Gravitational Radiation from a Naked Singularity. II: Even-Parity Perturbation

Abstract

A naked singularity occurs in the generic collapse of an inhomogeneous dust ball. We study the even-parity mode of gravitational waves from a naked singularity of the Lemaître-Tolman-Bondi spacetime. The wave equations for gravitational waves are solved by numerical integration using the single null coordinate. The result implies that the metric perturbation grows when it approaches the Cauchy horizon and diverges there, although the naked singularity is not a strong source of even-parity gravitational radiation. Therefore, the Cauchy horizon in this spacetime should be unstable with respect to linear even-parity perturbations. §1. IntroductionThe singularity theorems reveal that the occurrence of singularities is a generic property of spacetime in general relativity. 1) -3) However, these theorems state nothing about the detailed features of the singularities themselves; for example, we do not get information from these theorems about whether or not the predicted singularity is naked. Here, "naked" means that the singularity is in principle observable. A singularity is a boundary of spacetime. Hence, in order to obtain a solution of hyperbolic field equations for matter, gauge fields and spacetime itself in the causal future of a naked singularity, we need to impose a boundary condition on it. However, we do not yet know physically reasonable boundary conditions for singularities, and hence to avoid this difficulty, the cosmic censorship hypotheses (CCH) proposed by Penrose 4), 5) are often adopted in the analysis of physical phenomena involving strong gravitational fields.Unfortunately no one has ever succeeded in the proof of any version of the CCH. There is no precise statement of CCH which can be readily proved at this time. Given this situation it is worth trying to obtain counterexamples. Much effort has been made to search for naked singularity formation in gravitational collapse.In the Lemaître-Tolman-Bondi (LTB) spacetime, 6), 7) a naked shell-focusing singularity appears from generic initial data for spherically symmetric configurations of the rest mass density and a specific energy of the dust fluid. 8) -11) The initial functions in the most general expandable form have been considered. 12) The matter content in this spacetime may satisfy even the dominant energy condition. These results are summarized as follows: In this spacetime, a naked singularity appears * ) from generic initial data for spherically symmetric configurations of the rest mass density and a specific energy of the dust fluid. Shapiro and Teukolsky numerically studied evolution of collisionless gas spheroids with fully general relativistic simulations. 13) They found some evidence that prolate spheroids with sufficiently elongated initial configurations, and even with some angular momentum, may form naked singularities. Ori and Piran numerically examined the structure of self-similar spherical collapse solutions for a perfect fluid with a barotropic equation of state. 14), 15) They showed that there is a globally naked singularity in...

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