Einstein's hole argument and Schwarzschild singularities

According to the Einstein hole argument, vacuum metric solutions are equivalent only if they correspond to the same energy--momentum tensor in the source region. In this paper it is shown that singular coordinates that are used to show Schwarzschild geodesics completeness, introduce the fictive delta-like sources at the horizon. Then, metric tensors obtained by such singular transformations, cannot be considered as solutions of the same Einstein equations with the central source.