Introduction to dual-null dynamics  pp. 69-78

Introduction to dual-null dynamics

By S. A Hayward

Image View Previous Chapter Next Chapter

Abstract. Much of physics concerns temporal dynamics, which describes a spatial world (or Cauchy surface) evolving in time. In Relativity, the causal structure suggests that null dynamics is more relevant. This article sketches Lagrangian and Hamiltonian formalisms for dual-null dynamics, which describes the evolution of initial data prescribed on two intersecting null surfaces. The application to the Einstein gravitational field yields variables with recognisable geometrical meaning, initial data which divide naturally into gravitational and coordinate parts, and evolution equations which are covariant on the intersection surface and free of constraints.


The ADM or “3+1” formalism [1,2] is a natural approach to the Cauchy problem in General Relativity, and has been used widely both analytically and numerically. By comparison, null (or characteristic) evolution problems are more appropriate to the study of problems involving radiation, whether gravitational or otherwise, since radiation propagates in null directions. Null surfaces also have a central place in the causal structure of General Relativity which spatial surfaces do not.

A distinction should be drawn between the null-cone problem discussed elsewhere in this volume, in which the initial surface is a null cone, and the dual-null problem, in which there are two intersecting null initial surfaces. The latter problem was originally described by Sachs [3], with existence and uniqueness proofs being given by Müller zum Hagen and Seifert [4], Friedrich [5] and Rendall [6], and a general “2+2” formalism being developed by d'Inverno, Smallwood and Stachel [7–9].