Penrose inequality ($E\ge \sqrt{A/(16\pi )}$)

The Penrose inequality compares asymptotic energy and an appropriate horizon or enclosing area in geometrized units. Its physical motivation uses censorship, positive radiated energy, settling to a Kerr black hole, and Hawking's area theorem. The area variable requires care: the area of an arbitrary apparent horizon on non-time-symmetric data does not give a universally valid inequality. The Riemannian Penrose inequality uses the relevant outermost minimal surface under nonnegative scalar curvature.

For complete asymptotically flat three-dimensional Riemannian initial data with nonnegative scalar curvature and an outermost compact minimal surface boundary of area $A$, the Riemannian Penrose inequality gives the displayed bound. Equality is realized by the spatial exterior of Schwarzschild spacetime. Time-symmetric gravitational initial data have this form when the Einstein field equations and energy hypothesis give the required nonnegative curvature. This theorem does not justify substituting arbitrary non-time-symmetric apparent horizon area into the same formula.

The heuristic derivation of the Penrose inequality from Hawking's area theorem needs an initial area comparison, in addition to enclosure of the apparent horizon by the event horizon. Set inclusion alone is not an area inequality. An outer area-minimizing surface or appropriate enclosing-area hypotheses supply the needed comparison in formulations where it is valid; arbitrary slice-dependent apparent-horizon area need not do so.