An observation on the congruence of the escape velocity in classical mechanics and general relativity in a Schwarzschild metric

In classical mechanics, the escape velocity is defined as the minimum value of velocity that an object requires in order to leave a gravitational field asymptotically. This quantity may be used to measure the strength of the gravitational field. In general relativity, a suitable parameter characterizing the strength of a gravitational field is the Schwarzschild radius, RS, which corresponds to the singularity of Schwarzschild metrics. Classical mechanics itself is a weak-gravitational-field and small-velocity limit of general relativity and its predictions refer to the phenomena of that domain. But when the escape velocity is formally put equal to c, the critical radius defined in such a way coincides with an exact critical radius, namely the Schwarzschild radius. We discuss here the source of this coincidence.