# Axially Symmetric Null dust Spacetime, Naked Singularity and Cosmic Time Machine

## TL;DR

This paper introduces a new exact solution to Einstein's equations describing a gravitational collapse with a naked singularity and closed timelike curves, potentially functioning as a Cosmic Time Machine, with analysis of geodesics and lensing.

## Contribution

It presents a novel axially symmetric null dust spacetime solution exhibiting a naked singularity and CTCs, expanding understanding of cosmic time machines in general relativity.

## Key findings

- Spacetime is regular except on the axis where a naked singularity exists.
- Closed timelike curves develop causally well-behaved at a specific moment.
- The solution's physical interpretation involves geodesic deviation analysis.

## Abstract

In this article, we present a gravitational collapse null dust solution of the Einstein field equations. The spacetime is regular everywhere except on the symmetry axis where it possesses a naked curvature singularity, and admits one parameter isometry group, a generator of axial symmetry along the cylinder which has closed orbits. The space- time admits closed timelike curves (CTCs) which develop at some particular moment in a causally well-behaved manner and may represent a Cosmic Time Machine. The radial geodesics near to the singularity, and the gravitational Lensing (GL) will be discussed. The physical interpretation of this solution, based on the study of the equation of the geodesic deviation, will be presented. It was demonstrated that, this solution depends on the local gravitational field consisting of two components with amplitude $\Psi_2$, $\Psi_4$.