  The spherically symmetric null hypersurfaces in a Schwarzschild spacetime are
smooth away from the singularities and foliate the spacetime. We prove the
existence of more general foliations by null hypersurfaces without the
spherical symmetry condition. In fact we also relax the spherical symmetry of
the ambient spacetime and prove a more general result: in a perturbed
Schwarzschild spacetime (not necessary being vacuum), nearly round null
hypersurfaces can be extended regularly to the past null infinity, thus there
exist many foliations by regular null hypersurfaces in the exterior region of a
perturbed Schwarzschild black hole. A significant point of the result is that
the ambient spacetime metric is not required to be differentiable in all
directions.
