Null Surfaces and the Bach Equations

TL;DR

This paper demonstrates that the integrability conditions in the Null Surface Formulation of GR lead to a field equation equivalent to the vanishing of the Bach tensor, with explicit second-order perturbation expansion and implications for asymptotically flat space-times.

Contribution

It establishes a direct link between null surface integrability conditions and the Bach tensor, providing explicit equations and exploring their generalizations in GR.

Findings

The field equation is equivalent to the vanishing of the Bach tensor.

Explicit second-order perturbation expansion of the field equation.

The equation determines null surfaces in asymptotically flat, radiative space-times.

Abstract

It is shown that the integrability conditions that arise in the Null Surface Formulation (NSF) of general relativity (GR) impose a field equation on the local null surfaces which is equivalent to the vanishing of the Bach tensor. This field equation is written explicitly to second order in a perturbation expansion. The field equation is further simplified if asymptotic flatness is imposed on the underlying space-time. The resulting equation determines the global null surfaces of asymptotically flat, radiative space-times. It is also shown that the source term of this equation is constructed from the free Bondi data at future null infinity. Possible generalizations of this field equation are analyzed. In particular we include other field equations for surfaces that have already appeared in the literature which coincide with ours at a linear level. We find that the other equations do not yield null surfaces for GR.