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

We give a classification of spherically symmetric kinematic self-similarsolutions. This classification is complementary to that given in a previouswork by the present authors [Prog. Theor. Phys. 108, 819 (2002)]. Dustsolutions of the second, zeroth and infinite kinds, perfect-fluid solutions andvacuum solutions of the first kind are treated. The kinematic self-similarityvector is either parallel or orthogonal to the fluid flow in the perfect-fluidand vacuum cases, while the `tilted' case, i.e., neither parallel nororthogonal case, is also treated in the dust case. In the parallel case, thereare no dust solutions of the second (except when the self-similarity index$\alpha$ is 3/2), zeroth and infinite kinds, and in the orthogonal case, thereare no dust solutions of the second and infinite kinds. Except in these cases,the governing equations can be integrated to give exact solutions. It is foundthat the dust solutions in the tilted case belong to a subclass of the Lema{\^i}tre-Tolman-Bondi family of solutions for the marginally bound case. The flatFriedmann-Robertson-Walker (FRW) solution is the only dust solution of thesecond kind with $\alpha=3/2$ in the tilted and parallel cases and of thezeroth kind in the orthogonal case. The flat, open and closed FRW solutionswith $p=-\mu/3$, where $p$ and $\mu$ are the pressure and energy density,respectively, are the only perfect-fluid first-kind self-similar solutions inthe parallel case, while a new exact solution with $p=\mu$, which we call the``singular stiff-fluid solution'', is the only such solution in the orthogonalcase. The Minkowski solution is the only vacuum first-kind self-similarsolution both in the parallel and orthogonal cases. Some important correctionsand complements to the authors' previous work are also presented.Comment: 39 pages, 5 tables, 7 figures, revised version, comments on the Nariai solution are added, accepted for publication in Progress of Theoretical Physic