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 $\gamma$. We find that the total energy associated with this metric is $m \gamma$. 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 $\gamma < 1$, this singularity is globally visible along $\theta = 0$ as well as $\theta = \pi /2$. However, for $\gamma > 1$, this singularity is though globally naked along $\theta =\pi/2$, it is not visible (even locally) along $\theta = 0$. Thus, this exhibits ``directional nakedness'' for $\gamma > 1$. This could have implications for astrophysics.