Null Penrose inequality in a perturbed Schwarzschild spacetime

In this paper, we review the proof of the null Penrose inequality in a perturbed Schwarzschild spacetime. The null Penrose inequality conjectures that, on an incoming null hypersurface, the Hawking mass of the outmost marginally trapped surface is not greater than the Bondi mass at past null infinity. An approach to prove the null Penrose inequality is to construct a foliation on the null hypersurface starting from the marginally trapped surface to past null infinity, on which the Hawking mass is monotonically nondecreasing. However to achieve a proof, there arises an obstacle on the asymptotic geometry of the foliation at past null infinity. In order to overcome this obstacle, Christodoulou and Sauter proposed a strategy by varying the hypersurface to search for another null hypersurface where asymptotic geometry of the foliation becomes round. This strategy leads us to study the perturbation of null hypersurfaces systematically. Applying the perturbation theory of null hypersurfaces in a perturbed Schwarzschild spacetime, we carry out the strategy of Christodoulou and Sauter successfully. We find a one-parameter family of null hypersurfaces on which the null Penrose inequality holds. This paper gives a overview of our proof.