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,
is given [75
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with, from Eqs. (4.30) and (4.31
),
This leads immediately to
or, using the decomposition into real and imaginary parts, 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.
http://www.livingreviews.org/lrr-2012-1 |
Living Rev. Relativity 15, (2012), 1
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