Infall time in the Eddington–Finkelstein metric, with application to Einstein–Rosen bridges

Abstract

Eddington–Finkelstein metric is obtained from the Schwarzschild metric by a change of the time variable. It is well known that a test mass falling into a black hole does not reach the event horizon for any finite value of the Schwarzschild time variable [Formula: see text]. By contrast, we show that the event horizon is reached for a finite value of the Eddington–Finkelstein time variable [Formula: see text]. Then we study in Eddington–Finkelstein time the fate of a massive particle traversing an Einstein–Rosen bridge and obtain a different conclusion than recent proposals in the literature: we show that the particle reaches the wormhole throat for a finite value [Formula: see text] of the time marker [Formula: see text], and continues its trajectory across the throat for [Formula: see text]. Such a behavior does not make sense in Schwarzschild time since it would amount to continuing the trajectory of the particle “beyond the end of time.”