We begin with the Schwarzschild spacetime, treating the Schwarzschild mass, , as a zeroth-order 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
In linear theory, the complex (mass) dipole moment,, 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
with, from Eqs. (4.30) and (4.31),
This leads immediately to
Identifying [75, 53] the (intrinsic) angular momentum, either from the conventional linear identification or from the Kerr metric, as
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.
Living Rev. Relativity 15, (2012), 1
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