The Penrose inequality in general relativity is a conjectured inequality between the area of the horizon and the mass of a black-hole spacetime. The null Penrose inequality is the case where it concerns the area of the horizon and the Bondi mass at null infinity along a null hypersurface. An effective method to prove Penrose-type inequalities is to exploit the monotonicity of Hawking mass along certain foliations. The constant mass aspect function foliation is such a desired foliation, but the behavior of the foliation at past null infinity is an obstacle for the proof. An idea to overcome this difficulty is to vary the null hypersurface to achieve the desired behavior of the foliation at null infinity, leading to a spacetime version of the Penrose inequality. To formalise this idea, one need to study perturbations of null hypersurfaces.

I will describe the analytic method to represent a null hypersurface using a double null coordinate system, and present the system of equations, which consists of propagating and elliptic equations, to study the geometry of constant mass aspect function foliation. Then I will talk about my work on perturbations of null hypersurfaces in spacetimes close to Schwarzschild spacetime, through perturbation of solutions of the above mentioned system of equations, and its application to the null Penrose inequality.