Possible potentials responsible for stable circular relativistic orbits

Abstract

Bertrand's theorem in classical mechanics of the central force fields attracts us because of its predictive power. It categorically proves that there can only be two types of forces which can produce stable, circular orbits. In the present article an attempt has been made to generalize Bertrand's theorem to the central force problem of relativistic systems. The stability criterion for potentials which can produce stable, circular orbits in the relativistic central force problem has been deduced and a general solution of it is presented in the article. It is seen that the inverse square law passes the relativistic test but the kind of force required for simple harmonic motion does not. Special relativistic effects do not allow stable, circular orbits in presence of a force which is proportional to the negative of the displacement of the particle from the potential center. *

Galactic dark matter and Bertrand space-times

“…A Bertrand space-time [2] is defined as one in which each point in a spatial hypersurface admits closed, stable orbits. This generalizes the well known Bertrand's theorem in classical mechanics [3], [4] to a general relativistic scenario (for related work in the special relativistic case, see [5]). BSTs are static, spherically symmetric solutions of Einstein equations, which require a finite energy-momentum tensor, unlike the Schwarzschild solution.…”

“…(9) indicates that in general, away from the flat region, a Hernquist [13] profile is predicted from our analysis, in the Newtonian limit. 5 The appearance of the NFW profile is interesting. From eq.…”

“…These are seen to be positive definite, in line with figs. (4) and (5). The BST parameters have been chosen to be the same as in the previous discussion.…”

Astrophysics of Bertrand space-times

“…We will, for ease of presentation, refer to these as BSTs, throughout this paper. 2 This generalizes Bertrand's theorem in classical mechanics [7], [8] to a GR scenario (for related work in the special relativistic case, see [9]). …”

“…(23) is not satisfied then although the first fundamental form matches at the boundary, the second fundamental form does not, and this implies a thin shell of matter at the boundary. 9 In this case, a surface stress-energy tensor develops on both sides of Σ, given by [33] …”