Power, energy, and spectrum of a naked singularity explosion

TL;DR

This paper investigates the explosive radiation emitted during the formation of a naked singularity in gravitational collapse, revealing unbounded energy flux and potential implications for ultra high energy cosmic rays and gamma-ray bursts.

Contribution

It demonstrates that naked singularity formation leads to divergent energy flux and spectrum characteristics, providing new insights into high-energy astrophysical phenomena.

Findings

Energy flux diverges as (t_CH - t)^{-3/2} for minimally coupled scalar fields.

Total radiated energy can reach about 10^{54} erg for stellar mass clouds.

Spectrum above a characteristic frequency is nonthermal, following a (ν/ν_s)^{-1} distribution.

Abstract

Naked singularity occurs in the gravitational collapse of an inhomogeneous dust ball from an initial density profile which is physically reasonable. We show that explosive radiation is emitted during the formation process of the naked singularity. The energy flux is proportional to $(t_{\rm CH}-t)^{-3/2}$ for a minimally coupled massless scalar field, while is proportional to $(t_{\rm CH}-t)^{-1}$ for a conformally coupled massless scalar field, where $t_{\rm CH}-t$ is the `remained 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 the energy depends on parameters which characterize the initial density profile but do not depend on the gravitational mass of the cloud. In particular, there is characteristic frequency $\nu_{s}$ of singularity above which the divergent energy is radiated. The energy flux is dominated by particles of which the wave length is about $t_{\rm CH}-t$ 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 $10^{54}(M/M_{\odot})$ erg. The calculations are based on the geometrical optics approximation which turns out to be consistent as 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 ultra high energy cosmic rays or a central engine of gamma ray bursts.