Directional naked singularity in general relativity

Abstract

We consider a static, axially symmetric, and asymptotically flat exact solution of the Einstein vacuum equations, known as the gamma metric. This is characterized by two constant parameters $m$ and $γ$. We find that the total energy associated with this metric is $m γ$. Considering the total energy to be positive, we investigate the nature of a curvature singularity $r=2m$ ($r$ is the radial coordinate) in this metric. For $γ< 1$, this singularity is globally visible along $θ= 0$ as well as $θ= π/2$. However, for $γ> 1$, this singularity is though globally naked along $θ=π/2$, it is not visible (even locally) along $θ= 0$. Thus, this exhibits ``directional nakedness'' for $γ> 1$. This could have implications for astrophysics.