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 Ψ2, Ψ4.Comment: 19 pages ; accepted for publication in Adv. high energy phy
