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414 26 The various classes of algebraically special solutionsthe method of constructing the (non-twisting) solutions with this propertyis very similar to that for the Robinson–Trautman class outlined inChapter 27 (but, in virtue of σ ̸= 0, the calculations using the Newman–Penrose formalism are more lengthy).All vacuum metrics where the null congruence with κ =0,ω=0,σ̸=0, Θ ̸= 0 is a principal null congruence (Ψ 0 = 0) were obtained byNewman and Tamburino (1962); for details see also Carmeli (1977),p. 244. Metrics with σ ̸= 0,ρ= 0, are forbidden by the vacuum field equations,and solutions with σ ̸= 0,ω̸= 0, are possible only if ρρ = σσ (Untiand Torrence 1966). The twisting case has not been completely solved.There are two classes of Newman–Tamburino solutions:ρ 2 ̸=σσ ̸= 0(spherical class, x 1 +ix 2 = x+iy = ζ √ 2,x 3 = r, x 4 = u)g 11 = 2(2ζζ)3/2(r + a) 2 , g22 = 2(2ζζ)3/2(r − a) 2 , g34 = −1,g 12 = g 14 = g 24 = g 44 =0, R 2 = r 2 − a 2 , A = bu + c,( r − ag 13 =4A 2 (2ζζ) 3/2 xR 4+ r − 2a2a 2 R 2 − L2a 3 ), L = 1 2 ln r + ar − a , (26.21)( r + ag 23 =4A 2 (2ζζ) 3/2 yR 4 + r +2a2a 2 R 2 − L )2a 3 , a = A(2ζζ) 1/2 ,g 33 = 4A2 r 2 (2ζζ) 3/2R 4 − 4Ar3 (ζ 2 + ζ 2 )R 4 + 2r2 (2ζζ) 1/2R 2 − 2rLA ;ρ 2 = σσ (cylindrical class)ds 2 =2ω 1 ω 2 − 2ω 3 ω 4 , ω 2 = ω 1 , ω 3 =du,[ω 4 =dr − 2b 2 cn 2 (bx)+ c + b2 ln[r 2 ]cn(bx)]2cn 2 du,(bx)(26.22)ω 1 = r dx/2+4Y du + icn(bx)(dy +8buY dx +2b ln r du) / √ 2,Y = ± b ( 1 − cn 4 (bx) ) 1/42 √ ,2cn(bx)where b and c are real constants and cn(bx) is an elliptic function ofmodulus k =1/ √ 2. The metricds 2 = r 2 dx 2 +x 2 dy 2 − 4r[ ]x du dx−2du dr +x−2 c + ln(r 2 x 4 ) du 2 (26.23)can be obtained from (26.22) by a limiting procedure.