Analytic derivation of the map of null rays passing near a naked singularity

TL;DR

This paper analytically derives the mapping formula for null rays near a naked singularity in spherical dust collapse, clarifying the role of local geometry and extending previous numerical results.

Contribution

It provides an analytical derivation of the null ray mapping formula for a smooth density profile, previously obtained only numerically, highlighting the influence of local geometry.

Findings

Analytical derivation of the null ray mapping formula for smooth density profiles.

The singular nature of the mapping is governed by local geometry at the singularity.

Extension of previous numerical results to an analytical framework.

Abstract

Recently the energy emission from a naked singularity forming in spherical dust collapse has been investigated. This radiation is due to the particle creation in a curved spacetime. In this discussion, the central role is played by the mapping formula between the incoming and the outgoing null coordinates. For the self-similar model, this mapping formula has been derived analytically. But for the model with $C^{\infty}$ density profile, the mapping formula has been obtained only numerically. In the present paper, we argue that the singular nature of the mapping is determined by the local geometry around the point at which the singularity is first formed. If this is the case, it would be natural to expect that the mapping formula can be derived analytically. In the present paper, we analytically rederive the same mapping formula for the model with $C^{\infty}$ density profile that has been earlier derived using a numerical technique.