We begin with the Schwarzschild spacetime, treating the Schwarzschild mass, , as a zerothorder quantity, and integrate the linearized Bianchi identities for the linear Weyl tensor corrections. Though we could go on and find the linearized connection and metric, we stop just with the Weyl tensor. The radial behavior is given by the peeling theorem, so that we can start with the linearized asymptotic Bianchi identities, Eqs. (2.52) – (2.54).
Our main variables for the investigation are the asymptotic Weyl tensor components and the Bondi shear, , with their related differential equations, i.e., the asymptotic Bianchi identities, Eq. (2.52), (2.53) and (2.51). Assuming the gravitational radiation is weak, we treat and as small. Keeping only linear terms in the Bianchi identities, the equations for and (the mass aspect) become
The is small (first order), while the has the zerothorder Schwarzschild mass plus firstorder termsIn linear theory, the complex (mass) dipole moment,
is given [75], on a particular Bondi cut with a Bondi tetrad (up to dimensional constants), by the harmonic components of , i.e., from the in the expansion For a different cut and different tetrad, one needs the transformation law to the new and new . Under the tetrad transformation (a null rotation around ) to the asymptotically shearfree vector field, , Eq. (3.80),

with, from Eqs. (4.30) and (4.31),
the linearized transformation is given by [12] The extraction of the part of should, in principle, be taken on the new cut given by with constant . However, because of the linearization, the extraction can be taken on the constant cuts. Following the same line of reasoning that led to the definition of center of charge, we demand the vanishing of the part of .This leads immediately to
or, using the decomposition into real and imaginary parts, and ,Identifying [75, 53] the (intrinsic) angular momentum, either from the conventional linear identification or from the Kerr metric, as
and the mass dipole as we have By inserting Eq. (5.19) into Eq. (5.1), taking, respectively, the real and imaginary parts, using Eq. (5.7) and the reality of , we find the kinematic expression of linear momentum and the conservation of angular momentum.Finally, from the parts of Eq. (5.14), we have, at this approximation, that the mass and linear momentum remain constant, i.e., and . Thus, we obtain the trivial equations of motion for the center of mass,
The linearization off Schwarzschild, with our identifications, lead to a stationary spinning spacetime object with the standard classical mechanics kinematic and dynamic description. It was the linearization that let to such simplifications, and in Section 6, when nonlinear terms are included (in similar calculations), much more interesting and surprising physical results are found.
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Living Rev. Relativity 15, (2012), 1
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