Problems with Tunneling of Thin Shells from Black Holes

TL;DR

The paper demonstrates that the commonly used tunneling probability formula for black hole shells is not invariant under canonical transformations, leading to inaccuracies in black hole temperature calculations, and proposes a corrected invariant formula.

Contribution

It identifies the non-invariance issue of the tunneling probability formula and proposes a canonical-invariant extension, highlighting its limitations in black hole temperature estimation.

Findings

The standard tunneling probability is not invariant under canonical transformations.

The proposed invariant extension yields only half the correct black hole temperature.

Canonical invariance is crucial for accurate tunneling and temperature calculations.

Abstract

It is shown that $exp(-2 Im(\int p dr))$ is not invariant under canonical transformations in general. Specifically for shells tunneling out of black holes, this quantity is not invariant under canonical transformations. It can be interpreted as the transmission coefficient only in the cases in which it is invariant under canonical transformations. Although such cases include alpha decay, they do not include the tunneling of shells from black holes. The simplest extension to this formula which is invariant under canonical transformations is proposed. However it is shown that this gives half the correct temperature for black holes.