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Ho\v{r}ava--Lifshitz gravity has covariance only under thefoliation-preserving diffeomorphism. This implies that the quantities on theconstant-time hypersurfaces should be regular. In the original theory, theprojectability condition, which strongly restricts the lapse function, isproposed. We assume that a star is filled with a perfect fluid with no-radialmotion and that it has reflection symmetry about the equatorial plane. As aresult, we find a no-go theorem for stationary and axisymmetric star solutionsin projectable Ho\v{r}ava--Lifshitz gravity under the physically reasonableassumptions in the matter sector. Since we do not use the gravitational actionto prove it, our result also works out in other projectable theories andapplies to not only strong gravitational fields, but also weak gravitationalones.

A No-Go Theorem for Rotating Stars of a Perfect Fluid without Radial Motion in Projectable Ho\v{r}ava--Lifshitz Gravity