Covariant double-null dynamics: $(2+2)$-splitting of the Einstein equations

TL;DR

This work develops a covariant (2+2) embedding of spacetime adapted to a double null foliation, yielding a concise, geometrically transparent reformulation of the Einstein equations. By introducing two-dimensionally covariant extrinsic data (K_{Aab}, ω^a) and the normal derivatives D_A, the authors derive compact tetrad expressions for the Ricci tensor and establish the Bianchi identities in this framework. They also construct a Lagrangian formulation and propose a robust characteristic initial-value problem, detailing how data on intersecting null hypersurfaces uniquely determine a local vacuum solution. A key feature is the rationalized differential operators that preserve covariance and streamline curvature computations, enabling straightforward analysis of null-surface dynamics and horizon-related problems. The formalism, while limited to hypersurface-orthogonal normals, promises broad applicability to characteristic evolution, horizon dynamics, and related gravitational phenomena.

Abstract

The paper develops a $(2+2)$-imbedding formalism adapted to a double foliation of spacetime by a net of two intersecting families of lightlike hypersurfaces. The formalism is two-dimensionally covariant, and leads to simple, geometrically transparent and tractable expressions for the Einstein field equations and the Einstein-Hilbert action, and it should find a variety of applications. It is applied here to elucidate the structure of the characteristic initial-value problem of general relativity.