On the singular coordinate transformations of the Schwarzschild metric

TL;DR

This paper examines the limitations of common coordinate transformations in Schwarzschild spacetime, highlighting their low regularity and implications for the Einstein equations and quantum particle behavior at black hole horizons.

Contribution

It identifies the regularity issues of typical coordinate transformations and discusses their impact on the Einstein equations and quantum particle entry into black holes.

Findings

Standard transformations are only $C^0$, not $C^1$, affecting the metric's regularity.

Low regularity leads to fictitious delta-like sources in Einstein equations.

Implications for quantum particles attempting to enter black holes.

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.