### 5.1 Linearized off Schwarzschild

As a first example, we describe how the shear-free NGCs are applied in linear perturbations off the Schwarzschild metric. The ideas used here are intended to clarify the more complicated issues in the full nonlinear asymptotic theory. We will see that these linear perturbations greatly resemble our results from the previous section on the determination of the intrinsic center of charge in Maxwell theory, when there were small deviations from the Coulomb field.

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, Equations (56) – (58).

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. (56), (57) and (55). 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 zeroth-order Schwarzschild mass plus first-order terms

In linear theory, the complex (mass) dipole moment,

is given [59], 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 shear-free vector field, , Equation (149),

with, from Equations (192) and (193),

the linearized transformation is given by [6]
The extraction of the part of should, in principle, be taken on the new cut given by the real obtained from with constant in the expression, Equation (98), . 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 .

or, using the decomposition into real and imaginary parts, and ,

Identifying [5937] 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 Equation (230) into Equation (212), taking, respectively, the real and imaginary parts, using Equation (218) and the reality of , we find
the kinematic expression of linear momentum and
the conservation of angular momentum.

Finally, from the parts of Equation (213), 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,

It was the linearization that let to such simplifications. In Section 6, when nonlinear terms are included (in similar calculations), much more interesting and surprising physical results are found.