A Classification of Spherically Symmetric Kinematic Self-Similar Perfect-Fluid Solutions

We classify all spherically symmetric spacetimes admitting a kinematicself-similar vector of the second, zeroth or infinite kind. We assume that theperfect fluid obeys either a polytropic equation of state or an equation ofstate of the form $p=K\mu$, where $p$ and $\mu$ are the pressure and the energydensity, respectively, and $K$ is a constant. We study the cases in which thekinematic self-similar vector is not only ``tilted'' but also parallel ororthogonal to the fluid flow. We find that, in contrast to Newtonian gravity,the polytropic perfect-fluid solutions compatible with the kinematicself-similarity are the Friedmann-Robertson-Walker solution and general staticsolutions. We find three new exact solutions which we call the dynamicalsolutions (A) and (B) and $\Lambda$-cylinder solution, respectively.Comment: Revised version, a reference added, 36 pages, 4 tables, no figures, accepted for publication in Progress of Theoretical Physic