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, ξa0(τ) = τδa0;UpdateJump To The Next Update Information 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, Equation (79View Equation), τ ⇒ τ∗ = Φ (τ), to simplify the ‘velocity vector’,

a va(τ) = ξa′(τ) ≡ d-ξ-. (104 ) 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) = 𝔊 (τ ) + i𝔊 (τ), R I

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

a a a v (τ ) = vR (τ) + ivI(τ ).

By using the reparametrization of the world line, via τ∗ = Φ(τ ), with the decomposition

Φ (τ ) = ΦR (τ ) + iΦI(τ ),

the ‘velocity’ transforms as

v∗a(τ ∗) = va (τ )[Φ (τ)′]−1.

One can easily check that by the appropriate choice of ΦI (τ) we can make ∗a ∗ vR (τ ) and ∗a ∗ vI (τ ) orthogonal, i.e.,

ηabv∗Rav∗Ib= 0, (105 )
and by the choice of ΦR (τ), the ∗a ∗ vR (τ ) can be normalized to one,
ηabv∗Rav∗Rb = 1. (106 )

The remaining freedom in the choice of Φ(τ) is simply an additive complex constant, which is used shortly for further simplification.

We now write va(τ) = (v0(τ ),vi (τ)) R R R, which, with the slow motion approximation, yields, from the normalization,

0 ∘ --------- 1 i2 v R(τ) = 1 + (viR)2 ≈ 1 + --vR + .... (107 ) 2
UpdateJump To The Next Update Information From the orthogonality, we have
v0(τ) ≈ vi(τ )vi(τ), I R I

i.e., 0 v I(τ ) is second order. Since 0 0 ′ vI(τ) = ξI(τ) is second order, 0 ξI(τ) is a constant plus a second-order term. Using the remaining complex constant freedom in Φ (τ ), the constant can be set to zero:

ξ0(τ ) = second order. (108 ) I

Finally, from the reality condition on the uB, Equations (94View Equation), (97View Equation) and (96View Equation) yield, with τ = s + iλ and λ treated as small,

(R) a ˆ a ˆ -ξIb(s)ˆlb uB = ξR(s)la + vI(s)laξc′(s )ˆl , (109 ) R c -- ξbI(s)ˆlb- λ = Λ (s, ζ,ζ) = − c′ ˆ , (110 ) √- ξR(s)lc 22ξiI(s)Y10i = ----√2--i′-----0. 1 − -2-ξR(s)Y1i
Within this slow motion approximation scheme, we have from Equations (109View Equation) and (110View Equation),
(R) √ --(R) -1--i 0 a ˆ b ˆ uret = 2uB = s − √2-ξR(s)Y1i + 2vI(s)laξI(s)lb, (111 ) √ -- ( √ -- ) --2-i 0 --2-j 0 λ ≈ 2 ξI(s)Y1i 1 − 2 vR (s)Y 1j , (112 )
or, to first order, which is all that is needed,
√2-- λ = ----ξIi(s)Y 01i. 2

We then have, to linear order,

√ -- --2-i 0 τ = s + i 2 ξI(s)Y1i, (113 ) (R) 1 i 0 uret = s − √--ξR(s)Y 1i. 2

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