### 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 [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 two-component spinors [66]) The metric, Eq. (2.8), can be written compactly as
with

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

via the linear combinations

The 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 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. (2.3), (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 ().

• The Weyl tensor:
• The Maxwell tensor:
• The spin coefficients and metric variables:
• The functions of integration are determined, using coordinate conditions, as:
• The mass aspect,
satisfies the physically very important reality condition:
• Finally, from the asymptotic Bianchi identities, we obtain the dynamical (or evolution) relations:

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, , 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 asymptotically-flat solutions was Bondi’s [16] identifications, at , of the interior spacetime four-momentum (energy/momentum). Given the mass aspect, Eq. (2.50),

and the spherical harmonic expansion

Bondi identified the interior mass and three-momentum 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.