Bounds on the Bondi Energy by a Flux of Curvature

We consider smooth null cones in a vacuum spacetime that extend to future null infinity. For such cones that are perturbations of shear-free outgoing null cones in Schwarzschild spacetimes, we prove bounds for the Bondi energy, momentum, and rate of energy loss. The bounds depend on the closeness between the given cone and a corresponding cone in a Schwarzschild spacetime, measured purely in terms of the differences between certain weighted $L^2$-norms of the space-time curvature on the cones, and of the geometries of the spheres from which they emanate. A key step in this paper is the construction of a family of asymptotically round cuts of our cone, relative to which the Bondi energy is measured.