On the singular coordinate transformations of the Schwarzschild metric

Abstract

It is noted that the coordinate transformations usually used to demonstrate the continuity of geodesics at the Schwarzschild horizon are of class $C^0$, while the standard causality theory requires that the metric tensor to be at least $C^1$. Then this singular metric tensor leads to the appearance of the fictitious delta-like source in the Einstein equations, which prevents quantum particles to enter a black hole horizon.