A space-time calculus based on pairs of null directions

Abstract

A formalism is presented for the treatment of space-times, which is intermediate between a fully covariant approach and the spin-coefficient method of Newman and Penrose. With the present formalism, a pair of null directions only, rather than an entire null tetrad, is singled out at each point. The concept of a spin- and boost-weighted quantity is defined, the formalism operating entirely with such quantities. This entails the introduction of modified differentiation operators, one of which represents a natural extension of the definition of the operator ð which had been introduced earlier by Newman and Penrose. For suitable problems, the present formalism should lead to considerable simplifications over that achieved by the standard spin-coefficient method.

Quasilocal energy-momentum and energy flux at null infinity

“…Using the obtained results (17)(18)(19)(20) and the property that the spin weight of π is −1, it is straightforward to get…”

“…We will follow the method initiated by Newman, Penrose, and Tod [15,16]. Many additional useful results have been discussed in [16,17,18,19,20].…”

“…Moreover the variable α 0 is an abbreviation for α 0 := 1 2 √ 2 ζ, the spin-weight operator ∂ 0 is defined by ∂ 0 := P √ 2 ∂ ∂ζ + 2sᾱ 0 acting on a variable with spin-weight s [17], and P := 1 + ζζ = sin −2 θ 2 . For calculating the energy flux, we also need the asymptotic expansion for the other NP coefficients.…”

“…This paper is organized as follows: In the next section, we start from a basic review on the asymptotic behavior of space-time near null infinity including a discussion of the expansion of the Newman-Penrose coefficients which is more complete than the well-known ones in [9,14,15,16,17]. Our main result is contained in three parts: In Section III, we review the covariant Hamiltonian formalism and the associated boundary term approach to quasi-local energy momentum and energy flux.…”

Symmetry analysis of radiative spacetimes with a null isotropy using GHP formalism

“…(8) with z = iPQε, where the real zero-weighted parameter ε gives the permitted one degree of isotropy freedom. From the new basis (8), the definitions of the GHP operators and the transformation rules for the spin coefficients (see [21]), the GHP operators transform as…”