Abstract
The present work discusses motion of neutral and charged particles in Reissner–Nordström spacetime. The constant energy paths are derived in a variational principle framework using the Jacobi metric which is parameterized by conserved particle energy. Of particular interest is the case of particle charge and Reissner–Nordström black hole charge being of same sign, since this leads to a clash of opposing forces—gravitational (attractive) and Coulomb (repulsive). Our paper aims to complement the recent work of Pugliese et al. (Eur Phys J C 77:206. arXiv:1304.2940, 2017; Phys Rev D 88:024042. arXiv:1303.6250, 2013). The energy dependent Gaussian curvature (induced by the Jacobi metric) plays an important role in classifying the trajectories.
Highlights
The primary major difference is the following
The Lagrangian equation of motion is derived from the unrestricted variational principle with the charge and mass parameters of the source and probe dictating the particle motion for arbitrary particle energy, which is a derived quantity
In the present work we have considered particle trajectories that are parameterized by constant energy value
Summary
The primary major difference is the following. On the one hand, Pugliese et al [1,2,3] rely on the covariant framework where geodesic motion takes place in the Reissner– Nordström spacetime. In a Jacobi-metric framework, the equation of motion is obtained from a restricted variational principle with fixed energy where the Lagrangian depends on the Jacobi metric, which involves the particle energy explicitly. The Jacobi metric is manifestly non-covariant (on a spatial slice) [8,9] and the equation of motion is derived from a restricted (Maupertuis) variational principle with fixed particle energy. Apart from the importance of curvature in the Jacobi metric, one way in which the present work can complement the exhaustive analysis of [1,2] is the following: whereas [1,2] relies on an extremely detailed graphical analysis of the orbit structure keeping exact expressions we have provided analytic expressions of the orbits, albeit in a perturbative framework (of small charge of both the black hole and the probe).
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