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

Hideki Maeda 1 ) Tomohiro Harada 1 ) Hideo Iguchi 0 Naoya Okuyama 1 0 Department of Physics, Tokyo Institute of Technology , Tokyo 152-8550, Japan 1 Department of Physics, Waseda University , Tokyo 169-8555, Japan 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 , where p and 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 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 the -cylinder solution. - the best description of the central part of a generic collapsing gas sphere. 6) 11) For a polytropic equation of state, Yahil found the polytropic counterpart of the LarsonPenston solution describing a gravitationally collapsing sphere 12) (see also Refs. 8) and 13)). We hereafter refer to these solutions as the polytropic Larson-Penston solutions. In general relativity, self-similarity is defined by the existence of a homothetic Killing vector field. 14) Such self-similarity is referred to as that of the first kind (or homothety). Ori and Piran discovered the general relativistic counterpart of the Larson-Penston self-similar solution together with Hunters family of self-similar solutions for a perfect fluid obeying an equation of state p = K (0 < K < 0.036), where p and are the pressure and the energy density, respectively. 15) 17) They observed that a naked singularity forms in this solution for 0 < K < 0.0105. Harada and Maeda found that generic non-self-similar spherical collapse converges to the general relativistic Larson-Penston solution in an approach to a singularity for 0 < K < 0.036. 18), 19) Because a naked singularity forms for 0 < K < 0.0105, this implies the violation of cosmic censorship in the spherically symmetric case (see also Refs. 20) and 21)). This represents the strongest known counterexample of cosmic censorship. It also provides strong evidence for the self-similarity hypothesis in general relativistic gravitational collapse. The question then naturally arises whether collapsing self-similar solutions with a polytropic equation of state exist in general relativity. If such solutions do exist, they may play an important role in the final stage of generic collapse, as in the p = K case. In Newtonian gravity, the form of the dimensionless variable for self-similarity in the polytropic case is different from that in the isothermal case, because speed of sound is not constant in the former case. The self-similarity coordinate is given by t2 /r in the polytropic case and by t/r in the isothermal case. The scaling functions of physical quantities in the former case are also different from those in the latter case. 13) In general relativity, there exists a natural generalization of homothety called kinematic self-similarity, which is defined by the existence of a kinematic self-similar vector field 22) (see also earlier related works by Tomita 23)). Kinematic self-similarity is characterized by an index and classified into three kinds: the second, zeroth and infinite. One can show that an equation of state of the form p = K is the only barotropic one compatible with self-similarity of the first kind. 14) Self-similar perfect-fluid solutions of the first kind with an equation of state of this form have been classified for the dust case (K = 0) by Carr 24) and for the case 0 < K < 1 by Carr and Coley 25) (see also Ref. 26)). Special cases in which a homothetic Killing vector is not tilted (i.e., is either parallel or orthogonal to the fluid flow) have also been studied. 27), 28) Kinematic self-similar perfect-fluid solutions have been explored by several authors. 29) 32) Benoit and Coley have studied spherically symmetric spacetimes that admit a kinematic self-similar vector of either the second or zeroth kind. 30) Sintes, Benoit and Coley have considered spacetimes that admit a kinematic selfsimilar vector of the infinite kind. 31) In these works, the equation of state is not specified. We have previously investigated spherically symmetric spacetimes that contain a perfect fluid obeying a relativistic polytropic equation of state and admit a kinematic self-similar vector of the second kind in which the kinematic self-similar vector is tilted. 33) There, we assumed two kinds of polytropic equations of state in general relativity and showed that such spacetimes must be vacuums in both cases. Although a spherically symmetric spacetime that contains a relativistic polytropic perfect fluid is incompatible with kinematic self-similarity of the second kind, it could be compatible with other kinds of kinematic self-similarities (i.e., the zeroth or infinite kind), or with the case in which a kinematic self-similar vector is parallel or orthogonal to the fluid flow. In this paper, we extend our previous work in several important ways. We study spacetimes that contain a perfect fluid obeying either a polytropic equation of state or an equation of state p = K , and that admit a kinematic self-similar vector field of the second, zeroth or infinite kind. We assume two kinds of relativistic polytropic equations of state and study the case in which a kinematic self-similar vector is not only tilted but also parallel or orthogonal to the fluid flow. The organization of this paper is the following. In 2, the basic equations in a spherically symmetric spacetime are presented, and kinematic self-similarity is briefly reviewed. We treat the cases in which a kinematic self-similar vector is tilted, parallel and orthogonal to the fluid flow in 3, 4 and 5, respectively. Section 6 is devoted to a summary and discussion. Possible equations of state that are compatible with self-similarity are discussed in Appendix A. We adopt units such that c = 1. 2. Spherically symmetric spacetime and kinematic self-similarity The line element in a spherically symmetric spacetime is given by ds2 = e2(t,r)dt2 + e2(t,r)dr2 + R(t, r)2d 2, where U is the four-velocity of the fluid element. We have adopted comoving coordinates. Then, the Einstein equations and the equations of motion for the perfect fluid are reduced to the following simple form: m = R(1 + e2Rt2 e2 Rr2), where the subscripts t and r denote derivatives with respect to t and r, respectively, and m(t, r) is called the Misner-Sharpmass. We also write the auxiliary equations as + 2 RRt t + e22 2 RRrr 2 RRr r + tt + t2 tt + RRtt + RRtt RRtt p for EOS (II). For 1 < , EOS (II) is approximated by EOS (III) with K = 1 in the hig (...truncated)