Periodic orbits around a spherically symmetric naked singularity

The motion of time-like test particles in the Fisher/Janis-Newman-Winicour (F/JNW) spacetime is studied with the Hamiltonian formulation of the geodesic equations. The spacetime is characterized by its mass parameter ${r}_{g}$ and scalar field parameter $\ensuremath{\nu}$. The innermost bound and stable circular orbits are calculated and the effective potential is analyzed. Consistent with numerical results in earlier literature, for $\ensuremath{\nu}<\frac{1}{2}$, particles with nonzero angular momentum encounter an infinite potential barrier, preventing them from reaching the naked singularity at $r={r}_{g}$. Periodic orbits in the spacetime are also obtained. Compared to the periodic orbits around the Schwarzschild black hole, it is found that typically lower energies are required for the same orbits in the F/JNW spacetime.

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