4.4 Summary of Real Structures

To put the ideas of this section into perspective we collect the claims.

Example: the (charged) Kerr metric

Considering the Kerr or the charged Kerr metrics (or even more generally any asymptotically flat stationary metric), we have immediately that the Bondi shear σ0 vanishes and hence the associated ℋ-space is complex Minkowski space (cf. [3Jump To The Next Citation Point, 6Jump To The Next Citation Point] and Appendix F). From the stationarity and a real origin shift and rotation, the complex world line can be put into the form
ξa (τ ) = (τ,0, 0,ia ), (4.41 )
with a being the Kerr parameter. The complex cut function is then
a ˆ ¯ uB = ξ (τ)la(ζ,ζ) (4.42 ) = √τ--− iaY 0 (ζ, ζ), 2 2 1,3 -- √ -- -- Y0 (ζ,ζ) = − 21-−-ζζ-, 1,3 1 + ζζ
so that the angle fields of 3.33View Equation-3.34View Equation are
√ -- ζ- L = 2ia------, 1 + ζζ -- √ -- --ζ---- L = − 2ia -, 1 + ζζ ^L = √2ia --ζ---. 1 + ζζ

Using τ = s + iλ in Eq. (4.42View Equation), the reality condition (R) uB = uB on the cut function is that

√ -- 2 0 -- λ = Λ(s,ζ, ¯ζ) = ---aY 1,3(ζ,ζ), 2

so that on the τ-ribbon, λ ranges between ±√ -- 2 and the real slices from the ribbon becomes simply √ -- uB = s∕ 2.

Though we are certainly not making the claim that one can in reality ‘observe’ these complex world lines that arise from (asymptotically) shear-free congruences, we nevertheless claim that they can be observed in a different sense. In the next two sections our goal will be to show that, just as a complex center of charge world line in 𝕄 ℂ can be selected, so too can a complex center of mass world line be singled out in ℋ-space. As we will see, some surprising physical identifications arise from this program, and it is in this sense which the footprints of these complex world lines can be observed.


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