The Riemannian Penrose Inequality with Charge for Multiple Black Holes
We present the outline of a proof of the Riemannian Penrose in- equality with charge r ≤ m + m 2 − q 2 ,w hereA =4 πr 2 is the area of the outermost apparent horizon with possibly multiple connected components, m is the total ADM mass, and q the total charge of a strongly asymptotically flat initial data set for the Einstein-Maxwell equations, satisfying the charged dominant energy condition, with no charged matter outside the horizon. In a seminal paper (18), R. Penrose examined the validity of the cosmic censorship conjecture, and outlined a heuristic argument which shows how using also Hawking's area theorem, (10), implies a related inequality. In (17), he gen- eralized this heuristic argument leading to an inequality now referred to as the Penrose inequality. Consider a strongly asymptotically flat (SAF) Cauchy surface in a spacetime satisfying the dominant energy condition (DEC), with ADM mass m containing an event horizon of area A =4 πr 2 , which undergoes gravitational collapse and settles to a Kerr solution. Since the ADM mass m∞ of the final state is no greater than m, and since the area radius r∞ is no less than r, and since for the final state we must have m∞ ≥ 1 r∞ in order to avoid naked singularities, we must have had m ≥ 1 r also at the beginning of the evolution. The event horizon is indiscernible in the original slice without knowing the full evolution. However, one may replace the event horizon by the outermost minimal area enclosure of the apparent horizon, the boundary of the region admitting trapped surfaces, and obtain the same inequality. A counterexample to the Penrose inequality would therefore have suggested data which leads under the Einstein evolution to naked singularities, while a proof of the inequality could be viewed as evidence in support of cosmic censorship. The inequality further simplifies in the time-symmetric case, where the apparent horizon coincides with the outermost minimal area enclosure. The domi- nant energy condition reduces now to non-negative scalar curvature of the Cauchy hypersurface, leading to the Riemannian version of the inequality: the ADM mass and the area radius of the outermost compact minimal surface in a SAF 3-manifold