Contents

Table 11.4. Solutions with proper homothety groups Hr on V3, described using the conventions of Table 11.2Hr Metric Tab Homothety Reference(s)H4 (15.39) F t∂t + r∂r Henriksen and Wesson (1978), Bona (1988a)(15.72), A =e 2r ∂r Bogoyavlensky and Moschetti(1982)(15.75), a =0 F r∂r Collins and Lang (1987)(15.81), a =0 F ∑ a xa ∂xa Collins and Lang (1987)(15.81), b =0 F r∂r − D(x∂x + y∂y)(15.82) F ∂z Lorenz (1983c)(15.83), c =0 F t∂t +(D − 1)(y∂y + z∂z) Collins and Lang (1987)(15.84) F 2γt∂t +(3γ − 2)(y∂y + z∂z) Goode (1980)(15.85) F ∂t + ∂x − (y∂y + z∂z) Carot and Sintes (1997)(15.86), N = z F z∂z(15.90) F x n ∂nSee §16.2.2 F t∂t + r∂r Dyer et al. (1987), Havas (1992)(22.17) with c =0 E 4(r∂r + z∂z)+3θ∂θ − t∂t Koutras (1992b)(37.57) F r∂r Collins and Lang (1987)H3 (17.14) V ∂y − ∂t Phan (1993)(18.66), a = −1, N =e z F ∂z Bogoyavlensky and Moschetti(1982)(19.21), A =0,U = α ln ρ + z, V ∂z +2αϕ∂ϕ +2(α − 1)t∂t Godfrey (1972)k = α 2 ln ρ +2αz − ρ 2 /2(19.21), A =0,e U = ρ α (r + z) β , V ρ∂ρ + z∂z + α(2β + α)ϕ∂ϕ Godfrey (1972)e k = ρ α2 (r + z) 2β(α+β) /r 2β2 +(α − 1)(2β + α − 1)t∂t(21.61), µ = p F ∂z Hermann (1983)(21.72) F ∂z − σ[ϕ∂ϕ + t∂t) Hermann (1983)(23.1) F ∂x − 2ncz∂y − ab(y∂y + z∂z) McIntosh (1978a)