ar X iv : a st ro - p h / 99 05 29 8 v 1 2 3 M ay 1 99 9 Another view on the velocity at the Schwarzschild horizon

It is shown that a timelike radial geodesic does not become null at the event horizon. Recently an attempt was made to demonstrate that in the Schwarzschild geometry the radial geodesics of material particles become null at the event horizon [1]. For this purpose, was derived an expression that corresponds to the velocity of a material particle following a radial trajectory as measured by an observer, also on a radial trajectory, when they intersect. The observer mantains its spacelike Kruskal coordinates unchanged and for this reason we call it a Kruskal observer. For r > 2m, the expression is, (eq.(20) of [1] and eq.(8) of [2]), v = 1 + tanh(t/4m) dt dr (1− 2m/r) tanh(t/4m) + dt dr (1− 2m/r) , (1) where dt and dr refer to the movement t(r) of the particle. At the event horizon, where r = 2m and t = +∞, the value of eq.(1) is apparentely indetermined and in [1] it is stated that v = 1, independently of the precise relationship t(r). In [2] some manipulations were made maintaining the generality of the expression, i.e. without substituting for t(r). These permited to show (eq.(13) of [2]) that the velocity is always less than 1 along the way, until it obviously turns to 0/0 at r = 2m. So there is no a priori reason to think it is necessarily v = 1. This procedure was commented in a somewhat ungracious manner in [3] without any further explanations being made. The best way to avoid confusion and get a definitive result seems to be to consider a specific geodesic t(r), transforming (1) in a function of 1 variable. Let us then consider a material particle in an ingoing radial geodesic parametrized by its proper time τ . For this trajectory we can write, in Schwarzschild coordinates, ds = −dτ = − ( 1− 2m r ) dt + ( 1− 2m r ) −1 dr. (2) Inserting the conserved quantity for motion,