Bounds on the Bondi Energy by a Flux of Curvature

TL;DR

This paper establishes bounds on the Bondi energy, momentum, and energy loss for null cones in vacuum spacetimes, relating these bounds to curvature differences and geometric properties of the cones compared to Schwarzschild spacetime.

Contribution

It provides new bounds on Bondi energy and flux for null cones near Schwarzschild spacetime using curvature and geometric differences.

Findings

Bounds depend on weighted L^2-norm differences of curvature.

Constructs asymptotically round cuts for measuring Bondi energy.

Quantifies energy loss in terms of curvature perturbations.

Abstract

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.