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An apparent horizon is the outer boundary of the future-trapped region on a chosen spatial slice, under the usual regularity conditions. Its location depends on the slice; it is not the globally defined event horizon. At a regular boundary the outward null expansion vanishes while the inward null expansion is negative. Its area cannot be substituted into every version of the Penrose inequality without the appropriate area and initial-data hypotheses.

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.

The physical argument for the Penrose inequality combines weak cosmic censorship conjecture, the dominant energy condition, and relaxation to a stationary black hole. Work in geometrized units. Let $M_f$ and $A_f$ be the final Kerr black hole mass and horizon area. Positive energy radiated to infinity gives $E_{\mathrm{ADM}}\ge M_f$, where $E_{\mathrm{ADM}}$ is the initial ADM energy. For a Kerr black hole with $a_f=J_f/M_f$,If the initial apparent horizon obeys the necessary apparent-horizon area comparison with the enclosing event horizon, and Hawking's area theorem applies during the evolution, thenConsequently the anticipated answer, under those additional hypotheses, isThe bound is saturated by a nonrotating Schwarzschild black hole with no energy loss. Rotation or outgoing radiation makes the argument's inequalities stricter.

There is an essential qualification: inclusion inside an event horizon does not by itself compare areas. An arbitrary apparent horizon on general, non-time-symmetric initial data need not satisfy the displayed apparent-horizon area comparison; the unqualified version with its area is not universally true, even with the dominant energy condition. On time-symmetric data the relevant outermost minimal surface is an outer area-minimizing surface, as used in the Riemannian Penrose inequality, with nonnegative scalar curvature. In more general formulations an appropriate enclosing-area quantity is needed. The physical expectation is conditional on this area comparison, as well as on censorship, predictability, settling, and the energy assumptions; the mere presence of a trapped surface does not supply every step.

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.