Power, energy, and spectrum of a naked singularity explosion

It is well known that a naked singularity occurs in the gravitational collapse of an inhomogeneous dust ball from an initial density profile which is physically reasonable. In this paper we show that explosive radiation is emitted during the formation process of the naked singularity while we fix the background spacetime. The energy flux is proportional to ${(t}_{\mathrm{CH}}-{t)}^{-3/2}$ for a minimally coupled massless scalar field, while it is proportional to ${(t}_{\mathrm{CH}}-{t)}^{-1}$ for a conformally coupled massless scalar field, where $t_{\mathrm{CH}}-t$ is the “remaining time” until the distant observer could observe the singularity if the naked singularity was formed. As a consequence, the radiated energy grows unboundedly for both scalar fields. The amount of the power and energy depends on the parameters which characterize the initial density profile but do not depend on the gravitational mass of the cloud. In particular, there is a characteristic frequency ${\nu }_s$ of the singularity above which the divergent energy is radiated. The energy flux is dominated by particles of which the wavelength is about $t_{\mathrm{CH}}-t$ at each moment. The observed total spectrum is nonthermal, i.e., $\nu dN/d\nu \sim (\nu /{\nu }_s{)}^{-1}$ for $\nu >{\nu }_s.$ If the naked singularity formation could continue until a considerable fraction of the total energy of the dust cloud is radiated, the radiated energy would reach about ${10}^{54}{(M/M}_{\odot })\mathrm{erg}.$ The calculations are based on the geometrical optics approximation which turns out to be consistent for a rough order estimate. The analysis does not depend on whether or not the naked singularity occurs in its exact meaning. This phenomenon may provide a new candidate for a source of ultrahigh energy cosmic rays or a central engine of $\gamma$-ray bursts.