Static exteriors for nonstatic braneworld stars

We study possible static non-Schwarzschild exteriors for nonstatic spherically symmetric stars in a Randall $&$ Sundrum type II braneworld scenario. Thus, the vacuum region outside the surface of a star is assumed to be a static solution to the equation $^{(4)}R = 0$, where $^{(4)}R $ is the scalar curvature of the 4-dimensional Ricci tensor with spherical symmetry. Firstly, we show that for nonstatic spheres the standard matching conditions are much more restrictive than for static ones; they lead to a specific requirement on the vacuum region outside of a nonstatic star, that is absent in the case of static stars. Secondly, without making any assumption about the bulk, or the material medium inside the star, we prove the following theorem on the brane: for {\it any} nonstatic spherical star, without rotation, there are only two possible static exteriors; these are the Schwarzschild and the "Reissner-Nordstr{\"o}m-like" exteriors. This is quite distinct from the case of stars in hydrostatic equilibrium which admit a much larger family of non-Schwarzschild static exteriors.