(In)finiteness of spherically symmetric static perfect fluids

This work is concerned with the finiteness problem for static, spherically symmetric perfect fluids in both Newtonian gravity and general relativity. We derive criteria on the barotropic equation of state guaranteeing that the corresponding perfect fluid solutions possess finite/infinite extent. In the Newtonian case, for the large class of monotonic equations of state, and in general relativity we improve earlier results. Moreover, we are able to treat the two cases in a completely parallel manner, which is accomplished by using a relativistic version of Pohozaev's identity in the proof of the relativistic criterion. This identity and further generalizations are presented in detail.