Solar system tests and interpretation of gauge field and Newtonian prepotential in general covariant Hořava-Lifshitz gravity

Abstract

We study spherically symmetric, stationary vacuum configurations in general covariant theory (U(1) extension) of Hořava-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 $\lambda \ne 1$, 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 $\phi$ 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, $N\to N-\upsilon (A-\mathcal{A})/c^2$ and $N^i\to N^i+N{\nabla }^i\phi$, 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 $\mathcal{A}\equiv -\dot{\phi }+N^i{\nabla }_i\phi +N({\nabla }_i\phi {)}^2/2$. With this prescription, we show explicitly that the aforementioned solutions are consistent with solar system tests for both $\lambda =1$ and $\lambda \ne 1$, provided that $|\upsilon -1|<10^{-5}$. 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.

Comments: revtex4, 16 pages without figures. Phys. Rev. D86, 104024 (2012)