The NP version [39, 44
, 41] of the vacuum Einstein (or the Einstein–Maxwell) equations uses the
tetrad components
The complex spin coefficients, which play the role of the connection, are determined from the Ricci
rotation coefficients [39, 44
]:
The third basic variable in the NP formalism is the Weyl tensor or, equivalently, the following five complex tetrad components of the Weyl tensor:
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 threeRemark: 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 [39, 45]. With no
explanation of the integration process, except to mention that we use the Bondi coordinate and
tetrad system of Equations (7), (9
), and (11
) and asymptotic flatness, 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: 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: Using the mass aspect, , with Equations (51
) and (52
), the first of the asymptotic
Bianchi identities 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
, Equations (49
), (51
) and (52
) and its derivatives, all the asymptotic variables can now be determined
from Equations (56
) – (60
).
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 [18] and the peeling theorem [57]. The
peeling theorem is essentially given by the asymptotic behavior of the Weyl tensor in Equation (40
) (and
Equation (41
)). The Goldberg–Sachs theorem essentially states that for an algebraically-special metric, the
degenerate principle null vector field is the tangent field to a shear-free NGC. Both theorems are implicitly
used later.
One of the immediate physical interpretations arising from the asymptotically-flat solutions was
Bondi’s [8] identifications, at , of the interior spacetime four-momentum (energy/momentum). Given
the mass aspect, Equation (54
),
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and the spherical harmonic expansion
Bondi identified the interior mass and three-momentum with the The evolution of these quantities, (the Bondi mass/momentum loss) is then determined from
Equation (62). 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 In our case, where we consider only quadrupole gravitational radiation, the quadratic correction terms
do in fact vanish and hence Equation (66), modified by the Maxwell terms, is correct as it is
stated.
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