3.3 Approximations

Due to the difficulties involved in the intrinsic nonlinearities and the virtual impossibility of exactly inverting arbitrary analytic functions, it often becomes necessary to resort to approximations. The basic approximation will be to consider the complex world line a ξ (τ) as being close to the straight line, a a ξ0(τ) = τδ0; deviations from this will be considered as first order. We retain terms up to second order, i.e., quadratic terms. Another frequently used approximation is to terminate spherical harmonic expansions after the l = 2 terms.

It is worthwhile to discuss some of the issues related to these approximations. One important issue is how to use the gauge freedom, Eq. (3.8View Equation), ∗ τ → τ = F (τ), to simplify a ξ (τ ) and the ‘velocity vector’,

a va(τ) = ξa′(τ) ≡ d-ξ-. (3.39 ) dτ

A Notational issue: Given a complex analytic function (or vector) of the complex variable τ, say G (τ), then G(τ ) can be decomposed uniquely into two parts,

G(τ) = 𝔊R (τ ) + i𝔊I (τ),

where all the coefficients in the Taylor series for 𝔊R (τ) and 𝔊I (τ) are real. With but a slight extension of conventional notation we refer to them as real analytic functions.

With this notation, we also write

ξa(τ ) = ξaR (τ) + iξaI(τ) a a a v (τ ) = vR (τ) + ivI(τ).
By using the reparametrization of the world line, via τ∗ = F (τ ), we choose F (τ) = ξ0(τ), so that (dropping the ∗) we have
ξ0(τ) = ξ0 (τ) = τ, ξ0(τ) = 0 R I v0(τ) = v0R (τ) = 1, v0I(τ) = 0

Finally, from the reality condition on the uB, Eqs. (3.23View Equation), (3.26View Equation) and (3.25View Equation) yield, with τ = s + iλ and λ treated as small,

(R) ξb(s)ˆlb uB = ξaR(s)ˆla + vaI(s)ˆla-Ic′----, (3.40 ) ξR (s )ˆlc -- ξb(s)ˆl λ = Λ (s, ζ,ζ) = − -I---b-, (3.41 ) ξcR′(s)ˆlc √2-i 0 = ---2√ξI(s)Y1i--. 1 − --2ξi′R(s)Y10i 2
Within this slow motion approximation scheme, we have from Eqs. (3.40View Equation) and (3.41View Equation),
√ -- u(rRe)t = 2u(BR)= s − √1-ξiR(s)Y10i + 2vaI(s)ˆlaξbI(s)ˆlb, (3.42 ) ( 2 ) √ -- √ -- λ ≈ --2ξiI(s)Y10i 1 − --2vjR (s)Y 01j , (3.43 ) 2 2
or, to first order, which is all that is needed,
√ -- --2- i 0 λ = 2 ξI(s)Y 1i.

We then have, to linear order,

√2-- τ = s + i---ξiI(s)Y01i, (3.44 ) 2 u(R)= s − √1-ξi(s)Y 0. ret 2 R 1i

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