5.4 Asymptotically static and stationary spacetimes

By defining asymptotically static or stationary spacetimes as those asymptotically-flat spacetimes where the asymptotic variables are ‘time’ independent, i.e., uB independent, we can look at our procedure for transforming to the complex center of mass (or complex center of charge). This example, though very special, has the huge advantage in that it can be done exactly, without the use of perturbations [3].

Imposing time independence on the asymptotic Bianchi identities, Eqs. (2.52View Equation) – (2.54View Equation),

˙0 0 0 0 ψ 2 = − ∂ψ3 + σ ψ 4, ψ˙01 = − ∂ψ02 + 2σ0ψ03, ψ˙0 = − ∂ψ0 + 3σ0ψ0 , 0 1 2
and reality condition
-- Ψ ≡ ψ02 + ∂2σ-+ σ˙σ-= Ψ,

we have, using Eqs. (2.47View Equation) and 2.48View Equation) with 0 ˙σ = 0, that

ψ03 = ψ04 = 0, (5.34 ) ∂ ψ0 = 0, (5.35 ) 2 ∂ ψ01 = 3σ0ψ02, (5.36 ) 0 2- -0 -2 -- Ψ ≡ ψ 2 + ∂ σ = ψ2 + ∂ σ = Ψ. (5.37 )

From Eq. (5.37View Equation), we find (after a simple calculation) that the imaginary part of ψ0 2 is determined by the ‘magnetic’ [60Jump To The Next Citation Point] part of the Bondi shear (spin-weight s = 2) and thus must contain harmonics only of l ≥ 2. But from Eq. (5.35View Equation), we find that 0 ψ2 contains only the l = 0 harmonic. From this it follows that the ‘magnetic’ part of the shear must vanish. The remaining part of the shear, i.e., the ‘electric’ part, which by assumption is time independent, can be made to vanish by a supertranslation, via the Sachs theorem:

-- u^B = uB + α (ζ,ζ), (5.38 ) -- -- 2 -- ^σ(ζ,ζ) = σ(ζ, ζ) + ∂ α (ζ,ζ).
In this Bondi frame, (i.e., frame with a vanishing shear), Eq. (5.36View Equation), implies that
ψ0 = ψ0iY 1, (5.39 ) 1 1 √1i- √ -- 0i 6--2G- i 6--2G- i − 1 i ψ1 = − c2 D (grav) = − c2 (D (mass) + ic J ), (5.40 )
using the conventionally accepted physical identification of the complex gravitational dipole. (Since the shear vanishes, this agrees with probably all the various attempted identifications.)

From the mass identification, ψ02 becomes

√ -- ψ0 = − 2--2G-M . (5.41 ) 2 c2 B
Since the Bondi shear is zero, the asymptotically shear-free congruences are determined by the same GCFs as in flat spaces, i.e., we have
L (u ,ζ,ζ¯) = ∂ G (τ,ζ, ¯ζ) = ξa(τ)ˆm (ζ, ¯ζ), (5.42 ) B (τ) a uB = ξa (τ )ˆla(ζ, ¯ζ). (5.43 )

Our procedure for the identification of the complex center of mass, namely setting ψ∗10 = 0 in the transformation, Eq. (4.7View Equation),

ψ ∗0= ψ0 − 3Lψ0 + 3L2 ψ0 − L3 ψ0 1 1 2 3 4

leads, after using Eqs. (5.39View Equation), (5.34View Equation) and (5.42View Equation), to

ψ0 = 3Lψ0 , (5.44 ) 1 √2-- ψ0i= − 6--2G-Di , 1 c2 (grav) Di = M ξi. (grav) B

From the time independence, ξi, the spatial part of the world line is a constant vector. By a (real) spatial Poincaré transformation (from the BMS group), the real part of i ξ can be made to vanish, while by ordinary rotation the imaginary part of i ξ can be made to point in the three-direction. Using the the gauge freedom in the choice of τ we set ξ0(τ) = τ. Then pulling all these items together, we have for the complex world line, the UCF, L(uB, ζ, ¯ζ) and the angular momentum, Ji:

ξa(τ ) = (τ, 0,0,iξ3) (5.45 ) u = X (τ, ζ, ¯ζ) = ξa(τ)ˆl(ζ,ζ¯) ≡ √τ-− iξ3Y 0 , B a 2 2 1,3 ¯ 3 1 L (uB,ζ, ζ) = iξIY 1,3, J i = Si = MBc ξ3δi3 = MBc (0,0,ξ3) = MBc ξiI.
Thus, we have the complex center of mass on the complex world line, a a z = ξ (τ).

These results for the lower multipole moments, i.e., l = 0,1, are identical to those of the Kerr metric presented earlier! The higher moments are still present (appearing in higher r−1 terms in the Weyl tensor) and are not affected by these results.


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