Structure of the effective potential for a spherical wormhole

Abstract

The structure of the effective potential $V$ describing causal geodesics near the throat of an arbitrary spherical wormhole is analyzed. Einstein's equations relative to a set of regular coordinates covering a vicinity of the throat imply that any spherical wormhole can be constructed from solutions of an effective initial value problem with the throat serving as an initial value surface. The initial data involve matter variables, the area $A(0)$ of the throat, and the gradient $\ensuremath{\Lambda}(0)$ of the redshift factor on the throat. Whenever $\ensuremath{\Lambda}(0)=0$, the effective potential $V$ has a critical point on the throat. Conditions upon the data are derived ensuring that the critical point is a local minimum (respectively maximum). For particular families of quasi-Schwarzschild wormholes, $V$ exhibits a local minimum on the throat independently upon the energy $E$ and angular momentum ${L}^{2}$ of the test particles and thus such wormholes admit stable circular timelike and null geodesics on the throat. For families of Chaplygin wormholes, we show that such geodesics are unstable. Based on a suitable power series representation of the metric, properties of $V$ away from the throat are obtained that are useful for the analysis of accretion disks and radiation processes near the throat of any spherical wormhole.