Lower bound on the time-to-mass ratio of closed circular motions around spinning and charged naked singularities

Abstract

AbstractIt has recently been conjectured that circular trajectories (geodesic as well as non-geodesic) around central compact objects are characterized by the time-to-mass universal lower bound $$T_{\infty }/M>{{\mathcal {C}}}$$ T ∞ / M > C , where $$\{T_{\infty },M\}$$ { T ∞ , M } are respectively the orbital period as measured by flat-space asymptotic observers and the mass of the central compact object, and $$\mathcal{C}=O(1)$$ C = O ( 1 ) is a dimensionless constant of order unity. In the present paper we prove that this dimensionless bound is respected by circular orbits around central naked singularities. Intriguingly, it is explicitly proved that the orbital-time-to-mass ratio $$T_{\infty }/M$$ T ∞ / M around a central super-spinning singularity remains finite even in the $$r\rightarrow 0$$ r → 0 limit of circular orbits with infinitesimally small radial coordinates (which are characterized by a diverging time-to-radius ratio, $$T_{\infty }/r\rightarrow \infty $$ T ∞ / r → ∞ ). In particular, we reveal the fact that the shortest orbital period around a super-spinning naked singularity is given by the dimensionless relation $$T_{\infty }/M=2\pi $$ T ∞ / M = 2 π . In addition, we explore the functional behavior of the time-to-mass ratio of circular trajectories around super-charged (non-vacuum) naked singularities and prove that the dimensionless ratio $$T_{\infty }/M(r)$$ T ∞ / M ( r ) is bounded from below, where M(r) is the gravitational mass contained within the orbital radius of the test particle.