Gravitational collapse in Ho\v{r}ava-Lifshitz theory

We study gravitational collapse of a spherical fluid in nonrelativistic general covariant theory of the Ho\v{r}ava-Lifshitz gravity with the projectability condition and an arbitrary coupling constant $\lambda$, where $|\lambda - 1|$ characterizes the deviation of the theory from general relativity in the infrared limit. The junction conditions across the surface of a collapsing star are derived under the (minimal) assumption that the junctions be mathematically meaningful in terms of distribution theory. When the collapsing star is made of a homogeneous and isotropic perfect fluid, and the external region is described by a stationary spacetime, the problem reduces to the matching of six independent conditions. If the perfect fluid is pressureless (a dust fluid), it is found that the matching is also possible. In particular, in the case $\lambda = 1$, the external spacetime is described by the Schwarzschild (anti-) de Sitter solution written in Painlev\'e-Gullstrand coordinates. In the case $\lambda \not= 1$, the external spacetime is static but not asymptotically flat. Our treatment can be easily generalized to other versions of Ho\v{r}ava-Lifshitz gravity or, more generally, to any theory of higher-order derivative gravity.