Geodesic structure of a noncommutative black hole
Zihan Xi, Chen Wu, Wenjun Guo
TL;DR
This work computes the geodesic structure of Nicolini's noncommutative spherically symmetric black hole by deriving the metric's lapse function $f(r)$, formulating the Lagrangian, and obtaining the radial equation with the effective potential $V_{eff}^2=f(r)(\\eta+L^2/r^2)$. Transforming to the equatorial plane and using $u=1/r$ yields the orbit equation, enabling a full classification of time-like and null geodesics into bound, escape, and unstable circular categories. The authors analyze horizon structure as a function of mass $M$, showing no horizon for $M<1.9\\sqrt{\\theta}$, an extremal horizon at $M=1.9\\sqrt{\\theta}$, and two horizons for $M>1.9\\sqrt{\\theta}$, while finding that $L$ does not alter horizon count at fixed $M$. The key results indicate that increasing $M$ or $L$ reduces the perihelion precession rate, with the mass effect nonlinear and the angular-momentum effect linear, offering insights into particle and photon dynamics in noncommutative black hole spacetimes.
Abstract
This paper explores the metric of Piero Nicolini's noncommutative black hole spacetime, calculates its effective potential, and presents the corresponding potential curve. By analyzing this curve, we identify various orbit types for test particles and photons in this spacetime. Using the dynamical equations for particles and photons near the black hole, we plot the specific time-like and null geodesic structures. We analyze the impact of different values of the total mass of the source $M$ and angular momentum $L$ on time-like geodesics. Our results indicate that in the Piero black hole spacetime, increases in total mass and angular momentum reduce the perihelion precession rate of the orbit. Notably, the effect of total mass is nonlinear, while the effect of angular momentum is linear.