In this note we examine whether spherically symmetric solutions in covariant Horava-Lifshitz gravity can reproduce Newton’s Law in the IR limit $\lambda \to 1$. We adopt the position that the auxiliary field $A$ is independent of the space-time metric [J. Alexandre and P. Pasipoularides, Phys. Rev. D 83, 084030 (2011).][J. Greenwald, V. H. Satheeshkumar, and A. Wang, J. Cosmol. Astropart. Phys. 12 (2010) 007.], and we assume, as in [A. M. da Silva, Classical Quantum Gravity 28, 055011 (2011).], that $\lambda$ is a running coupling constant. We show that under these assumptions, spherically symmetric solutions fail to restore the standard Newtonian physics in the IR limit $\lambda \to 1$, unless $\lambda$ does not run, and has the fixed value $\lambda =1$. Finally, we comment on the Horava and Melby-Thompson approach [P. Horava and C. M. Melby-Thompson, Phys. Rev. D 82, 064027 (2010).] in which $A$ is assumed as a part of the space-time metric in the IR.

• Received 18 August 2011

DOI:https://doi.org/10.1103/PhysRevD.84.084020