[Paper Review] Sign Errors in "The Four Laws of Black Hole Mechanics"

Richard P. Behiel|arXiv (Cornell University)|Mar 26, 2026

Relativity and Gravitational Theory0 citations

TL;DR

The paper identifies two compensating sign errors in the 1973 BCH formulation and shows that correcting them preserves the original conclusions. It explains the impact on equations (33)-(34) and the definitions of N and S.

ABSTRACT

In 1973, Bardeen, Cater, and Hawking published "The Four Laws of Black Hole Mechanics", establishing the mathematical framework that would later be understood as the thermodynamics of black holes. Central to the paper is equation (33), which writes the variation of the total energy-momentum integral in terms of physically meaningful quantities: angular momentum, particle number, and entropy. Equation (33) feeds into the differential mass formula, equation (34), which is the first law of black hole mechanics. This note identifies two compensating sign errors in the BCH paper. The first error, demonstrated by a derivation from equation (32), is that equations (33) and (34) should carry minus signs rather than plus signs on the last two integrals, those involving the redshifted chemical potential and the redshifted temperature. he second error is that the definitions of total particle number N and total entropy S given after equation (20) are missing minus signs that are required for these quantities to be positive. These two errors cancel, in that reversing the signs in the definitions of N and S to ensure positive quantities makes equations (33) and (34) correct. All conclusions of the BCH paper remain valid. This note is intended merely as a guide for readers who, in working through the derivation step by step, might otherwise be puzzled by the sign discrepancies. Numbered equations refer to the BCH paper; lettered equations are introduced in this note.

Motivation & Objective

• Clarify the sign conventions used in the BCH derivation of the first law of black hole mechanics.
• Pinpoint where sign errors occur in the derivation of the energy-momentum variation and the first law.
• Explain how compensating sign corrections affect the definitions of total particle number and entropy.

Proposed method

• Refer to equation numbering in the BCH paper to trace sign conventions.
• Derive from equation (32) that equations (33) and (34) should have minus signs on the last two integrals.
• Identify missing minus signs in the definitions of total particle number N and total entropy S after equation (20).
• Show that applying the correct signs makes equations (33) and (34) consistent without altering the paper's conclusions.

Experimental results

Research questions

• RQ1 What specific sign discrepancies appear in equations (33) and (34) of the BCH paper?
• RQ2 How do the proposed sign corrections affect the interpretation of the redshifted chemical potential and temperature?
• RQ3 Do the sign corrections change the overall conclusions of BCH regarding black hole thermodynamics?

Key findings

• There are two compensating sign errors in the BCH paper: plus signs in certain integrals should be minus signs.
• The definitions of total particle number N and total entropy S require minus signs to be positive.
• Correcting these signs aligns equations (33) and (34) with the intended thermodynamic interpretation.
• The overall conclusions of the BCH paper remain valid after accounting for the sign corrections.