Someone found a few 'plus and minus' math errors in one of Stephen Hawking’s big papers on black holes. Even the GOATs mess up sometimes.
March 27, 2026
Original Paper
Sign Errors in "The Four Laws of Black Hole Mechanics"
arXiv · 2603.25171
The Takeaway
The 1973 paper 'The Four Laws of Black Hole Mechanics' is a pillar of modern physics, yet it contained two minus-sign errors that went unnoticed for 50 years. Fortunately, the errors canceled each other out, meaning the famous conclusions remain correct despite the basic arithmetic mistakes made by the legends who wrote it.
From the 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 ho