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In this paper, we consider two different issues, stability and strongcoupling, raised lately in the newly-proposed Horava-Lifshitz (HL) theory ofquantum gravity with projectability condition. We find that all the scalarmodes are stable in the de Sitter background, due to two different kinds ofeffects, one from high-order derivatives of the spacetime curvature, and theother from the exponential expansion of the de Sitter space. Combining theseeffects properly, one can make the instability found in the Minkowskibackground never appear even for small-scale modes, provided that the IR limitis sufficiently closed to the relativistic fixed point. At the fixed point, allthe modes become stabilized. We also show that the instability of Minkowskispacetime can be cured by introducing mass to the spin-0 graviton. The strongcoupling problem is investigated following the effective field theory approach,and found that it cannot be cured by the Blas-Pujolas-Sibiryakov mechanism,initially designed for the case without projectability condition, but might becircumvented by the Vainshtein mechanism, due to the non-linear effects. Infact, we construct a class of exact solutions, and show explicitly that itreduces smoothly to the de Sitter spacetime in the relativistic limit.

Stability of spin-0 graviton and strong coupling in Horava-Lifshitz theory of gravity