On the geometry of null cones to infinity under curvature flux bounds

Abstract

Abstract. The main objective of this paper is to control the geometry of a future outgoing truncated null cone extending smoothly toward infinity in an Einstein-vacuum spacetime. In particular, we wish to do this under minimal regularity assumptions, namely, at the (weighted) L 2 -curvature level. We show that if the curvature flux and the data on an initial sphere of the cone are sufficiently close to the corresponding values in a standard Minkowski or Schwarzschild null cone, then we can obtain quantitative bounds on the geometry of the entire infinite cone. The same bounds also imply the existence of limits at infinity, along the null cone, of naturally scaled geometric quantities. In [1], we will apply these results in order to control various physical quantitiese.g., the Bondi energy and (linear and angular) momenta-associated with such infinite null cones in vacuum spacetimes.

Bounds on the Bondi energy by a flux of curvature

“…These weighted curvature fluxes appear in [2,5,7]. Moreover, this is the main motivation for our choice of weights in [1] and in this paper.…”

“…In particular, we impose no conditions on the existence or the structure of a null infinity on the spacetime. This paper relies heavily on [1], which proved, in the aforementioned setting, that the instrinsic and extrinsic geometry of N is controlled uniformly up to infinity. Moreover, several techniques and estimates in this paper depend on [18], which developed many of the tools needed for working with N at the L 2 -curvature level.…”

“…see Section 2.1.3 and Definition 2.6. 6 χ is defined more precisely in (1.8) and in [1]. 7 Note that L on N is uniquely determined by its values on the initial sphere S.…”

“…To accomplish this, we note an additional degree of freedom in our setup: the results from [1] outlined above hold for any geodesic foliation of N for which the deviation from N S is sufficiently small. In particular, any other geodesic foliation of N that is "sufficiently close" to the current one would satisfy the small deviation condition.…”

“…Remark. In Theorem 1.1 and in [1], we elected to work with shear-free null hypersurfaces in Schwarzschild spacetimes primarily because the values of the connection and curvature quantities on these hypersurfaces are explicitly given and well-known. It is possible that analogous results can be proved for null cones near Kerr spacetimes, or for some other null hypersurfaces near Schwarzschild spacetimes.…”