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 Section 3.4 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, MSch ≡ MB, 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. (2.52View Equation) – (2.54View Equation).

Our main variables for the investigation are the asymptotic Weyl tensor components and the Bondi shear, σ0, with their related differential equations, i.e., the asymptotic Bianchi identities, Eq. (2.52View Equation), (2.53View Equation) and (2.51View Equation). Assuming the gravitational radiation is weak, we treat σ0 and σ˙0 as small. Keeping only linear terms in the Bianchi identities, the equations for ψ0 1 and Ψ (the mass aspect) become

˙0 3--0 ψ1 + ∂Ψ = ∂ σ , (5.1 ) Ψ˙ = 0, (5.2 ) -- Ψ = Ψ, -- (5.3 ) Ψ ≡ ψ02 + ∂2 σ0. (5.4 )
The 0 ψ1 is small (first order), while the
0 i 0 ij 0 Ψ = Ψ + Ψ Y 1i + Ψ Y2ij + ... (5.5 )
has the zeroth-order Schwarzschild mass plus first-order terms
√ -- 2 2G Ψ0 = − ---2--MSch + δΨ0, (5.6 ) c Ψi = − 6G-P i. (5.7 ) c3

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

Di = Di + ic−1J i (5.8 ) (grav) (mass)
is given [75Jump To The Next Citation Point], on a particular Bondi cut with a Bondi tetrad (up to dimensional constants), by the l = 1 harmonic components of ψ01, i.e., from the ψ0i1 in the expansion
0 0i 1 0ij 1 ψ1 = ψ1 Y1i + ψ1 Y 2ij + ... (5.9 )
For a different cut and different tetrad, one needs the transformation law to the new ψ ∗0 1 and new ψ ∗0i 1. Under the tetrad transformation (a null rotation around a n) to the asymptotically shear-free vector field, ∗a l, Eq. (3.80View Equation),
a ∗a a ¯L- a L- a −2 l → l = l − r m − rm¯ + O (r ),

with, from Eqs. (4.30View Equation) and (4.31View Equation),

a ˆ -- ij 0 -- uB = ξ (τ)la(ζ,ζ) + ξ (τ)Y2ij(ζ,ζ) + ... (5.10 ) 1---0 1-i 0 -- ij 0 -- = √2-ξ (τ) − 2ξ (τ)Y1i(ζ, ζ) + ξ (τ)Y2ij(ζ,ζ) + ... (5.11 ) -- i 1 -- ij 1 -- L (uB,ζ,ζ) = ξ (τ)Y1i(ζ,ζ) − 6ξ (τ)Y2ij(ζ,ζ) + ... (5.12 )
the linearized transformation is given by [12Jump To The Next Citation Point]
0∗ 0 ψ1 = ψ1 − 3L Ψ. (5.13 )
The extraction of the l = 1 part of ψ01∗ should, in principle, be taken on the new cut given by -- -- uB = ξa(τ)ˆla(ζ,ζ ) + ξij(τ)Y0 (ζ,ζ) + ... 2ij with constant τ. However, because of the linearization, the extraction can be taken on the uB constant cuts. Following the same line of reasoning that led to the definition of center of charge, we demand the vanishing of the l = 1 part of ψ01∗.

This leads immediately to

ψ01|l=1 = 3L Ψ|l=1, (5.14 )
or, using the decomposition into real and imaginary parts, ψ0i= ψ0i + iψ0i 1 1R 1I and ξi(uret) = ξi(uret) + iξi(uret) R I,
√ -- 6 2G ψ01iR = − ---2--MSch ξiR(uret), (5.15 ) √c-- 0i 6--2G- i ψ 1I = − c2 MSch ξI(uret). (5.16 )

Identifying [75, 53Jump To The Next Citation Point] the (intrinsic) angular momentum, either from the conventional linear identification or from the Kerr metric, as

J i = Si = MSchc ξiI (5.17 )
and the mass dipole as
i i D (mass) = MSch ξR, (5.18 )
we have
√ -- √ -- 0i 6--2G- i 6--2G- i −1 i ψ 1 = − c2 D (grav) = − c2 (D (mass) + ic J ). (5.19 )
By inserting Eq. (5.19View Equation) into Eq. (5.1View Equation), taking, respectively, the real and imaginary parts, using Eq. (5.7View Equation) and the reality of Ψ, we find
i i′ i P = MSchξR ≡ MSchv R, (5.20 )
the kinematic expression of linear momentum and
J i′ = 0, (5.21 )
the conservation of angular momentum.

Finally, from the l = (0,1 ) parts of Eq. (5.14View Equation), we have, at this approximation, that the mass and linear momentum remain constant, i.e., M = M = M Sch B and δΨ0 = 0. Thus, we obtain the trivial equations of motion for the center of mass,

i′′ MSch ξR = 0. (5.22 )

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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