Abstract

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 $\varphi$ do not directly couple to matter fields, the theory is inconsistent with solar system tests for $\lambda\not=1$, no matter how small $|\lambda-1|$ 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 $\varphi$ 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 $\upsilon$ is a dimensionless coupling constant to be constrained by observations, $N$ and $N^i$ are, respectively, the lapse function and shift vector, 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 $\lambda\not=1$, provided that $|\upsilon-1|<10^{-5}$. From this result, the physical and geometrical interpretations of the fields $A$ and $\varphi$ become clear. However, it still remains to be understood how to obtain such a prescription from the action principle.