[论文解读] Sign Errors in "The Four Laws of Black Hole Mechanics"

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

Relativity and Gravitational Theory被引用 0

一句话总结

该论文在1973年BCH公式中识别出两个互相抵消的符号错误，并表明更正后保持原结论不变。它解释了对方程(33)-(34)的影响以及N和S的定义。

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.

研究动机与目标

• 澄清在BCH推导黑洞力学第一定律时使用的符号约定。
• pinpoint 在能量-动量变化和第一定律的推导中定位符号错误所在。
• 解释符号修正对总粒子数N和总熵S定义的影响。

提出的方法

• 参阅BCH论文中的方程编号以追踪符号约定。
• 从方程(32)推导出方程(33)和(34)在最后两个积分上应带负号。
• 在方程(20)之后的总粒子数N和总熵S的定义中识别缺失的负号。
• 证明应用正确的符号使方程(33)和(34)在不改变论文结论的前提下保持一致。

实验结果

研究问题

• RQ1 BCH论文的方程(33)和(34)中出现了哪些具体的符号不一致？
• RQ2 提出的符号修正如何影响对红移化学势和温度的解释？
• RQ3 符号修正是否改变BCH关于黑洞热力学的总体结论？

主要发现

• 在BCH论文中存在两处互相抵消的符号错误：某些积分中的正号应为负号。
• 总粒子数N和总熵S的定义需要负号以确保正值。
• 更正这些符号后，方程(33)和(34)与所期望的热力学解释保持一致。
• 在考虑符号修正后，BCH论文的总体结论仍然成立。