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

The 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 $t$. By contrast, we show that the event horizon is reached for a finite value of the Eddington-Finkelstein time variable $t'$. 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 $t'_1$ of the time marker $t'$, and continues its trajectory across the throat for $t'>t'_1$. 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."