The NP version [55, 60, 57] of the vacuum Einstein (or the Einstein–Maxwell) equations uses the tetrad components
rather than the metric, as the basic variable. (An alternate version, not discussed here, is to use a pair of twocomponent spinors [66]) The metric, Eq. (2.8), can be written compactly as withThe complex spin coefficients, which play the role of the connection, are determined from the Ricci rotation coefficients [55, 60]:
via the linear combinationsThe third basic variable in the NP formalism is the Weyl tensor or, equivalently, the following five complex tetrad components of the Weyl tensor:
Note that we have adopted the sign conventions of [55], which differ from those in [66].When an electromagnetic field is present, we must include the complex tetrad components of the Maxwell field into the equations:
as well as the Ricci (or stress tensor) constructed from the three , e.g., , with .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 farfield 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 firstorder 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 asymptoticallyflat 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. (2.3), (2.5), and (2.7) and asymptotic flatness ( 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., , , …, are ‘functions of integration’, i.e., functions only of ().
Remark 3. These last five equations, the first of which contains the beautiful Bondi energymomentum loss theorem, play the fundamental role in the dynamics of our physical quantities.
Remark 4. Using the mass aspect, , with Eqs. (2.47) and (2.48), the first of the asymptotic Bianchi identities, Eq. (2.52), can be rewritten in the concise form,
From these results, the characteristic initial problem can roughly be stated in the following manner. At we choose the initial values for (, i.e., functions only of (). The characteristic data, the complex Bondi shear, , is then freely chosen. Since and are functions of , Eqs. (2.45), (2.47) and (2.48) and its derivatives, all the asymptotic variables can now be determined from Eqs. (2.52) – (2.56).
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 [29] and the peeling theorem [73]. The peeling theorem is essentially given by the asymptotic behavior of the Weyl tensor in Eq. (2.36) (and Eq. (2.37)). 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 asymptoticallyflat solutions was Bondi’s [16] identifications, at , of the interior spacetime fourmomentum (energy/momentum). Given the mass aspect, Eq. (2.50),

and the spherical harmonic expansion
Bondi identified the interior mass and threemomentum with the and harmonic contributions;The evolution of these quantities, (the Bondi mass/momentum loss) is then determined from Eq. (2.58). 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
However, in the nonlinear treatment, correction terms quadratic in and its derivatives are often included [75]. In the presence of a Maxwell field, this is again modified by the addition of electromagnetic multipole terms [42, 4].In our case, where we consider only quadrupole gravitational radiation, the quadratic correction terms do in fact vanish and hence Eq. (2.62), modified by the Maxwell terms, is correct as it is stated.
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Living Rev. Relativity 15, (2012), 1
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