4.3 Real cuts from the complex good cuts, II

The construction of real structures from the complex structures, i.e., finding the complex values of τ that yield real values of uB and the associated real cuts, is virtually identical to the flat-space construction of Section 3. The only difference is that we start with the GCF
-- a ˆ ¯ -- uB = G (τ, ζ,ζ) = ξ (τ)la(ζ,ζ ) + Gl≥2 (τ, ζ,ζ) (200 )
rather than the flat-space
-- a ˆ ¯ uB = G (τ,ζ,ζ) = ξ (τ)la(ζ,ζ).

Using τ = s + iλ, Equation (200View Equation) is written

u = G (s,λ, ζ,ζ) + iG (s,λ,ζ, ζ). (201 ) B R I
UpdateJump To The Next Update Information The reality of uB, i.e.,
-- GI (s,λ,ζ,ζ ) = 0, (202 )
again leads to
-- λ = Λ(s,ζ,ζ ) (203 )
and the real slicing,
(R) ( -- -) uB = GR s,Λ (s,ζ,ζ),ζ,ζ . (204 )

Using Equation (191View Equation) and expanding to first order in λ, we have the expressions:

√ -- √ -- 2 0 2 0 ′ 1 i i ′ 0 -- uB = ----ξR(s) − ---ξI(s)λ − -[ξR(s) − ξI(s) λ]Y1i(ζ,ζ ) (205 ) 2 ij ij2 ′ 0 2- + [ξR (s) − ξI (s)λ ]Y 2ij(ζ,ζ) [ √-- √-- ] -- +i -2-ξ0(s) + -2-ξ0(s)′λ − i[ξi(s) + ξi(s)′λ]Y 0(ζ,ζ) 2 I 2 R 2 I R 1i ij ij ′ 0 -- +i [ξI (s) + ξR (s)λ]Y2ij(ζ, ζ), (R) -- -- u B = G√R-(s,Λ (s,ζ,√ζ),ζ,ζ) (206 ) --2- 0 --2-0 ′ 1- i i ′ 0 -- = 2 ξR(s) − 2 ξI(s)λ − 2[ξR(s) − ξI(s) λ]Y1i(ζ,ζ ) ij ij ′ 0 -- + [ξR (s) − ξI (s)λ√ ]Y-2ij(ζ,ζ), -- -- -- 2ξ0(s) + ξi(s)Y 0(ζ,ζ) + 2ξij(s)Y 0 (ζ,ζ) λ = Λ (s,ζ,ζ) = − -√---0I------iI----1i0---------Iij----2ij0-------. (207 ) { 2ξR(s)′ − ξR (s)′Y1i(ζ,ζ) + 2ξR(s)′Y 2ij(ζ,ζ)}
Since in all applications -- Λ (s,ζ,ζ) is multiplied by a first-order quantity, only the first-order expression for -- Λ (s,ζ,ζ) is needed. Using the slow motion assumption, we have,
√ -- √ -- λ = --2-ξi(s)Y 0(ζ,ζ) − 2ξij(s)Y 0 (ζ, ζ). (208 ) 2 I 1i I 2ij
Substituting this into Equation (206View Equation), via the slow motion assumption, leads to the real cut function -- -- uB = GR (s,Λ(s,ζ,ζ ),ζ, ζ):
√ -- √2-- -- √ -- -- u(rRe)t = 2u(BR)= s − ---ξiR(s)Y 01i(ζ,ζ) + 2ξijR(s)Y20ij(ζ, ζ) (209 ) 2 + 1ξivi + 24-vijξij+ 12(ξivki− viξki)Y 0+ 1ξkviY 0 − vijξkjY 0 . 3 I I 5 I I 5 I I I I 1k 6 I I 2ij I I 2ik

Later the linear versions are extensively used:

√ -- √2-- -- √ -- -- u (Rret)= 2u (RB) = s − ---ξiR (s)Y01i(ζ,ζ) + 2ξiRj(s )Y 02ij(ζ,ζ) (210 ) ( √ -- 2 ) 2 i 0 -- √ --ij 0 -- τ = s + i ---ξI(s)Y1i(ζ,ζ) − 2ξI (s)Y2ij(ζ, ζ) . (211 ) 2


  Go to previous page Go up Go to next page