[Paper Review] Second comment to "Invariance of the tunneling method"

Marco Pizzi|ArXiv.org|Sep 21, 2009

Black Holes and Theoretical Physics8 references3 citations

TL;DR

This paper challenges the validity of a proposed derivation of Hawking radiation via the tunneling method, arguing that key steps in the derivation—particularly the treatment of the action near the horizon—are mathematically flawed. The author contends that the action remains real and constant along null geodesics in a locally inertial frame, contradicting claims of an imaginary part due to curvature invariants, and identifies a sign error in a related calculation that invalidates the claimed result.

ABSTRACT

I reply to the four points raised by S. A. Hayward, R. Di Criscienzo, M. Nadalini, L. Vanzo, S. Zerbini (arXiv:0909.2956v1) against my comment (arXiv:0907.2020v1) to their previous article. I maintain my position on the wrongness of their paper, reporting also another mistake.

Motivation & Objective

• To refute the claim that the tunneling method provides a valid derivation of Hawking radiation as presented in related works.
• To demonstrate that the action along a null geodesic is constant and real in a locally inertial frame, contradicting assertions of an imaginary part.
• To identify and correct a critical algebraic error in a key calculation of the cited paper, showing the action's imaginary part vanishes.
• To argue that curvature invariants involving second derivatives of the metric do not affect the action in the principal WKB approximation.
• To reassert that the semiclassical derivation of Hawking radiation remains unconvincing and requires further investigation.

Proposed method

• Uses the principal WKB approximation to argue that the dominant contribution to the action comes from the classical complexified trajectory, which is constant along null geodesics.
• Applies the equivalence between the geodesic equation and the Hamilton-Jacobi equation to validate the form of the action equation (3) in the author’s prior work.
• Demonstrates that the relativistic invariant interval $ ds $, which defines the action, depends only on the metric and connection coefficients (first derivatives), not second derivatives.
• Shows that since $ ds $ is real and invariant in any coordinate system, it cannot acquire an imaginary part in any frame, including near the horizon.
• Identifies a sign error in the radial and temporal action components in the cited paper’s Section 5.4, correcting $ ext{Im}[I_{+}] $ to zero after algebraic reevaluation.
• Argues that the absence of poles in the action integrand renders standard regularization procedures like the $ i heta $ prescription irrelevant.

Experimental results

Research questions

• RQ1 Can the tunneling method correctly derive the imaginary part of the action required for Hawking radiation?
• RQ2 Does the action along a null geodesic near the horizon acquire an imaginary component due to curvature invariants?
• RQ3 Is the calculation of the action in the cited paper mathematically consistent, particularly in the Lemaître-Rylov gauge?
• RQ4 Can the apparent derivation of Hawking radiation be invalidated by a simple algebraic error in the action components?
• RQ5 Is the semiclassical derivation of Hawking radiation still viable given these identified flaws?

Key findings

• The action along a null geodesic is constant and real in a locally inertial frame, and since it is a relativistic invariant, it remains real in all coordinate systems.
• The claim that the action acquires an imaginary part due to second derivatives of the metric is incorrect, as the invariant interval $ ds $ depends only on first derivatives.
• A sign error in the cited paper’s formula (V.68) invalidates the derivation, leading to $ \text{Im}[I_{+}] = 0 $ upon correction.
• The absence of poles in the action integrand renders the use of $ i\epsilon $ regularization unjustified and irrelevant.
• The tunneling method’s derivation in the cited work is fundamentally flawed, both mathematically and physically.
• The semiclassical derivation of Hawking radiation remains unconvincing and requires further rigorous investigation.