# Axial symmetry Type N space-time with a naked curvature singularity and Closed Time-like Curves

## TL;DR

This paper presents a new exact solution to Einstein's equations describing a type N space-time with a naked singularity, closed time-like curves, and unique geometric properties, expanding understanding of such exotic configurations.

## Contribution

It introduces a novel family of type N solutions with naked singularities and CTCs, generalizing Misner space in curved space-time, and analyzes their geometric and physical properties.

## Key findings

- Presence of naked curvature singularity on the symmetry axis.
- Existence of closed time-like curves after a certain time.
- Space-time is geodesically complete along the radiation direction.

## Abstract

A family of type N exact solution of the Einstein's field equations, regular everywhere except on the symmetry axis where it possesses a naked curvature singularity, is present. The stress-energy tensor is of the anisotropic fluid coupled with pure radiation field satisfy the different energy conditions and the physical parameters diverge $r \rightarrow 0$. The space-time admitting a non-expanding, non-twisting, and shear-free geodesic null congruence and belongs to a special class of type N Kundt metrics. The space-time is geodesically complete along the radiation direction in the constant $z$-planes and exhibits geometrically different properties from the known pp-waves. The present family of solution admits closed time-like curves (CTC) which appear after a certain instant of time and the space-time is a four-dimensional generalization of the Misner space metric in curved space-time.