Paper
Angular Momentum Penrose Inequality
Authors
Da Xu
Abstract
We prove the Angular Momentum Penrose Inequality, a fundamental result connecting the total mass of an isolated gravitational system to the size and spin of any black holes it contains. The inequality states that the mass of a spacetime must be at least as large as a specific combination of the black hole's horizon area and angular momentum, with the bound being tight precisely for the Kerr family of rotating black holes.
The proof combines four techniques: solving a geometric equation that straightens out the initial data, applying a conformal transformation that encodes angular momentum, tracking how angular momentum is preserved through the construction, and invoking known bounds that prevent black holes from spinning too fast. A central innovation is a new notion of mass that incorporates both the standard Hawking mass and angular momentum, and which increases monotonically along a natural geometric flow from the black hole horizon out to infinity.
As an application, we also prove the Charged Penrose Inequality for non-rotating charged black holes, showing that electric charge contributes to the mass bound in a manner analogous to angular momentum.