Black holes, compact objects and solar system tests in non-relativistic
  general covariant theory of gravity
  We study spherically symmetric static spacetimes generally filled with an
anisotropic fluid in the nonrelativistic general covariant theory of gravity.
In particular, we find that the vacuum solutions are not unique, and can be
expressed in terms of the $U(1)$ gauge field $A$. When solar system tests are
considered, severe constraints on $A$ are obtained, which seemingly pick up the
Schwarzschild solution uniquely. In contrast to other versions of the
Horava-Lifshitz theory, non-singular static stars made of a perfect fluid
without heat flow can be constructed, due to the coupling of the fluid with the
gauge field. These include the solutions with a constant pressure. We also
study the general junction conditions across the surface of a star. In general,
the conditions allow the existence of a thin matter shell on the surface. When
applying these conditions to the perfect fluid solutions with the vacuum ones
as describing their external spacetimes, we find explicitly the matching
conditions in terms of the parameters appearing in the solutions. Such matching
is possible even without the presence of a thin matter shell.
