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

TL;DR

This paper proves a no-go theorem for stationary, axisymmetric perfect fluid star solutions without radial motion in projectable Hořava--Lifshitz gravity, applicable to both strong and weak gravitational fields.

Contribution

It establishes a fundamental restriction on star solutions in projectable Hořava--Lifshitz gravity without relying on the gravitational action, extending to other projectable theories.

Findings

No stationary, axisymmetric perfect fluid star solutions without radial motion exist in the theory.

The no-go theorem applies to both strong and weak gravitational fields.

The result holds under physically reasonable assumptions in the matter sector.

Abstract

Ho\v{r}ava--Lifshitz gravity has covariance only under the foliation-preserving diffeomorphism. This implies that the quantities on the constant-time hypersurfaces should be regular. In the original theory, the projectability condition, which strongly restricts the lapse function, is proposed. We assume that a star is filled with a perfect fluid with no-radial motion and that it has reflection symmetry about the equatorial plane. As a result, we find a no-go theorem for stationary and axisymmetric star solutions in projectable Ho\v{r}ava--Lifshitz gravity under the physically reasonable assumptions in the matter sector. Since we do not use the gravitational action to prove it, our result also works out in other projectable theories and applies to not only strong gravitational fields, but also weak gravitational ones.