Mathematical Journal of Okayama University

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1 Mathematical Journal of Okayama University Volume 42 Issue 2000 Article 6 JANUARY 2000 Certain Metrics on R 4 + Tominosuke Otsuki Copyright c 2000 by the authors. Mathematical Journal of Okayama University is produced by The Berkeley Electronic Press (bepress.

2 Certain Metrics on R 4 + Tominosuke Otsuki Abstract We studied the pseudo-riemannian metric on R 3 R + where γ 2 = Σ 3 b=x b x b a = constant which satisfies the Einstein condition (in [] [2] [3]. The purpose of this work is to find Einstein metrics including the above metric as a special one which is analogous to the relation between the Schwarzschild metric the Kerr one in the theory of relativity.

3 Otsuki: Certain Metrics on Math. J. Okayama Univ. 42 ( CERTAIN METRICS ON R 4 + TOMINOSUKE OTSUKI Abstract. We studied the pseudo-riemannian metric ds 2 = nx 3 δ bc ax bx c o dx x 4 x 4 bc= + ar 2 b dx c dx 4 dx 4 + ax 4 x 4 on R 3 R + where r 2 = P 3 b= x bx b a = constant which satisfies the Einstein condition (in [] [2] [3]. The purpose of this work is to find Einstein metrics including the above metric as a special one which is analogous to the relation between the Schwarzschild metric the Kerr one in the theory of relativity.. Preliminaries Curvature Tensor Using the polar coordinates (r ϑ ϕ of R 3 : x = r sin ϑ cos ϕ x 2 = r sin ϑ sin ϕ x 3 = r cos ϑ setting x 4 = t the above metric can be written as (. ds 2 = { t 2 + ar 2 dr2 + r 2 (dϑ 2 + sin 2 ϑdϕ 2 + at 2 dt2}. Now setting the formal coordinates (u i as u = r u 2 = ϑ u 3 = ϕ u 4 = t we consider a pseudo-riemannian metric (.2 ds 2 = 4 g ijdu i du j g ij = g ji = F ij ij= u 4 u 4 with (.3 F β = F β (u u 2 F λµ = F λµ (u u 2 u 4 F 2 = F λ = 0 where we set β γ = 2 λ µ ν = 3 4 in the paper. Christoffel symbols made by g ij : { j i h } = 2 are given exactly as follows : { β γ } = 2 F ( δβ F + δγ u γ k g ik( g jk + g kh g jh (g ij = (g ij u h u j u k F u β 6 The δ βγ F ββ { β λ γ } = u 4 F 4λ F βγ Produced by The Berkeley Electronic Press 2000

4 Mathematical Journal of Okayama University Vol. 42 [2000] Iss. Art TOMINOSUKE OTSUKI (.4 { β µ } = u 4 δ β δ 4µ { β λ µ } = 2 ν F λν F νµ u β (F ij = (F ij { µ ν } = 2 F F µν λ λ { µ ν } = { µ ν } u Λ (δ u µδ λ 4ν + δν λ δ 4µ F 4λ F µν 4 where λ { 3 3 } Λ = 2 F λ4 F 33 λ { 3 4 } u Λ = 4 2 F λ3 F 33 λ { 4 4 } Λ = 2 F λ4 F 44 + F λ3 F 34 { β γ } becomes exacty as { } = 2 F F u { 2 } = 2 F 22 F { 2 } = 2 F F 2 { 2 2 } = 2 F 22 F 22 { 2 2 } = 2 F F 22 2 { 2 } = u 2 F 22 F 22. u Using these expressions the components of the curvature tensor of the metric g ij : R j i hk = { j i k} u h are given as follows : R β 2 = 2 F { δ { i j h } + u k l { l i h }{ j l k } l { l i k }{ j l h } 2 F + δ 2 F 2 F ββ 2 F } ββ 2 + δ β δ β2 u β u β u u + 4 δ F { F log(f F ββ + F log F 22 u β u β δ β2 γ F γγ F u γ F ββ u γ (.5 4 } F F 44 F β2 u 4 u 4 4 δ 2 F { F log(f F ββ + F log F u β u u u β δ β γ F γγ F u γ F ββ u γ 4 u 4 u 4 F F 44 F β } + 4 δ β2f { log F F ββ + log F F ββ + log F 22 F } 22 u u u 4 δ βf { log F F ββ + log F 22 F ββ + log F F } R β λ 2 = u 4 δ β { (F λ4 F + 2 F νµ F λν F µ4 F νµ 2

5 Otsuki: Certain Metrics on (.6 2 F F λ4 F ββ F ββ + 2 F 22 F λν F µ4 F νµ u νµ (.7 R λ 2 = { F 34( δ 2u 4 (.8 R λ µ 2 = 4 Then (.9 R β νγ = 2u 4 ( δ γ ( where CERTAIN METRICS ON R F λ4 F ββ } + u 4 δ β2 { u (F λ4 F 22 2 F 22F λ4 F ββ F ββ u 2 F λ4 F ββ u } F 3λ δ F 3λ 2 + F 44( δ F 4λ δ F } 4λ 2 u u ( F µν ν u F 4 F ν u β R β λ νγ = 2u 4 ( F λ4 δ 4 ν + ελ4 F 3ν F λ( F ν u β F ββ F ββ u γ F νλ F µν F νλ. u F F βγ 2 u 4 F 44 δ λ ν F λ 2 F ν F λ F ν u β u γ 4 u γ u β + F ν u γ F γγ F γγ u β F 4 F ν F βγ F ν u F F ββ δ βγ = F 33 F 44 F 34 F 34 ε 33 = ε 44 = 0 ε 34 = ε 43 = R λ νγ = δ γ { 4 F ( F u F F λν u + F F 22 F λν (. + F 34( F λ3 + F 3ν + F 44( F λ4 + F 4ν 2u 4 u ν u λ 2u 4 u ν u λ + } F 44 F λν + F F λν u 4 u 4 4 u γ 4 F ρ R λ µ νγ = u γ { λ µ ν } Λ F 2 F λν u γ F ρ F ρν F λ u γ 4 F F γγ F γγ F λν u γ ρ { ρ ν } Λ F µρ F λ u γ 2 ρ F λν { λ ρ ν } Λ F µ F ρ u γ Produced by The Berkeley Electronic Press

6 Mathematical Journal of Okayama University Vol. 42 [2000] Iss. Art TOMINOSUKE OTSUKI ( Last F µ 2 F λ ( F 4µ F λν + F 4µ F λν u ν u γ 2u 4 u γ u γ 2u 4 δ µ ν (.3 R β 34 = 4 F ρ R β λ 34 = 2 ( ρ F 4 F λ u γ + 2u 4 F 4µ F λν u γ. F ρ( F ρ4 F 3 F ρ3 F 4 u β u β F λ F 3 u β 2 F ρ( { ρ λ 3 } Λ F 4 u β F 4( δ3 λ 2u 4 (.5 R λ 34 = 2 F 2 F λ3 2 F ρ F λ 2 F 3 u β λ F 3 { ρ 4 } Λ u β F 4 δ λ F 3 4 u β u β ( Fρ3 { λ ρ 4 } Λ F ρ4 { λ ρ 3 } Λ 2u 4 F ρ F 4ρ( F ρ3 F λ4 F ρ4 F λ3 R λ µ 34 = { λ µ 3} Λ + ({ µ 3 } Λ { λ 4 } Λ { µ 4 } Λ { λ 3 } Λ (.6 u 4 δ µ 3 4 ({ λ 4 } Λ + F 44 F λ4 + δ µ 4 4 ({ λ 3 } Λ + F 44 F λ3 u 4 u 4 u 4 + F µ 4 4 (F λ3 { 4 } Λ F λ4 { 3 } Λ + F F µ u 4 4 ( Fλ3 F 4 F λ4 F

7 Otsuki: Certain Metrics on CERTAIN METRICS ON R Ricci Tensor From (.5 (.6 we compute the components of the Ricci tensor R ij = 4 k= R i k kj of the metric (.2 they are written as followsin dividing three groups. First R βγ = j R β j jγ = R β 2 δ 2γ R β 2 2 δ γ + R β 3 3γ + R β 4 4γ becomes R = 2 F 22( 2 F F + u u 4 F 22 F 22 log(f F 22 u u ( F 22 F log(f F µν µν F µν 2 F µν F µν F µν u u 4 u µν u F µν( F µν F F F µν F 22 F u u + ( F 44 + F 33 6 F 44 F 2u 4 u 4 R 22 = 2 F ( 2 F + 2 F 22 + u u 4 F F log(f F 22 ( F F 22 u log(f F 22 u µν µν F µν 2 F µν F µν F µν 4 u µν 2 F µν( F µν F 22 F 22 F µν F F 22 u u + ( F 44 + F 33 6 F 44 F 22 2u 4 u 4 (2.3 R 2 = F µν 2 F µν F µν F µν 2 u µν 4 u µν 2 u + F µν( F µν log F + F µν log F u u Second µν R λµ = R λ µ + R λ µ = R λ µ + R λ 3 34 δ 4µ R λ 4 34 δ 3µ Produced by The Berkeley Electronic Press

8 Mathematical Journal of Okayama University Vol. 42 [2000] Iss. Art TOMINOSUKE OTSUKI becomes R λµ = β (2.4 log F u β F ββ F λµ u β F F λµ 2 u 4 F 2 F λµ + 4 F 4( F λ + F µ F λµ u µ u λ u ρ F log F F [ λµ + δ 4µ { λ 3 3} Λ + { 3 4 } Λ { λ 3 } Λ u 4 { λ 4 4 } Λ + 4 ( F F ρ F ρµ F λ { 3 3 } Λ { λ 4 } Λ F F λ3 F 3 F 4 F F λ4 F 3 F ] [ 3 δ 3µ { λ 4 3} Λ + 4 { 3 } Λ { λ 4 } Λ 4 { 4 } Λ { λ 3 } Λ + 4 { λ 3 } Λ + ( F F λ3 F 4 F 4 u 4 4 F F λ4 F 4 F ] ( F 44 + F 33 F λµ u 4 u 4 2u 4 2u 4 F λ3f µ3 F ρ F ρ. Third ρ R λ = R λ + R 2 2λ + R 3 34 δ 4λ R 4 34 δ 3λ becomes R λ = F 4 F λ ( u 4 2 F F 33 4δ + ( 2 F F 34 3λ ( ( + F 33 F 4λ + { F ( 33 F 33 F 34 u λ 4 2 F 4λ F 44 + F 34 F 3λ F 44 2F 4λ F 34 ( F 43 F } 33 F 33 F 34. These expressions of R ij are computed under the condition (.3 for F ij here we set further restrictions as (2.6 F 33 = F 34 =

9 Otsuki: Certain Metrics on CERTAIN METRICS ON R In fact we consider F 44 depends on u u 2 u 4 other F ij depend only u u 2. Then some of { j i h } given by (.4 becomes as (2.7 { 3 λ 3 } Λ = { 3 λ 4 } Λ = 0 { 4 λ 4 } Λ = 2 F λ4 F 44. Since we have F 4 F λ = u F 4 F λ = = ( F34 F 3λ ( F 34 F 33 F 3λ F 4λ ( F33 F 4λ + δ 4λ (2.5 is turned into the expression (2.8 R λ = ( F 34 F 33 F 3λ F 4λ + ( ( F 34 F 3λ u 4 2 F 33 F 4λ u 4 δ 4λ F F ( 44 F 34 F 33 3λ F 33 F 34 by (2.6. Now denoting (2.9 Y = (2.8 is written as ( F 34 F 33 F 33 F 34 Z = ( F 34 F 33 F 34 F 44 R 3 = Y + Y + Y u R 4 = Z + Z u 4 2 u 4 Now we suppose that (2.0 R λ = 0. log + 4 F 34 From R λ = 0 we consider the two cases as follows Case I : Y = Y 2 = 0 Case II : (Y Y 2 (0 0. F 44 Y. Case I. Y = Y 2 = 0 implies that F 34 /F 33 = b constant therefore we have = F 33 F 44 F 34 F 34 = F 33 (F 44 b 2 F 33 Z = (F 44 b 2 F 33 F 33 hence Z = log F 33 Produced by The Berkeley Electronic Press

10 Mathematical Journal of Okayama University Vol. 42 [2000] Iss. Art TOMINOSUKE OTSUKI which implies Z / = 0 by (2.6 R 4 = ( Z + log u 4 = ( log F 33 log(f 33 (F 44 b 2 F 33 u 4 = u 4 log(f 44 b 2 F 33. Therefore R 4 = 0 implies that (2. F 44 = b 2 F 33 + φ(u 4 F 34 = bf 33 = φ(u 4 F 33 where φ = φ(u 4 is an integral free function of u 4. Cade II. Since we have R 3 = Y { log(u log(y 2 + } log 4 = Y ( Y 2 log 4 (u 4 4 R 3 = 0 implies that the function Y 2 (u 4 4 = ( F 34 F 33 2 (u 4 4 F 33 F 34 must depend only on u u 2. Therefore by (2.6 we see that (2.2 (u 4 4 = f(u u 2 or F 44 = F 34F 34 F 33 + (u 4 4 f F 33 where f is a function of u u 2. From R 4 = 0 we obtain Z 2 u 4 Z = 2 u 4 log where we set = u 4 f F 34 F 44 Y 2 F 34 ( 4 f ( F 34 F 33 (u 4 5 F 33 F 34 F 33 = f u 4 f (u 4 3( F 34 log F 33 F 34 F 34 F 34 = u 4 A + (u 4 3 B A = log f 2 B = f (F 34 2 ( F34 2 log F

11 Otsuki: Certain Metrics on CERTAIN METRICS ON R they depend only on u u 2. Regarding the above expression as an ordinary differential equation on u 4 we see that Z = 2 A + 2 (u 4 4 B + C (u 4 2 where C is some function of u u 2. Then we have Z = ( F 34 F 33 F 34 F 44 = (u 4 4 ( F 34 F 34 F 34F 34 F 33 f F 33 f F 33 (u 4 4 F 33 = log f + (u 4 4 ( f F F 34 34F 34 F 33 + C (u 4 2 F 34 F 33 which implies By the assumption (2.6 it must be log F 33/f + C (u 4 2 = 0. log F 33 /f = 0 C = 0. Therefore we obtain for Case II (2.3 F 33 = af F 44 = a ( F34 F 34 f + (u 4 4 = F 33 F 44 F 34 F 34 = f (u 4 4 where f = f(u u 2 a is an integral constant. In the following sections we shall investigate the solution g ij = u 4 u 4 F ij which satisfies the Einstein condition : (2.4 R ij = 4 Rg ij R = ij g ij R ij under the conditions (.3 (2.6. Produced by The Berkeley Electronic Press

12 Mathematical Journal of Okayama University Vol. 42 [2000] Iss. Art TOMINOSUKE OTSUKI 3. Solutions for Case I We can express the Einstein condition (2.4 by (3. R ij = λf ij. Since F λ = 0 it must be R λ = 0. Thus we divide the argument into two cases I II. In this section we search for solutions for Case I. Since we have (3.2 F 34 = bf 33 F 44 = b 2 F 33 + φ = F 33 F 44 F 34 F 34 = φf 33 F 33 = F 44 = + b2 F 33 φ F 34 = F 34 = b φ F 44 = φ φ = φ(u 4 we obtain from (2.4 after long computations (3.3 R 33 = ( log(f /F 22 F 33 log(f /F 22 F 33 4 u F u F 22 2 F 33 + F 33 F 33 2 F 4F 33 F ( dφ 2u 4 φ F 33 u 4 u 4 φ that is R 34 = 4 2 ( log(f /F 22 F 34 log(f /F 22 F 34 u F u F 22 ( F33 2 F 34 + F u u 4 b F 44 + F 34 F 34 + φ φ 2u 4 2 F F F 33 F 34 3 u 4 u 4 φ F 34 = b 4 b 2 ( log(f /F 22 F 33 u F F 2 F 33 + ( 2u 4 φ 2 dφ + 3 u 4 u 4 φ {( + b2 F33 F 34 F F 33 φ F 34 F } 44 log(f /F 22 F 33 u F 22 b F 33 F 33 4F 33 F bf 33 = br 33 (3.4 R 34 = br

13 Otsuki: Certain Metrics on CERTAIN METRICS ON R R 44 = 4 2 ( log(f /F 22 F 44 u F F 43 F 2 F b F 44 + F 44 F } 44 φ φ 3 u 4 u 4 φ F 44 = b 2 R 33 3 dφ 2u 4 φ log(f /F 22 F 44 u F 22 {( + b F43 F 43 F F 33 φ F 44 (3 + F 34 F 34 2u 4 φ 3 u 4 u 4 that is (3.5 R 44 = b 2 R 33 3 dφ 3. 2u 4 φ u 4 u 4 Regarding R β we obtain from (2.(2.2 (2.3 R = 2 F 33 ( 2 F + 2 F 22 + { ( F F 2F 33 u u 2F 22 u u 4F 22 F (3.6 + F F 22 + ( F F 22 + F 22 F } 22 u u F 22 u u + { F 33 F 33 + F F 33 F F } 33 4F 33 F 33 u u F u u F 22 ( 2u 4 φ 2 dφ + 3 u 4 u 4 φ F R 22 = 2 F 33 ( 2 F + 2 F 22 + { ( F F 2F 33 2F u u 4F F (3.7 + F F 22 + ( F F 22 + F 22 F } 22 u u F 22 u u + { F 33 F 33 + F 22 F 33 F 22 F } 33 4F 33 F 33 F 22 F u u ( 2u 4 φ 2 dφ + 3 u 4 u 4 φ F 22 R 2 = 2F 33 2 F 33 u + 4F 33 { F 33 F 33 u F 33 + F F F 33 u (3.8 + F 22 F } 33 = 2 log F 33 + { log F 33 log F 33 F 22 u 2 u 4 u Produced by The Berkeley Electronic Press 2000

14 Mathematical Journal of Okayama University Vol. 42 [2000] Iss. Art TOMINOSUKE OTSUKI + log F log F 33 + log F 22 log F } 33. u u Now we suppose (3. holds for F ij satisfying (3.2 (3.8. (3.5 we have R 44 = λb 2 F 33 3 dφ 3 = λf 44 λφ 3 dφ 3 2u 4 φ u 4 u 4 2u 4 φ u 4 u 4 which implies (3.9 λ = 3 dφ 2u 4 φ 2 3 u 4 u 4 φ. From (3.3(3.9 R 33 = λf 33 we obtain 2 F 33 (3.0 ( log(f /F 22 F 33 F 2 u F u From log(f /F 22 F 33 F 33 F 33 2 dφ F 22 2F 33 F u 4 φ 2 F 33 = 0. R 34 = λf 34 R 44 = λf 44 are satisfied automatically from the above arguments. From R 2 = 0 we obtain 2 F 33 (3. ( F 33 F 33 + F F 33 + F 22 F 33 = 0. u 2 F 33 u F u F 22 u Finally from (3.6(3.7(3.9 R = λf we obtain 2 F 33 + ( 2 F + 2 F 22 { ( F F F 33 u u F 22 u u 2F 22 F (3.2 + F F 22 + ( F F 22 + F 22 F } 22 u u F 22 u u { F 33 F 33 + F F 33 F F } 33 2F 33 F 33 u u F u u F 22 2 u 4 φ 2 dφ F = 0 2 F 33 + ( 2 F + 2 F 22 { ( F F F 33 F u u 2F F (3.3 + F F 22 + ( F F 22 + F 22 F } 22 u u F 22 u u { F 33 F 33 + F 22 F 33 F 22 F } 33 2F 33 F 33 F 22 F u u 2 u 4 φ 2 dφ F 22 =

15 Otsuki: Certain Metrics on CERTAIN METRICS ON R Using the formula : 2 Φ = 2 log Φ + log Φ Φ u β u β log Φ u β the equalities (3.0 (3.3 can be written as follows { 2 log F 33 + ( log F33 log F 33 log(f /F 22 log F } 33 F u u 2 u u u u (3.0 + { 2 log F 33 + ( log F33 log F 33 log(f 22/F log F } 33 F dφ u 4 φ 2 = 0 (3. 2 log F 33 + ( log F33 log F 33 log F 22 log F 33 u 2 u u log F log F 33 = 0 u { 2 log F log F 22 + ( log F33 log F 33 F u u u u 2 u u (3.2 + log F 22 log F 22 log F log F 22 log F log F } 33 u u u u u u + F 22 { 2 log F + 2 ( log F log F log F log F 22 + log F log F 33 } 2 u 4 φ 2 dφ F 22 { 2 log F log F + 2 = 0 ( log F33 log F 33 (3.3 + log F log F log F log F 22 log F 22 log F 33 } + F { 2 log F 22 u u log F 22 log F 33 u supposing F 0 F 22 0 F ( log F22 log F 22 log F log F 22 u u u u } 2 dφ u u 4 φ 2 = 0 If we can find F (u u 2 F 22 (u u 2 F 33 (u u 2 φ = φ(u 4 satisfying (3.0 (3.3 we obtain the metric g ij = u 4 u 4 F ij satisfying the Produced by The Berkeley Electronic Press

16 Mathematical Journal of Okayama University Vol. 42 [2000] Iss. Art TOMINOSUKE OTSUKI Einstein condition (3.. In the following we show that we can obtain such solutions under the additional restriction : F (3.4 = F 22 = 0 F 33 = ψ(u sin 2 u 2. Then (3.0 is reduced to { d 2 log ψ (3.5 F du 2 + ( d log ψ d log ψ d log(f /F 22 2 du du du 2 2 dφ F 22 u 4 φ 2 = 0 (3. is reduced to ( d log ψ (3.6 d log F 22 cos u2 = 0. du du sin u 2 (3.2 (3.3 are reduced respectively to { d 2 log ψ F du 2 + d2 log F 22 du ( d log ψ du d log ψ du d log ψ du } (3.7 + d log F 22 d log F 22 d log F d log F 22 d log F d log ψ } du du du du du du 2 dφ u 4 φ 2 = 0 { d 2 log ψ (3.8 F du 2 + ( d log F22 d log F 22 d log F d log F 22 2 du du du du + d log F 22 d log ψ } 2 2 dφ du du F 22 u 4 φ 2 = 0. Then we see firstly from (3.6 that (3.9 ψ(u = cf 22 (u c = constant. Using this relation we obtain from (3.5(3.7 (3.8 respectively (3.5 { d 2 log F 22 F du 2 + ( d log F22 d log F 22 d log(f /F 22 d log F } 22 2 du du du du (3.7 2 F 22 2 u 4 φ 2 dφ = 0 { 2 d2 log F 22 F du 2 + d log F 22 d log F 22 d log F d log F } 22 du du du du 4

17 Otsuki: Certain Metrics on CERTAIN METRICS ON R (3.8 2 u 4 φ 2 dφ = 0 { d 2 log F 22 F du 2 + d log F 22 d log F 22 d log F d log F } 22 du du 2 du du (3.7 (3.5 becomes 2 F 22 2 u 4 φ 2 dφ = 0. { d 2 log F 22 F du 2 d log F d log F } = 0 2 du du F 22 (3.7 (3.5 2 becomes (3.20 F ( d log F22 du F u 4 φ 2 dφ = 0. Thus we see that to solve the system of differential equations (3.5 (3.7 (3.8 on F F 22 φ they can be replaced by (3.7 d 2 F ( 22 df 22 2 df df 22 F 22 du F = 0 F 22 du 2F F 22 du du F 22 ( (3.20 df 22 2 F F dφ F 22 du F 22 u 4 φ 2 = 0 (3.8 Now setting d 2 F 22 df df 22 F 22 du 2 2 F dφ 2F 2F F 22 du du F 22 u 4 φ 2 = 0. q = q(u 4 := 2 u 4 φ 2 dφ y = F 22 z = df 22 du the above expressions become respectively ( z 2 F z y z y 2 F y + 2F y = 0 ( z 2 F + 4 y y + qf = 0 y z F z 2F y 2F y F q = 0. From the third one of the above equations we obtain (3.2 y = z z = qf y + F 2F z + 2F Produced by The Berkeley Electronic Press

18 Mathematical Journal of Okayama University Vol. 42 [2000] Iss. Art TOMINOSUKE OTSUKI the second first ones become (3.22 z 2 = F (qy 2 + 4y (3.23 z = z2 y + F z 2F 2F respectively. Now let y z satisfy (3.2 then we obtain d du {z 2 F (qy 2 + 4y} = 2zz F (qy 2 + 4y F (2qyz + 4z = 2z(qF y + F 2F z + 2F F (qy 2 + 4y 2F (qyz + 2z = F df du {z 2 F (qy 2 + 4y} which implies the equation z 2 F (qy 2 + 4y = F constant hence we see that if y z satisfy (3.22 at some value of u then it holds for all its near values. We see easily that (3.23 the second expression of (3.2 imply (3.22. Therefore it is sufficient to treat only (3.22 for our purpose. From (3.2 (3.22 q does not depend on u 4 hence we obtain φ = c 0 c = constants q = 4c. c 0 + c u 4 u 4 then (3.22 can be written as (3.24 z 2 = 4F y( c y. Hence it must be F y( c y 0 therefore dy du = ±2 F y( c y or dy εy( c y = ±2 εf du where ε = for F > 0 ε = for F < 0. We denote indefinite integrals of the left right h sides by dy εf (3.25 Φ(y = h(u = (u du εy( c y we obtain Φ(y = ±2(h(u + c 2 c 2 = constant. Denoting the inverse function of Φ by Ψ we obtain the following theorem. 6

19 Otsuki: Certain Metrics on CERTAIN METRICS ON R Theorem. The system (3.9 (3.2 satisfying the Einstein condition is given by a solution of (3.26 F = F (u F 22 = Ψ(±2(h(u + c 2 F 33 = cf 22 sin 2 u 2 F 34 = bf 33 F 44 = b 2 F 33 + c 0 + c u 4 u 4 where b c c 0 c c 2 are constants F is any smooth function of u. Example. Let b = 0 c = c 0 = c = 0 F = then ε = dy Φ(y = = 2 y h(u = du = u 2 y = ±2(u + c 2 y hence y = (u + c 2 2. Putting c 2 = 0 so y = F 22 = u 2 F 33 = u 2 sin2 u 2 F 34 = 0 F 44 =. Therefore we obtain the metric (. with a = 0 : ds 2 = u 4 u 4 (du 2 + u 2 du u 2 sin 2 u 2 du Example 2. Let b = 0 c = c 0 = c = F = +u u then ε = Φ(y = h(u = du +u u dy y( + y = log(2y y( + y (with Φ(0 = 0 = log(u + + u u (with h(0 = 0 log(2y y( + y = 2 log(u + + u u (with c 2 = 0 from which we obtain 2y y( + y = 2u u + + 2u + u u 2y + 2 y( + y = 2u u + 2u + u u hence we obtain y = F 22 = u 2 F 33 = u 2 sin 2 u 2 F 34 = 0 F 44 =. + u 4 u 4 Therefore we obtain the metric (. with a = : ds 2 = ( du 2 + u 2 du u 2 sin 2 u 2 du u 4 u 4 + u u + u 4 u 4 Analogously if we set b = 0 c = c 0 = c = F = u u we obtain the metric (. with a =. Produced by The Berkeley Electronic Press

20 Mathematical Journal of Okayama University Vol. 42 [2000] Iss. Art TOMINOSUKE OTSUKI 4. Analysis a Conclusion for Case II For Case II described in Section 2 we have (2.3 we set F 34 = fh then we obtain F 33 = af F 34 = fh F 44 = ( fh 2 + a (u 4 4 (4. F 33 = F 44 = ( (u 4 4 h 2 + F 34 = F 34 a f = (u 4 4 h F 44 = F 33 = a(u 4 4 = F 33 F 44 F 34 F 34 = f (u 4 4 where a 0 constant f h are functions of u u 2. Using (4. for (2.3 we obtain (4.2 R 2 = 2 f + f f 2f u 4f 2 + log F f u 4f u + log F 22 f 4f u 2 (u 4 4 f h h. u If we consider the Einstein condition (3. it must be R 2 = 0 hence from (4.2 we have f h h = 0. u Since 0it must be f 0 h is not constant for Case II therefore we have (i h u 0 h = 0 or h h (ii = 0 0. u In the following we consider the case (i. By (4. (2.7 we obtain from (2.4 (4.3 R 33 = a 4 F log(f /F 22 f a u u 4 F 22 log(f /F 22 f a F ( 2 f f f + a 2 2f 2 (u 4 4 F f 2 h h au 4 u 4 F 33. u u Next analogously from (2.4 we obtain R 34 = 4 F log(f /F 22 ( h f + f h u u u 4 F 22 log(f /F 22 h f (4.4 2 F {( 2 f f f h + f u u 2f u u 2 h + 3 f h } u u 2 u u 8

21 Otsuki: Certain Metrics on CERTAIN METRICS ON R F 22( 2 f f f h + 2f 2 (u 4 4 F f 2 h h h u u Finally from (2.4 we obtain au 4 u 4 F 34. R 44 = 4a F log(f /F 22 ( h 2 f + 2fh h u u u 4a F 22 log(f /F 22 h 2 f 2a F { h 2( 2 f f f u u 2f u u (4.5 +3h f h ( 2 h + 2fh + u u u u 2h h h } u u 2a F 22 h 2( 2 f f f 2f + 2a (u 4 4 F f 2 h 2 h h + 4 au 4 u 4 F 44. u u u 4 u 4 Now considering the Einstein condition (3. we put (4.6 λ = R 33 /F 33 = R 33 /(af = 4 F log(f /F 22 log f u u 4 F 22 log(f /F 22 F ( 2 f log f 2 f 2f 2 f f + 2 (u 4 4 F f h u h u au 4 u 4 we obtain from (4.3 (4.4 (4.7 R 34 λf 34 = R 34 h a R 33 = 4 F log(f /F 22 f h u u 2 F ( 2 h f + 3 u u 2 Since F 0 R 34 λf 34 = 0 is equivalent to log(f /F 22 (4.8 f h 2f u u from which we obtain the relation 2 h 3 f h = 0 u u u u F /F 22 = f 3( h u 2ρ(u2 by integration where ρ is a function of u 2. f h. u u Produced by The Berkeley Electronic Press

22 Mathematical Journal of Okayama University Vol. 42 [2000] Iss. Art TOMINOSUKE OTSUKI Next we obtain from (4.3 (4.5 the equality : ( fh 2 + (u 4 4 = R 44 R 44 λf 44 = R 44 λ a ( a 2 f R 33 fh 2 + (u 4 4 = R 44 h2 a 2 R 33 a 2 f(u 4 4 R 33 (4.9 = 2a F { log(f /F 22 fh h + 3h f h + 2fh 2 h u u u u u u + 2f h u h u } + 4 u 4 u 4 + a(u 4 4 { + 4 F 22 log(f /F 22 log f F log(f /F 22 u F ( f log f u 2 f 2f 2 f f } which cannot vanish as a function of u u 2 u 4. Hence we see that there exist no solutions for the case (i. For the case (ii : h h u = 0 (i we obtain the equalities: 0 analogously to the case (4.3 a 4 F 22 log(f /F 22 R 33 = a 4 F log(f /F 22 u f a 2 F ( f u + a 2 (u 4 4 F 22 f 2 h h au 4 u 4 F 33 R 34 = 4 F log(f /F 22 h f u u (4.4 4 F 22 log(f /F 22 ( h f + f h f f h 2f u u 2 F 22{( 2 f 2f 2 h +f + 3 f h } 2 2 f f f 2f 2 F ( 2 f u u f f h + 2 (u 4 4 F 22 f 2 h h h au 4 u 4 F

23 Otsuki: Certain Metrics on CERTAIN METRICS ON R R 44 = 4a F log(f /F 22 h 2 f u u 4a F 22 log(f /F 22 ( h 2 f + 2fh h 2a F h 2( 2 f u u (4.5 f f 2f u u 2a F 22{ h 2( 2 f f f 2f (4.6 Now we put +3h f h ( 2 h + 2fh + 2h h h } + 2a (u 4 4 F 22 f 2 h 2 h h + 4 u 4 u 4 au 4 u 4 F 44. λ = R 33 /F 33 = R 33 /(af = 4 F log(f /F 22 log f u u 4 F 22 log(f /F 22 log f (u 4 4 F 22 f h h au 4 u 4 F ( 2 f f f f 2f 2 we obtain from (4.3 (4.4 (4.5 the equations (4.7 R 34 λf 34 = 4 F 22 log(f /F 22 f h 2 F 22( 2 h f + 3 f h 2 R 44 λf 44 = 2a F 22{ log(f /F 22 fh h + 3h f h (4.9 +2fh 2 h + 2f h h } + 4 u 4 u 4 Produced by The Berkeley Electronic Press

24 Mathematical Journal of Okayama University Vol. 42 [2000] Iss. Art TOMINOSUKE OTSUKI { a(u F log(f /F 22 u F ( f 2 f 2f 2 f f } log f + u 4 F 22 log(f /F 22 log f which cannot vanish as a function of u u 2 u 4.Hence we see that there exist no solutions for the case (ii. Thus we obtain the following conclusion. Theorem 2. We cannot find solutions g ij satisfying the Einstein condition for Case II: F 34 /F 33 constant where F β = F β (u u 2 F 2 = 0 F λ = 0 β = 2; λ = 3 4 F 3λ = F 3λ (u u 2 F 44 = F 44 (u u 2 u 4. References [] T. OTSUKI: Killing vector fields of a spacetime SUT Journal of Math. 35 ( [2] T. OTSUKI: On a 4-space with certain general connection related with a Minkowski-type metric on R 4 + Math. J. Okayama Univ. 40 ( [2000]. [3] T. OTSUKI: Killing vector fields of a spacetime on R 4 + Yokohama Math. J. 49 ( TOMINOSUKE OTSUKI Kaminomiya Tsurumi-ku Yokohama Japan address: totsuki@iris.ocn.ne.jp (Received February