A spacetime calculus based on a single null direction

Abstract

A space-time calculus based on a single null direction is developed. The fundamental objects are totally symmetric spinors formed from the Ricci rotation coe cients together with four new di erential operators which only depend upon the choice of o A up to a complex scaling. The formalism combines features of the GHP formalism with those of one which is invariant under null rotations. Einstein's equations and the full Bianchi identities are given in this new formalism.

Taming the Nonlinearity of the Einstein Equation

“…Such cases might be better understood by instead adopting a version of the GHP formalism which does not require that the tetrad be normalized [21]. An approach where only one tetrad direction is fixed [27] might also be useful.) Letting χ ≡ Ωð1 þ 1 2 hÞ 1=2 , the directions of l a and k a can then be identified with the first two elements of a background GHP tetrad vial a ¼ χl a andn a ¼ −2ðχhÞ −1 k a .…”

Invariant differential operators and the Karlhede classification of type-N non-vacuum solutions

“…The Weyl spinor in the generalized GHP notation is defined in [6]. We proceed to define the Ricci spinor in generalized GHP notation:…”

“…In a recent paper by Machado Ramos and Vickers [4], the bound on differentiation in the vacuum type-N case was reduced from the Collins bound [5] of six to five. The approach used here consisted in using the generalized GHP formalism [6] to express the components of the Weyl spinor and its derivatives. The generalized GHP notation is very convenient in dealing with Petrov type-N spacetimes since it is invariant under the group of null rotations and vacuum type-N spacetimes have a Weyl spinor which admits null rotations as its invariance group.…”

“…Thus the calculation of all terms relating to ∇ n R is simplified. This result makes it possible to express all quantities in ∇ n R, which are symmetric, in terms of generalized GHP invariants which are also symmetric [6]. All such quantities relating to the first covariant derivative of R are given.…”

Integration using invariant operators: conformally flat radiation metrics

“…In a recent paper [1] Machado Ramos and Vickers introduced some new operators which are invariant under null rotations. In a subsequent paper [2] this was generalized to incorporate spin and boost transformations so that the resulting formalism depends only upon a choice of a single null direction. Not surprisingly this formalism combines many features of the Geroch-Held-Penrose (GHP) [3] and null rotation invariant formalisms.…”