2.4 The Newman–Penrose formalism

Though the NP formalism is the basic working tool for our analysis, this is not the appropriate venue for its detailed exposition. Instead we will simply give an outline of the basic ideas followed by the results found, from the application of the NP equations, to the problem of asymptotically-flat spacetimes.

The NP version [55Jump To The Next Citation Point, 60Jump To The Next Citation Point, 57] of the vacuum Einstein (or the Einstein–Maxwell) equations uses the tetrad components

a a a a a λi = (l,n ,m , ¯m ), (2.27 )
(i = 1,2,3,4) rather than the metric, as the basic variable. (An alternate version, not discussed here, is to use a pair of two-component spinors [66Jump To The Next Citation Point]) The metric, Eq. (2.8View Equation), can be written compactly as
ab ij a b g = η λiλj, (2.28 )
with
( ) 0 1 0 0 || || ij || 1 0 0 0 || η = | | . (2.29 ) |( 0 0 0 − 1 |) 0 0 − 1 0

The complex spin coefficients, which play the role of the connection, are determined from the Ricci rotation coefficients [55Jump To The Next Citation Point, 60Jump To The Next Citation Point]:

i b a i b i γjk = λ jλk∇a λb ≡ λjλb;k, (2.30 )
via the linear combinations
1 α = 2(γ124 − γ344), λ = − γ244, κ = γ131, β = 1(γ − γ ), μ = − γ , ρ = γ , 2 123 343 243 134 (2.31 ) γ = 1(γ122 − γ342), ν = − γ242, σ = γ133, 2 šœ€ = 12(γ121 − γ341), π = − γ241, τ = γ132.

The third basic variable in the NP formalism is the Weyl tensor or, equivalently, the following five complex tetrad components of the Weyl tensor:

ψ0 = − Cabcdlamblcmd, ψ1 = − Cabcdlanblcmd, (2.32 )
( ) ψ2 = − 1- Cabcdlanblcnd − Cabcdlanbmcmd- , (2.33 ) 2
ψ = C lanbncmd, ψ = C nambncmd. (2.34 ) 3 abcd 4 abcd
Note that we have adopted the sign conventions of [55Jump To The Next Citation Point], which differ from those in [66Jump To The Next Citation Point].

When an electromagnetic field is present, we must include the complex tetrad components of the Maxwell field into the equations:

Ļ•0 = Fablamb, (2.35 ) 1 ( -- ) Ļ•1 = -Fab lanb + mamb , 2 a--b Ļ•2 = Fabn m ,
as well as the Ricci (or stress tensor) constructed from the three Ļ• i, e.g., T lalb = Ļ• Ļ•- ab 0 0, with −4 Rab = kTab,k = 2Gc.

Remark 2. We mention that much of the physical content and interpretations in the present work comes from the study of the lowest spherical harmonic coefficients in the leading terms of the far-field expansions of the Weyl and Maxwell tensors.

The NP version of the vacuum (or Einstein–Maxwell) equations consists of three sets (or four sets) of nonlinear first-order coupled partial differential equations for the variables: the tetrad components, the spin coefficients, the Weyl tensor (and Maxwell field when present). Though there is no hope that they can be solved in any general sense, many exact solutions have been found from them. Of far more importance, large classes of asymptotic solutions and perturbation solutions can be found. Our interest lies in the asymptotic behavior of the asymptotically-flat solutions. Though there are some subtle issues, integration in this class is not difficult [55, 61]. With no explanation of the integration process, except to mention that we use the Bondi coordinate and tetrad system of Eqs. (2.3View Equation), (2.5View Equation), and (2.7View Equation) and asymptotic flatness (− 5 ψ0 ∼ O(r ) and certain uniform smoothness conditions on sideways derivatives), we simply give the final results.

First, the radial behavior is described. The quantities with a zero superscript, e.g., 0 σ, 0 ψ2, …, are ‘functions of integration’, i.e., functions only of (-- uB, ζ,ζ).

Remark 3. These last five equations, the first of which contains the beautiful Bondi energy-momentum loss theorem, play the fundamental role in the dynamics of our physical quantities.

Remark 4. Using the mass aspect, Ψ, with Eqs. (2.47View Equation) and (2.48View Equation), the first of the asymptotic Bianchi identities, Eq. (2.52View Equation), can be rewritten in the concise form,

-- -0 Ė™Ψ = Ė™σĖ™σ + kĻ•02Ļ•2. (2.58 )

From these results, the characteristic initial problem can roughly be stated in the following manner. At uB = uB0 we choose the initial values for (0 0 0 ψ0,ψ1,ψ 2), i.e., functions only of (-- ζ, ζ). The characteristic data, the complex Bondi shear, -- σ0 (uB,ζ,ζ), is then freely chosen. Since ψ03 and ψ04 are functions of σ0, Eqs. (2.45View Equation), (2.47View Equation) and (2.48View Equation) and its derivatives, all the asymptotic variables can now be determined from Eqs. (2.52View Equation) – (2.56View Equation).

An important consequence of the NP formalism is that it allows simple proofs for many geometric theorems. Two important examples are the Goldberg–Sachs theorem [29Jump To The Next Citation Point] and the peeling theorem [73]. The peeling theorem is essentially given by the asymptotic behavior of the Weyl tensor in Eq. (2.36View Equation) (and Eq. (2.37View Equation)). The Goldberg–Sachs theorem is discussed in some detail in Section 2.6. Both theorems are implicitly used later.

One of the immediate physical interpretations arising from the asymptotically-flat solutions was Bondi’s [16] identifications, at + ℑ, of the interior spacetime four-momentum (energy/momentum). Given the mass aspect, Eq. (2.50View Equation),

-- -0 Ψ = ψ02 + ∂2σ0 + σ0 Ė™σ ,

and the spherical harmonic expansion

0 i 0 ij 0 Ψ = Ψ + Ψ Y1i + Ψ Y 2ij + ..., (2.59 )
Bondi identified the interior mass and three-momentum with the l = 0 and l = 1 harmonic contributions;
c2 MB = − -√----Ψ0, (2.60 ) 2 2G
i c3 i P = − ---Ψ . (2.61 ) 6G

The evolution of these quantities, (the Bondi mass/momentum loss) is then determined from Eq. (2.58View Equation). The details of this will be discussed in Section 5.

The same clear cut asymptotic physical identification for interior angular momentum is not as readily available. In vacuum linear theory, the angular momentum is often taken to be

√ -- k --2c3 0k J = − 12G Im (ψ 1 ). (2.62 )
However, in the nonlinear treatment, correction terms quadratic in 0 σ and its derivatives are often included [75Jump To The Next Citation Point]. In the presence of a Maxwell field, this is again modified by the addition of electromagnetic multipole terms [42Jump To The Next Citation Point, 4Jump To The Next Citation Point].

In our case, where we consider only quadrupole gravitational radiation, the quadratic correction terms do in fact vanish and hence Eq. (2.62View Equation), modified by the Maxwell terms, is correct as it is stated.


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