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Newtonian analysis of gravitational waves due to the formation of a naked singularity

Spherical dust collapse generally forms a shell focusing naked singularity at the symmetric center. This naked singularity is massless. Further, the Newtonian gravitational potential and speed of the dust fluid elements are everywhere much smaller than unity until the central shell focusing naked singularity formation and the formation of the Cauchy horizon associated with it if an appropriate initial condition is set up. Although such a situation is highly relativistic, the analysis by the Newtonian approximation scheme is available even in the vicinity of the space-time singularity as long as we focus on the chronological past of the Cauchy horizon which does not suffer the causal influence of the central naked singularity. This remarkable feature makes the analysis of such singularity formation process very easy. We investigate nonspherical even-parity matter perturbations in this scheme by complementary use of numerical and semianalytical approaches, and estimate linear gravitational waves generated just before the formation of the central shell focusing naked singularity by the quadrupole formula. The result shows good agreement with the relativistic perturbation analysis recently performed by Iguchi et al. The energy flux of the gravitational waves is finite but the space-time curvature carried by them diverges.