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

We classify all spherically symmetric spacetimes admitting a kinematic self-similar vector of the second, zeroth or infinite kind. We assume that the perfect fluid obeys either a polytropic equation of state or an equation of state of the form $p=K\mu$, where $p$ and $\mu$ are the pressure and the energy density, respectively, and $K$ is a constant. We study the cases in which the kinematic self-similar vector is not only ``tilted’’ but also parallel or orthogonal to the fluid flow. We find that, in contrast to Newtonian gravity, the polytropic perfect-fluid solutions compatible with the kinematic self-similarity are the Friedmann-Robertson-Walker solution and general static solutions. We find three new exact solutions which we call the dynamical solutions (A) and (B) and $\mathrm{\Lambda }$-cylinder solution, respectively.