[Paper Review] Directional naked singularity in general relativity

K. S. Virbhadra|ArXiv.org|Jun 4, 1996

Advanced Differential Geometry Research7 references7 citations

TL;DR

This paper investigates the gamma metric, a static, axially symmetric solution to the Einstein vacuum equations, and demonstrates that for γ > 1, the curvature singularity at r = 2m is globally naked along θ = π/2 but not visible even locally along θ = 0, exhibiting 'directional nakedness'. This anisotropic visibility of a singularity challenges the cosmic censorship hypothesis and suggests potential astrophysical implications for observable singularities under specific geometric conditions.

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.

Motivation & Objective

• To analyze the visibility of curvature singularities in the gamma metric, a solution to the Einstein vacuum equations.
• To investigate whether the singularity at r = 2m is globally visible in all directions or only in specific angular regions.
• To determine under what conditions the singularity becomes 'directionally naked', i.e., visible in some directions but not others.
• To explore the implications of such directional nakedness for the cosmic censorship hypothesis and general relativity.

Proposed method

• The gamma metric is used as a static, axially symmetric, and asymptotically flat solution to the Einstein vacuum equations.
• The total energy of the spacetime is derived as mγ, with m and γ as constant parameters.
• The visibility of the singularity at r = 2m is analyzed by examining null geodesics and the behavior of light rays in different angular directions (θ = 0 and θ = π/2).
• The analysis distinguishes between global visibility (from infinity) and local visibility (in the neighborhood of the singularity) along different symmetry axes.
• The parameter γ is varied to compare cases γ < 1 and γ > 1, assessing how it affects the causal structure near the singularity.
• The study relies on geometric and differential-geometric analysis of the metric to determine whether future-directed null geodesics escape to infinity from the singularity in specific directions.

Experimental results

Research questions

• RQ1 Under what conditions is the curvature singularity at r = 2m globally visible in the gamma metric?
• RQ2 Does the visibility of the singularity depend on the direction in space, particularly along θ = 0 versus θ = π/2?
• RQ3 For γ > 1, why is the singularity globally naked along θ = π/2 but not even locally visible along θ = 0?
• RQ4 How does the directional visibility of the singularity challenge the cosmic censorship hypothesis?
• RQ5 What are the astrophysical implications of a singularity being visible in some directions but not others?

Key findings

• For γ < 1, the curvature singularity at r = 2m is globally visible along both θ = 0 and θ = π/2.
• For γ > 1, the singularity remains globally naked along θ = π/2, indicating visibility from infinity in that direction.
• However, for γ > 1, the singularity is not even locally visible along θ = 0, meaning no future-directed null geodesics escape from the singularity in that direction.
• This directional difference in visibility constitutes a 'directional nakedness' phenomenon, where the singularity is visible in one direction but not in another.
• The total energy of the spacetime is mγ, and positivity of this energy is assumed to ensure physical relevance.
• The result implies that the cosmic censorship hypothesis may not hold universally, as singularities can be visible in a directional, anisotropic manner.