[Paper Review] Tunnelling from black holes in the Hamilton Jacobi approach

B. Chatterjee, Amit Ghosh+1 more|arXiv (Cornell University)|Apr 13, 2007

Quantum Electrodynamics and Casimir Effect4 citations

TL;DR

This paper applies the Hamilton-Jacobi method to model Hawking radiation as quantum tunnelling across black hole horizons, demonstrating consistent tunnelling rates for non-rotating and rotating charged black holes across different coordinate systems. It resolves discrepancies with prior work by enforcing appropriate boundary conditions, confirming the tunnelling interpretation of Hawking radiation with precise temperature predictions.

ABSTRACT

It has recently been shown that it is possible to understand Hawking radiation as tunnelling across black hole horizons using appropriate Hamilton-Jacobi boundary conditions. The procedure is applied to the non-rotating black hole in different coordinate systems and to the rotating charged black hole. Differences with the earlier literature are pointed out.

Motivation & Objective

• To re-express Hawking radiation as quantum tunnelling using the Hamilton-Jacobi approach.
• To analyze tunnelling in non-rotating black holes across different coordinate systems.
• To extend the tunnelling framework to rotating, charged black holes.
• To resolve inconsistencies with earlier literature by enforcing proper boundary conditions.
• To validate the tunnelling mechanism as a consistent description of black hole radiation.

Proposed method

• Uses the Hamilton-Jacobi equation to derive the action for scalar particles tunnelling across black hole horizons.
• Imposes specific boundary conditions at the horizon to model outgoing and ingoing wave solutions.
• Applies the method to Schwarzschild and Kerr-Newman black holes in various coordinate systems.
• Evaluates the tunnelling rate by computing the imaginary part of the action.
• Derives the Hawking temperature from the tunnelling rate using the relation Γ ∝ exp(−Im I).
• Compares results with standard Hawking temperature predictions to verify consistency.

Experimental results

Research questions

• RQ1 Can the Hamilton-Jacobi method consistently describe Hawking radiation as tunnelling in non-rotating black holes across different coordinate systems?
• RQ2 How does the tunnelling rate for charged, rotating black holes compare with standard Hawking temperature predictions?
• RQ3 What role do boundary conditions play in resolving discrepancies with earlier tunnelling models?
• RQ4 Does the tunnelling approach yield the same temperature for different coordinate representations of the same black hole?
• RQ5 How does the method handle the distinction between outgoing and ingoing waves at the horizon?

Key findings

• The Hamilton-Jacobi method successfully reproduces the correct Hawking temperature for non-rotating black holes in multiple coordinate systems.
• For rotating, charged black holes, the tunnelling rate yields a temperature consistent with the standard Hawking temperature.
• Proper boundary conditions at the horizon are essential to eliminate inconsistencies present in earlier tunnelling treatments.
• The imaginary part of the action directly leads to the exponential tunnelling rate, confirming the thermal nature of radiation.
• The method provides a unified framework for understanding Hawking radiation as tunnelling across different black hole geometries.