Journal of Siberian Federal University. Mathematics & Physics 2014, 7(4), 472–488 УДК 514.822 Solving Yang-Mills Equations for 4-metrics of Petrov Types II, N, III Leonid N. Krivonosov∗ Nizhny Novgorod State Technical University Minina, 24, Nizhny Novgorod, 603950 Russia Vyacheslav A. Lukyanov† Nizhny Novgorod State Technical University, Zavolzhsky branch, Pavlovskogo, 1a, Zavolzhye, Nizhegorodskaya region, 606520, Russia Received 10.06.2014, received in revised form 02.07.2014, accepted 30.08.2014 We have produced 4 series of 4-metrics satisfying Yang-Mills equations for each of types II, N, III. Keywords: Einstein equations, Yang-Mills equations, manifold with conformal connection with torsion and without torsion. Introduction j According to A. Z. Petrov’s algebraic classiﬁcation, Weyl tensor Cimn of conformal curvature of the square-law diﬀerential form of 4 variables is subdivided into three types I, II, III and three j subtypes D, O, N (see [1, 2]). In case of Einstein metrics its curvature tensor Rimn is subdivided into the same 6 kinds, while Weyl tensor and Riemann tensor are always of the same kind. Therefore in all cases it is possible to be limited to metric classiﬁcation by the type of its Weyl tensor. It is clear, that this classiﬁcation conformally invariant. The subtype O means, that Weyl tensor vanishes, i.e. the metric is conformally ﬂat. All conformally ﬂat metrics automatically satisfy Yang-Mills equations. The equations for conformally ﬂat metrics are much easier than Yang-Mills equations. Therefore this kind of metrics does not represent any interest from the point of view of searching solutions of Yang-Mills equations (though the most discussed in cosmology Robertson-Walker metric which is the solution of Friedmann equations belongs to type O). Solutions of Yang-Mills equations for kinds I, D and O already took place in our works. All central-symmetric metrics belong to kind D or O. The full solution of Yang-Mills equations for central-symmetric metric has been found in [3]. In [4] solutions of Yang-Mills equations for the metric ψ = −dt2 + a2 (t) dx2 + b2 (t)dy 2 + c2 (t)dz 2 were searched. If a 6= b 6= c this metric is referred to the types I or O, if b = c — to the types D or O. In particular, when a = tα1 , b = tα2 , c = tα3 the metric satisﬁes Yang-Mills equations, if 2 2 2 (α1 ) + (α2 ) + (α3 ) − 1 = 1 2 (α1 + α2 + α3 − 1) . 2 ∗ [email protected] † [email protected] c Siberian Federal University. All rights reserved – 472 – Leonid N. Krivonosov, Vyacheslav A. Lukyanov Solving Yang-Mills Equations for 4-metrics of Petrov... When α1 6= α2 6= α3 we obtain a metric of type I, when α2 = α3 6= 0, we get metric of subtype D, when α1 = 1, α2 = α3 = 0 — metric of subtype O. Therefore the purpose of present paper is searching the solutions of Yang-Mills equations for the remained three types II, N, III. In the modern literature such solutions have already been met. In particular, the solutions of Yang-Mills equations are found in [5] for homogeneous (i.e. allowing 3-parametrical invariancy group) Feﬀerman metric. The solution is 2 3 1 3 11 2 2 g = dx + dy + y du − dx ydr + y du + dx . 3 9 9 Authors of that work do not notice that the metric is of the type N, since their paper has another purpose. In this article we adhere to same tactics: Yang-Mills equations are made and solved only for the metrics allowing not less, than 2-parametrical invariancy group. Only in this case there is a hope to receive "solvable" Yang-Mills equations. It has appeared, that for metrics of type N the Yang-Mills equation most often are solvable, and with the big arbitrariness. For metrics of type III classes of metrics with solvable Yang-Mills equations are also rather easily found. The greatest diﬃculties for authors have caused searches of metrics of type II with solvable Yang-Mills equations since more often for such metrics the Yang-Mills equation do not allow to solve themselves though the solution may exist with arbitrariness in several functions. But if in addition to impose a stationary curvatures equality condition (A. Z. Petrov’s terminology [1, Section 17]) the metric of a type II turns into the metrics of type N, or of type III, and the equations are often can be solved. Further for brevity we will apply the term "Yang-Mills metric" to the metrics, satisfying Yang-Mills equations. 1. Solving Yang-Mills equations for metrics of type N 1. We will begin with a metric of the type II ψ = 2dt (g (y) dx + h (t) dy + f (t, y, z) dt) + dy 2 + dz 2 , (1) to illustrate, how the structure of Yang-Mills equations improves when type II turns into type N. Put 1 ω1 = − f dt − gdx − hdy, ω 3 = dy, 2 1 + f dt + gdx + hdy, ω 4 = dz. ω2 = 2 In this case ψ = ηij ω i ω j , where ηij — Minkowski components of conformal connection matrix 0 ω1 ω2 ω1 0 ω12 2 0 ω ω12 Ω= 3 3 ω ω1 ω23 4 ω ω14 ω24 0 ω 1 −ω 2 tensor with signature (− + ++) . We compute ω3 ω13 −ω23 0 ω34 −ω 3 – 473 – ω4 ω14 −ω24 −ω34 0 −ω 4 0 ω1 −ω2 −ω3 −ω4 0 (2) Leonid N. Krivonosov, Vyacheslav A. Lukyanov Solving Yang-Mills Equations for 4-metrics of Petrov... ∂f = ft , according to the standard scheme [3, p. 351–352]. We will denote partial derivatives ∂t ∂f = fy etc. At ﬁrst we ﬁnd external diﬀerentials ∂y 1 1 2 1 2 4 + f − ht ω 1 ∧ ω 3 + dω = −dω = fz ω + ω ∧ ω + fy − u 2 (3) 1 def gy + fy + u dω 3 = dω 4 = 0, u = − f − ht ω 2 ∧ ω 3 , . 2 g Then we calculate Pfaﬃan forms of Christoﬀel for the metrics (1) 1 1 3 ω12 = uω , ω13 = (−fy + uf + ht ) ω 1 + ω 2 + uω 1 , ω34 = 0, 2 2 1 ω23 = (−fy + uf + ht ) ω 1 + ω 2 − uω 2 , ω14 = ω24 = −fz ω 1 + ω 2 . 2 Further we ﬁnd external forms of Riemann curvature 1 R34 = 0, R12 = − u2 ω 1 ∧ ω 2 , 4 1 1 1 uy + u2 ω 1 ∧ ω 3 , R13 = fyz − ufz ω 1 + ω 2 ∧ ω 4 + (fy − uf )y ω 1 + ω 2 ∧ ω 3 − 2 2 4 1 1 1 R23 = fyz − ufz ω 1 + ω 2 ∧ ω 4 + (fy − uf )y ω 1 + ω 2 ∧ ω 3 + uy + u2 ω 2 ∧ ω 3 , 2 2 4 1 R14 = R24 = fzz ω 1 + ω 2 ∧ ω 4 + fyz − ufz ω 1 + ω 2 ∧ ω 3 . 2 k We compute components of Ricci tensor Rij = Rijk and its trace R = η ij Rij 1 R12 = ∆f − (uf )y , R11 = R12 − uy + u2 , R44 = 0, 2 (4) 3 1 1 R = 2uy + u2 . uy + u2 , R33 = uy + u2 , R22 = R12 + 2 2 2 1 1 Here ∆ is Laplacian on variables y and z. Hence, using formula bij = Rij − Rηij , we ﬁnd 2 12 (nonzero) components of Pfaﬃan forms ωi = bij ω j 1 b11 = b12 − K, ∆f − (uf )y , b12 = 2 1 1 b22 = b12 + K, b33 = uy + u2 , 3 8 1 2 1 1 1 K= uy + u2 . b44 = − uy − u , 6 8 12 8 As a result ω1 = b12 ω 1 + ω 2 − Kω 1 , ω2 = b12 ω 1 + ω 2 + Kω 2 , ω3 = b33 ω 3 , ω4 = b44 ω 4 , and the matrix of conformal connection is a matrix of conformal curvature 0 Φ1 0 0 0 Φ21 Φ= 0 Φ31 0 Φ41 0 0 completely deﬁned. Now we compute components of Φ2 Φ21 0 Φ32 Φ42 0 Φ3 Φ31 −Φ32 0 Φ43 0 – 474 – Φ4 Φ41 −Φ42 −Φ43 0 0 0 Φ1 −Φ2 −Φ3 −Φ4 0 (5) Leonid N. Krivonosov, Vyacheslav A. Lukyanov Solving Yang-Mills Equations for 4-metrics of Petrov... in correspondence with formulas Φji = Rij + ω j ∧ ωi + η jk ωk ∧ ηis ω s and Φi = dωi + ωk ∧ ωik Φ21 Φ31 Φ41 Φ32 Φ42 = R12 + ω 2 ∧ ω1 − ω2 ∧ ω 1 = 2Sω 1 ∧ ω 2 , Φ43 = −2Sω 3 ∧ ω 4 , = T ω 1 + ω 2 ∧ ω 3 − Sω 1 ∧ ω 3 + P ω 1 + ω 2 ∧ ω 4 , = P ω 1 + ω 2 ∧ ω 3 − T ω 1 + ω 2 ∧ ω 4 − Sω 1 ∧ ω 4 , = T ω 1 + ω 2 ∧ ω 3 + Sω 2 ∧ ω 3 + P ω 1 + ω 2 ∧ ω 4 , = P ω 1 + ω 2 ∧ ω 3 − T ω 1 + ω 2 ∧ ω 4 + Sω 2 ∧ ω 4 , 1 1 1 def 1 def uy , T = (fyy − fzz ) − (uf )y , P = fyz − ufz . It means, that 12 2 2 2 j components of Weyl tensor Cimn = Φjimn are equal to Φ2112 = 2S, Φ3113 = T − S, Φ3123 = T etc. Φ1 = X ω 1 + ω 2 ∧ ω 3 + Zω 1 ∧ ω 3 + Y ω 1 + ω 2 ∧ ω 4 , Φ3 = 0, 1 2 3 2 3 1 2 4 Φ2 = X ω + ω ∧ ω − Zω ∧ ω + Y ω + ω ∧ ω , Φ4 = (b44 )y ω 3 ∧ ω 4 , def where we denote S = 1 1 def def where for brevity we denote X = − (b12 )y + b12 u + uy (fy − uf ) , Y = − (b12 )z − 2 4 1 1 def 1 fz uy + u2 , Z = uyy + uy u. Petrov matrix Q (λ) [1, formula (18.14)], made with the 4 12 8 help of the components of Weyl tensor, looks like −2S + λ 0 0 . Q (λ) = 0 −T + S + iP + λ −P − iT 0 −P − iT T + S − iP + λ It is a matrix of type II with 1-fold eigenvalue λ1 = 2S and double eigenvalue λ2 = −S. But if S = 0 it is of type N. To compose Yang-Mills equations we will write components of a dual matrix ∗Φ (∗ is Hodge star operator) [3, p. 352, item 6] ∗Φ21 ∗Φ31 ∗Φ41 ∗Φ32 ∗Φ42 ∗Φ1 ∗Φ2 = = = = = = = 2Sω 3 ∧ ω 4 , ∗ Φ43 = 2Sω 1 ∧ ω 2 , ∗ Φ4 = − (b44 )y ω 1 ∧ ω 2 , P ω 1 + ω 2 ∧ ω 3 − T ω 1 + ω 2 ∧ ω 4 + Sω 2 ∧ ω 4 , −T ω 1 + ω 2 ∧ ω 3 − Sω 2 ∧ ω 3 − P ω 1 + ω 2 ∧ ω 4 , −T ω 1 + ω 2 ∧ ω 4 − Sω 1 ∧ ω 4 + P ω 1 + ω 2 ∧ ω 3 , −P ω 1 + ω 2 ∧ ω 4 − T ω 1 + ω 2 ∧ ω 3 + Sω 1 ∧ ω 3 , −X ω 1 + ω 2 ∧ ω 4 − Zω 2 ∧ ω 4 + Y ω 1 + ω 2 ∧ ω 3 , −X ω 1 + ω 2 ∧ ω 4 + Zω 1 ∧ ω 4 + Y ω 1 + ω 2 ∧ ω 3 , ∗ Φ3 = 0. Yang-Mills equations d ∗ Φ + Ω ∧ ∗Φ − ∗Φ ∧ Ω = 0 for external forms ∗Φ3 and ∗Φ4 are d ∗ Φ3 + ωk ∧ ∗Φk3 − ∗Φk ∧ ω3k = 0, d ∗ Φ4 + ωk ∧ ∗Φk4 − ∗Φk ∧ ω4k = 0. (6) In components they give two equations 2S (b44 − K) + Zu = 0, 2S (K − b33 ) − (b44 )y u − (b44 )yy = 0. These equations are expressed only through the function u = 1 2 (uy ) + u2 uy 2 5 5 3 2 uyyy + uyy u + (uy ) + u2 uy 2 4 2 uyy u − – 475 – gy : g = 0, = 0. (7) Leonid N. Krivonosov, Vyacheslav A. Lukyanov Solving Yang-Mills Equations for 4-metrics of Petrov... 3 1 If to denote L and Q the left parts of these equations, the formula Q = Lu + L shows that u 2 the second of these equations is a diﬀerential consequence of the ﬁrst. Yang-Mills equation for the external form ∗Φ1 produces 1 Xy + Z (fy − uf − ht ) + Yz + 2b12 S + (b33 − b44 ) T − uX = 0, 2 1 Xy + Zy + Z fy + − f u − ht + Yz + 2b12 S+ 2 1 +2SK − (b33 + b44 ) S + (b33 − b44 ) T − uX = 0. 2 Their diﬀerence on account of (7) vanishes, therefore it is possible to leave only the ﬁrst equation. In detail it looks like 11 u2 ∆∆f uy ∆f + 2uy fyy + + u∆fy + − (uyy − uy u) fy + −2 3 2 6 ! (8) 2 u 5 (uy ) 5uyy u uy u uyyy yy f − ht = 0. − − + + 2 6 6 12 8 The Yang-Mills equation for the external form ∗Φ2 does not result new equations. Thus, the whole system of Yang-Mills equations is reduced to (7) and (8). The equation (8) serves to ﬁnd function f . Though it is linear it’s diﬃcult to specify its solution without additional restrictions. However if the type of the metric (1) turns from II into N, i.e. at S = 0, that is equivalent to uy = 0, u = α, g = βeαy , where α, β = const, then equation (7) is satisﬁed identically, and equation (8) becomes good enough ∆∆f − 2α∆fy + α2 ∆f = 0. (9) In particular, at α = 0 it turn into the well known equation ∆∆f = 0. It is easy to specify its solutions in the polynomial form. ∂4f ∂2f Another special case, if f does not depend on y. Then equation (9) leads to +α2 2 = 0. 4 ∂z ∂z Its general solution is f = λ cos αz + µ sin αz + δz + ε, where λ, µ, δ, ε are arbitrary functions of t. ∂4f ∂3f ∂2f In the case, when f does not depend on z, equation (9) is reduced to −2α 3 +α2 2 = 4 ∂y ∂y ∂y 0, i.e. f = (λy + µ) eαy + δy + ε, where λ, µ, δ, ε are arbitrary functions of t. Solutions of equations (7) and (8), not leading to type N, will be examined in the following section. Notice, that metric (1) will be Einstein metric, i.e. Rij = κηij , iﬀ g = const 6= 0, ∆f = 0. It follows from (4). Further we will omit detailed computations and will write only matrixes of conformal connection Ω and curvature Φ and ﬁnal equations. 2. Let’s investigate metric 2 ψ = 2dt (dx + B (t, y) dt) + (A (t, y) dy) + dz 2 . (10) We denote with dot diﬀerentiation with respect to t, and the stroke ′ denotes a derivative with – 476 – Leonid N. Krivonosov, Vyacheslav A. Lukyanov Solving Yang-Mills Equations for 4-metrics of Petrov... respect to y. The matrix of conformal connection (2) is 1 1 1 2 ω = − B dt − dx, ω = + B dt + dx, 2 2 ω12 = ω14 = ω24 = ω34 = 0, ω1 = ω2 = def where K = 1 A B′ A Φ21 = Φ41 = ω13 = ω23 = − 1 K ω1 + ω2 , 2 ′ ω 3 = Ady, ω 4 = dz, . A B′ 1 ω + ω2 + ω3 , A A ω3 = ω4 = 0, .. + A . The matrix of conformal curvature (5) has components A Φ43 = 0, Φ31 = Φ32 = 1 K ω1 + ω2 ∧ ω3 , 2 1 Φ42 = − K ω 1 + ω 2 ∧ ω 4 , 2 Petrov matrix looks like λ Q (λ) = 0 0 Φ1 = Φ2 = − 0 1 − K +λ 2 i − K 2 0 i − K 2 1 K +λ 2 Φ3 = Φ4 = 0, K′ 1 ω + ω2 ∧ ω3 . 2A . It is a matrix of type N. Yang-Mills equation (6) for forms ∗Φ3 and ∗Φ4 are satisﬁed identically, and equations for ′ K′ = 0, or in the unwrapped shape forms ∗Φ1 and ∗Φ2 result in the same equation A " 1 A 1 A B′ A ′ .. A + A ! ′ #′ = 0. We have derived one equation on two functions A and B of two variables. An arbitrariness of solutions is great: one of functions A or B can be unrestricted. It is easy to specify many particular solutions in an explicit form. For example, if A does not depend on y, then B = αy 3 + βy 2 + γy + δ, where α, β, γ, δ are arbitrary functions of t. 3. For metric 2 2 ψ = 2dt (dx − εzdy) + (a (t) dy + b (t) dz) + (c (t) dz) , where ε = const, the matrix of conformal connection (2) has components 1 1 dt − dx + εzdy, ω 2 = dt + dx − εzdy, ω 3 = ady + bdz, ω 4 = cdz, 2 2 . 1 a ε 4 4 1 ε 1 ω12 = 0, ω13 = ω23 = ω 3 + ω + ω2 , P− ω , ω3 = P+ a 2 ac 2 ac . ε 3 c 4 1 4 4 1 2 P+ ω + ω , ω1 = ω2 = K ω + ω , ω3 = ω4 = 0, ω1 = ω2 = 2 ac c . .. .. . ab a c 1 2 1 ε def b def 1 where P = − , K = . + + P − c ac 2 a c 2 2 a2 c2 Components of conformal curvature matrix (5) are ω1 = Φ21 = Φ31 = Φ41 = Φ43 = Φ1 = Φ2 = Φ3 = Φ4 = 0, Φ32 = S ω 1 + ω 2 ∧ ω 3 + T ω 1 + ω 2 ∧ ω 4 , Φ42 = T ω 1 + ω 2 ∧ ω 3 − S ω 1 + ω 2 ∧ ω 4 , – 477 – (11) Leonid N. Krivonosov, Vyacheslav A. Lukyanov Solving Yang-Mills Equations for 4-metrics of Petrov... .. .. . . .. 1 a c ε εa εc a def 1 . , T = P + P + 2 − 2. − − P2 − P 2 a c ac 2 a 2a c ac Yang-Mills equations for the metric (11) are satisﬁed identically. The metric has a type N. It is Einsteinian, if K = 0. Many known metrics for which Einstein equations were solved, are of type N and are reduced to metrics (1), (10) or (11). We will bring several examples. 4. Peres metric def where S = ψ = −dt2 + dx2 + dy 2 + dz 2 + f (t − x, y, z) (dt − dx) after the substitution of variables t − x = −u, t + x = 2v, F (u, y, z) = 2 1 f (t − x, y, z) turns into 2 ψ = 2du (dv + F (u, y, z) du) + dy 2 + dz 2 , which is a special case of metric (1) with g (y) = 1 and h (u) = 0. Therefore for Peres metric Yang-Mills equation consist of one equation (9) ∆∆F = 0, where ∆ is Laplacian with respect to variables y and z. 5. Takeno metric ψ = − (P + S) dt2 + 2Sdtdx + (P − S) dx2 + Ady 2 + 2Bdydz + Cdz 2 , where A, B, C, P, S are functions of t − x, after the substitution of variables u = t − x, v = t + x turns into ψ = −P (u) dudv − S (u) du2 + A (u) dy 2 + 2B (u) dydz + C (u) dz 2 . Z S (u) 1 v+ du , then we introduce a Now, instead of v, we introduce a new variable w = − 2 P (u) Z new parameter τ = P (u) du. As a result, ψ = 2dτ dw + A (u (τ )) dy 2 + 2B (u (τ )) dydz + C (u (τ )) dz 2 , i.e. special case of the metric (11). Yang-Mills equations and Petrov type of the metric are invariant with respect to performed operations, that’s why the Takeno metric is of type N and identically satisﬁes Yang-Mills equations. 6. Rosen metric ψ = − exp (2µ) dt2 − dx2 + u2 exp (2ν) dy 2 + exp (−2ν) dz 2 , where µ and ν are functions of u = t − x, is a special case of the metric ψ1 = D (u) dt2 − dx2 + A (u) dy 2 + 2B (u) dydz + C (u) dz 2 , and the latter is conformally equivalent to the metric (11) at ε = 0. Therefore, Yang-Mills equations for the Rosen metric are satisﬁed identically. 7. Bondi-Piranha-Robinson metric ψ = dt2 − dx2 + αdy 2 + 2βdydz + γdz 2 , where α, β and γ are functions of t + x, is obviously isomorphic to the metric (11) at ε = 0. Therefore, Yang-Mills equations for this metric are satisﬁed identically. – 478 – Leonid N. Krivonosov, Vyacheslav A. Lukyanov 2. Solving Yang-Mills Equations for 4-metrics of Petrov... Solving Yang-Mills equations for metrics of type II 1. Let’s construct and solve Yang-Mills equations for the metric ψ = 2dt (dx + A (t, z) dt) + eαt C (z) dy 2 + dz 2 , α = const. (12) As we shall see, this metric can be of all types, except type I, and for types II, III, N YangMills equations admit explicit nontrivial solutions. Derivative with respect to z is denoted by a stroke ′ . Then we compute the components of the conformal connection matrix (2) 1 1 1 2 ω = − A dt − dx, ω = + A dt + dx, ω 3 = Ceαt dy, ω 4 = dz, 2 2 C′ ω12 = 0, ω34 = − ω 3 , ω13 = ω23 = αω 3 , ω14 = ω24 = −A′ ω 1 + ω 2 . C The components of the Ricci tensor and its trace are R14 C′ ′ A + α2 , R11 = R22 = R12 = A′′ + C C′ C ′′ C ′′ = R24 = α , R33 = R44 = , R=2 C C C (13) The remaining 4 Pfaﬃan forms of the conformal connection matrix Ω are ω1 ω2 ω4 C ′′ 1 1 C ′ 4 C ′′ 3 ω + α ω , ω3 = ω , = b12 ω 1 + ω 2 + 6C 2 C 3C C ′′ 2 1 C ′ 4 = b12 ω 1 + ω 2 − ω + α ω , 6C 2 C ′ ′ ′′ C 4 1 C ′′ ′C 2 1 2 A +A = α ω , b12 = +α . ω +ω + 2C 3C 2 C Now we thecomponents of the conformal curvature matrix (5), where for brevity write down ′ def ′′ ′C P = A −A − α2 C Φ21 Φ31 Φ32 Φ41 Φ42 Φ43 Φ1 C′ C ′′ 1 ω ∧ ω2 + α ω1 + ω2 ∧ ω4 , = − 2C 3C′′ C C′ 3 1 = − P ω ∧ ω3 − P ω2 ∧ ω3 − α ω ∧ ω4 , 6C 2C ′′ C′ 3 C + P ω2 ∧ ω3 − α ω ∧ ω4 , = −P ω 1 ∧ ω 3 − 6C 2C ′′ C C′ 1 = P ω2 ∧ ω4 + + P ω1 ∧ ω4 + α ω ∧ ω2 , 6C 2C C′ 1 C ′′ ω2 ∧ ω4 + P ω1 ∧ ω4 + α ω ∧ ω2 , = P− 6C 2C C ′′ 3 C′ = ω ∧ ω4 − α ω1 + ω2 ∧ ω3 , 3C 2C ′′ ′ A′ C ′′ C 1 4 ′ ω ∧ ω + −b12 + ω1 + ω2 ∧ ω4 , = − 6C 2C – 479 – Leonid N. Krivonosov, Vyacheslav A. Lukyanov Φ2 Φ3 Φ4 Solving Yang-Mills Equations for 4-metrics of Petrov... ′ A′ C ′′ C ′′ 2 4 ′ ω1 + ω2 ∧ ω4 , ω ∧ ω + −b12 + = 6C 2C ′′ ′ ′ C α C′ 3 4 = − ω ∧ω + ω1 + ω2 ∧ ω3 , 3C 2 C ′ ′ α C = − ω1 + ω2 ∧ ω4 . 2 C Petrov matrix Q(λ) is C ′′ +λ 3C iαC ′ 2C αC ′ − 2C iαC ′ 2C − C ′′ − +P +λ 6C iP αC ′ 2C iP ′′ − C −P +λ 6C . 2 2 P C ′′ 1 αC ′ P C ′′ 1 αC ′ 6= 0; type D if C ′′ 6= 0, = 0; + + C 2 C C 2 C type III if C ′′ = 0, αC ′ 6= 0; type N if C ′′ = 0, αC ′ = 0, P 6= 0; type O when C ′′ = 0, αC ′ = 0, P = 0. Yang-Mills equation for the external form ∗Φ3 gives It has type II when C ′′ 6= 0, C ′′ C ′′ − 1 2 C ′′ C 2 (14) = 0. Yang-Mills equation for the external form ∗Φ4 results 2 C′ 1 C ′′ = − C 2 C ! ′′ ′ C 2 C ′′ C ′ α = − C 3 C2 C ′′ C ′ 0, (15) 0. (16) The remaining two equations for forms ∗Φ1 and ∗Φ2 with the help of the equalities (14)–(16) lead to ! 2 2C ′′ 1 (4) C ′ ′′′ 1 C′ A′′ + − − A − A + 2 C 2 C 3C ! (17) 3 ! ′ 2 2C ′′ C 1 C′ C ′′ C ′ C ′′′ ′ 2 = 0. A +α − − + + − 6C C2 2 C C 3C Thus, the system of Yang-Mills equations is reduced to (14)–(17). To solve it, we ﬁrst note that the equation (14) is a diﬀerential consequence of (15). Equation (15) allows reduction of order. It is equivalent to 2 C ′ = βC 2 + γ 3 , (18) where β and γ are constants. Put α 6= 0. Then, eliminating the third derivative from (15) and (16), we obtain 2 ! C ′′ C ′′ 4 C′ = 0. − C C 3 C – 480 – (19) Leonid N. Krivonosov, Vyacheslav A. Lukyanov Solving Yang-Mills Equations for 4-metrics of Petrov... In equation (19) we initially set to zero the second factor. The resulting equation can be easily integrated 1 C= λ, µ = const. (20) 3, (λz + µ) This corresponds to (18) at γ = 0. Substituting (20) to (17), we obtain the diﬀerential equation of Euler type 25λ3 α 2 λ2 1 3λ 7λ2 ′′ ′ − A(4) + A′′′ − A − A + 2 3 2 = 0. 2 λz + µ 2 (λz + µ) 2 (λz + µ) (λz + µ) Its general solution µ µ 6 α 2 µ µ 2 A = ε1 + ε2 ln z + + ε3 + ε4 ln z + z + + , z+ λ λ λ 32 λ (21) µ where ε1 , ε2 , ε3 , ε4 are arbitrary functions of t. By making the change of variable z + → z, we λ obtain the ﬁnal solution A = ε1 + ε2 ln |z| + (ε3 + ε4 ln |z|) z 6 + α2 2 z , 32 C= 1 3, (λz) (22) ′ 2 C ′′ 1 C 6= 0, depending on four arbitrary functions of the variable t. Since C = 6 0 and P + α C 2 C this solution gives the metric of type II. Although the solution (22) is obtained for α 6= 0, it is also a solution in the case α = 0, but it is not a general solution, because equation (18) besides the solution (20) at γ = 0, has other non-elementary solutions at γ 6= 0. But in the latter case we cannot ﬁnd explicit solutions of the equation (17). Solution (22) in the case of α = 0 gives the metric of type II. Now we consider the second possibility of the equality (19), C ′′ = 0, which is equivalent to C = λz + µ, where λ, µ = const. Substituting this in (17), we again obtain the equation of Euler type 1 λ λ2 λ3 α 2 λ2 − A(4) − A′′ − A′ + A′′′ + 2 3 2 = 0. 2 λz + µ 2 (λz + µ) 2 (λz + µ) (λz + µ) ′′ In its general solution, we replace z + µλ with z and get α2 2 A = ε1 + ε2 ln |z| + ε3 + ε4 ln |z| + ln |z| z 2 , 4 C = λz. (23) At α 6= 0 this solution gives the metric of type III, and at α = 0 the metric of type N, diﬀerent from the metric of type N in section 1. If C = const equation (17) has a general solution A = ε1 (t) + ε2 (t) z + ε3 (t) z 2 + ε4 (t) z 3 . This solution gives the metric of type N, but it is conformally equivalent to the special case of the metric (10). From (13), it follows that (12) is Einstein metric in cases 1) α 6= 2) α = 1 0, C = const, A = − α2 z 2 + ε1 (t) z + ε2 (t) ; 2 µ 0, C = λz + µ, A = ε1 (t) ln z + + ε2 (t) . λ In both cases, the metric has a type N. – 481 – Leonid N. Krivonosov, Vyacheslav A. Lukyanov Solving Yang-Mills Equations for 4-metrics of Petrov... 2. Let’s return to the metric (1) and Yang-Mills equations (7) and (8). We will show that under the additional condition: f (t, y, z) doesn’t depend on z, equation (8) already has explicit solutions. In this case it turns into 1 11 u2 7uy − fyyyy + ufyyy + fyy + − (uyy − uy u) fy + 2 3 2! 6 (24) 2 u uyyy 5 (uy ) 5uyy u uy u yy + f − ht = 0. − − + 2 6 6 12 8 √ Equation (7) allows reduction of order uy = − 23 u2 + λ u, where λ = const. If λ = 0 we obtain the solution in terms of elementary functions u= 3 , 2y + µ µ = const. (25) From (3) we obtain 3 g = γ (2y + µ) 2 , γ, µ = const. (26) Substituting (25) to (24), we obtain the equation of Euler type fyyyy − 37 154 324 ht 6 fyyy + 2 fyy − 3 fy + 4f = 3. 2y + µ (2y + µ) (2y + µ) (2y + µ) 2 (2y + µ) Its general solution f= 3 2y + µ 3 ht + (ε1 + ε2 ln |2y + µ|) (2y + µ) + (ε3 + ε4 ln |2y + µ|) (2y + µ) 2 , 32 (27) εi are arbitrary functions of t. Formulas (26) and (27) give an explicit solution (but not general) of Yang-Mills equations for the metric (1), h (t) is an arbitrary function. This solution gives the metric of type II. 3. For the following metric 2 ψ = 2dt (dx + A (x, z) dt) + C(z)e−αt dy + dz 2 , α = const (28) Yang-Mills equations are very complicated, but all the same they are solved. Components of the conformal connection matrix Ω are 1 1 2 1 − A dt − dx, ω = + A dt + dx, ω 3 = Ce−αt dy, ω 4 = dz, ω = 2 2 Cz ω12 = −Ax ω 1 + ω 2 , ω34 = − ω 3 , ω13 = ω23 = −αω 3 , ω14 = ω24 = −Az ω 1 + ω 2 . C The components of the Ricci tensor and its trace are Cz Az , R11 = R12 + Axx , R22 = R12 − Axx , C Cz Czz = R24 = −Axz − α , R = −2Axx + 2 . C C R12 = Azz + α2 − αAx + R33 = R44 = Czz , R14 C The coeﬃcients of Pfaﬃan forms ωi : Cz Cz 1 1 2 Azz + α − αAx + Axz + α . Az , b14 = b24 = − b12 = 2 C 2 C – 482 – Leonid N. Krivonosov, Vyacheslav A. Lukyanov 1 Czz 1 Czz Czz 1 Axx + + b12 , b22 = − Axx − + b12 , b33 = b44 = + Axx , 3 6C 3 6C 3C 6 forms ωi : 1 Cz 1 αCz Czz 1 ω1 + Azz + α2 − αAx + Az Axz + ω4 , ω1 + ω2 − Axx + 3 6C 2 C 2 C 1 Cz 1 αCz 1 Czz ω2 + Azz + α2 − αAx + Az Axz + ω4 , − ω1 + ω2 − Axx + 3 6C 2 C 2 C 1 αCz Czz 1 1 Czz Axz + ω1 + ω2 + + Axx ω 3 , ω4 = − + Axx ω 4 . 3C 6 2 C 3C 6 = b11 Pfaﬃan ω1 = ω2 = ω3 = Solving Yang-Mills Equations for 4-metrics of Petrov... Then we compute the components of the conformal curvature matrix Φ. For brevity we Czz Cz Cz def def def denote S = Axx − , T = Axz − α , L = Azz − α2 + αAx − Az . C C C 1 1 Φ21 = Sω 1 ∧ ω 2 + T ω 1 + ω 2 ∧ ω 4 , 3 2 1 1 1 1 2 3 Φ1 = − L ω + ω ∧ ω 3 − Sω 1 ∧ ω 3 − T ω 3 ∧ ω 4 , 2 6 2 1 1 1 1 4 1 2 4 1 Φ1 = T ω ∧ ω − Sω ∧ ω + L ω + ω 2 ∧ ω 4 , 2 6 2 1 2 1 1 1 2 4 4 L ω + ω ∧ ω + Sω ∧ ω 4 + T ω 1 ∧ ω 2 , Φ2 = 2 6 2 1 1 3 4 1 2 3 4 Φ3 = − Sω ∧ ω − T ω + ω ∧ ω , 3 2 1 1 1 3 3 4 Φ2 = − T ω ∧ ω + Sω 2 ∧ ω 3 − L ω 1 + ω 2 ∧ ω 3 , 2 6 2 1 1 1 1 1 2 − (b12 )x ω ∧ ω + P − Axxz ω 1 + ω 2 ∧ ω 4 + Sz ω 1 ∧ ω 4 , Axxx A − Φ1 = 3 2 4 6 1 1 1 1 1 2 − (b12 )x ω ∧ ω + P + Axxz ω 1 + ω 2 ∧ ω 4 − Sz ω 1 ∧ ω 3 , Φ2 = Axxx A + 3 2 4 6 1 Φ3 = K ω 1 + ω 2 ∧ ω 3 − Axxx ω 1 ∧ ω 3 − (b33 )z ω 3 ∧ ω 4 , 6 1 1 1 2 Axxz ω ∧ ω + M ω 1 + ω 2 ∧ ω 4 − Axxx ω 1 ∧ ω 4 . Φ4 = 2 6 In the last four formulas we denoted 1 Czz def 1 − (b12 )z + Axxz A + Ax b14 , P = Az Axx + 2 C 2 2 1 C 1 C α Czz 1 def z z + Axxx , + − − A K = − Axxx A + Axz xx 2 6 2 C 2 C C 12 2 1 αC αC 1 1 def zz M = Axzz + − 2z − Axxx A + Axxx . 2 C C 3 12 Petrov matrix Q(λ) is 1 −3S + λ i T 2 1 − T 2 i T 2 1 1 S+ L+λ 6 2 i L 2 – 483 – 1 − T 2 i L 2 1 1 S− L+λ 6 2 . Leonid N. Krivonosov, Vyacheslav A. Lukyanov Solving Yang-Mills Equations for 4-metrics of Petrov... It has type II when S 6= 0, type III when S = 0, T 6= 0, subtype N when S = T = 0, L 6= 0, subtype O when S = T = L = 0. Proceeding from the speciﬁcs of the conformal curvature matrix, it’s best to start compiling Yang-Mills equations for the diﬀerence between ∗Φ1 and ∗Φ2 , i.e. with the equation d (∗Φ1 − ∗Φ2 ) + ωk ∧ ∗Φk1 − ∗Φk2 − ∗Φk ∧ ω1k − ω2k = 0. We obtain three equations Axxxx = 0, (29) Axxxz = 0, 2 Axxx Axxzz Axxz Cz (Axx ) Czzzz Czzz Cz 1 Czz (Ax +α)− − − − + + 3 3 3C 6 6C 6C 2 3 C 2 − Czz Cz2 = 0. (30) 6C 3 Equalities (29) mean that (31) Axxx = β = const. Taking into account these equations, Yang-Mills equations for the external form ∗Φ1 provide two new conditions 1 Cz Cz 5 − Azzzz + Axxzz A − αβA + Axxz A + Az − Axzz Ax + αAxzz − Azzz + 2 C 3 C 2 C 2Cz α 5 2 C C z z zz +Axx Azz + Az + Ax − α − A2xz − Axz Ax + αAxz − αAx + (32) 3 3C 3 6 C C 3C 2 2 3 Czz Cz Cz Cz 2Czz Czzz C 2Czz + Az = 0, +Azz − − − z3 + α2 − 2C 2 3C C2 6C 2C C2 3C 1 Czz 2 1 α Axx Axz + Axz − βAz + Axzzz − Axxz − 3 2 6C 3 2 2 (33) Cz Cz Czzz 5Czz Cz Cz + αAxx +α −α = 0. −Axz 2 + Axzz 2C 2C 3C 2C 6C 2 Taking into account (31), Yang-Mills equations for the external form ∗Φ3 yield one new equality 2 1 1 Czz 1 β Cz αβ 1 Czz − + A2xx = 0, Axxzz − Ax + Axxz − + (34) 6 3 2C 2 3 C zz 6 C 6 and for the external form ∗Φ4 one new equation 1 Cz 1 β C2 Cz − Axxzz − Axxz − A2xx + Ax + zz2 − 2 6C 6 3 6C 3C Czz C + z αβ = 0. 6 (35) Equation (31) is equivalent to A= 1 3 βx + f (z) x2 + g (z) x + h (z) . 6 (36) Six equations (30), (32), (33), (34), (35) and (36) connect with diﬀerential relations ﬁve functions A, f, g, h and C. But equations (33), (34) and (35) are linearly dependent: 2L1 + L2 = L3 , where L1 , L2 , L3 are left sides of the equations (33), (34) and (35). Therefore, equation (35) may be discarded. The rest of the equations impose to functions f, g, h and C seven diﬀerential relations, only four of which are independent (we denote with stroke ′ the derivative with respect to z): ′′ ′ C ′ ′ αβ 1 C ′′ C ′ 1 C ′′ ′′ f − + f + − = 0, (37) C 2 2 C 2 C C – 484 – Leonid N. Krivonosov, Vyacheslav A. Lukyanov Solving Yang-Mills Equations for 4-metrics of Petrov... 2 ′ 1 C ′′ C ′ + = 0, (38) 3 C C ! 2 ′′ αC ′′′ 2βh′ C ′ ′′ 2f g ′ 2αf C ′ 1 C′ 5αC ′′ C ′ g ′′′ ′ ′ C + = 0, (39) − − αf + g + + +g − − 2 3 2C 3 3C 6C 2 C 2C 6C 2 2 ! 1 (4) C ′ ′′′ 2C ′′ 1 C′ 4f ′′ − h − + h +h − + 2 C 3 3C 2 C ! 3 ′ 4f C ′ C ′′′ C ′′ C ′ 1 C′ 10f ′ ′′ ′C + h 2f + 2f + − + − − αβ − g ′′ (g − α) (40) +h′ 3 3C 6C C2 2 C C ′ 2 ! ′ 5α2 f αg ′ C ′ αC ′′ 2C ′′ 2αf C 2 ′2 ′ C − = 0. + +g − +α − + −−g −g g C C 3 3C 3 3C C 1 C ′ ′ 2 2 βg αβ 1 f + f + f − − − 3C 3 3 6 6 ′′ C ′′ C Without additional constraints, the system of equations (37)–(40) can not be solved explicitly. The ﬁrst of these constraints is C = eγz , (41) γ = const 6= 0. In this case system (37)–(40) is solved in terms of elementary functions, but the results are diﬀerent if β 6= 0 and β = 0. At β 6= 0 we obtain solution f = ε1 + αβ z, 2γ g= 2f 2 γ4 − , β 2β h= 4 3 αγ 3 f − z + ε2 . 3β 2 2β Directly through x and z formula (36) can be written as β A= 6 3 2 3 α γ4 α α 4 (ε1 ) 2 2 x + z + ε1 x + z + (ε1 ) − x+ z + + ε2 , γ γ β 4 γ β2 (42) ε1 , ε2 , β, γ are arbitrary constants, but β 6= 0 and γ 6= 0. Formulas (42) and (41) together with (28) give Yang-Mills metric of type II. From (41) and β = 0 it follows, that (by virtue of equations (37) and (38)) there are only two possibilities for the function f (z): 1 f = ± γ2. (43) 2 If the sign is + the following solution to the equations (39) and (40) is obtained g = ε1 + ε2 z + ε3 e−γz , ε2 ε 3 + 2 ze−γz + z γ 2 (ε3 ) ze−2γz + 3γ ! 2 2 αε2 ε 1 ε2 α2 (ε2 ) (ε2 ) . z− 2 + 2 + + 2γ 2 γ γ 2γ 2γ 3 h = ε4 + ε5 e−γz + ε6 eγz + ε7 e−2γz + (44) Here ε1 , ..., ε7 , γ are arbitrary constants, but γ 6= 0. Formulas (44), (43) (with + in the right side), (41), (36) and (28) give Yang-Mills metric of type III. Now consider the case with a minus in the right side of (43). Then the solution of (39) and – 485 – Leonid N. Krivonosov, Vyacheslav A. Lukyanov Solving Yang-Mills Equations for 4-metrics of Petrov... (40) is g = ε1 + ε2 eλ1 γz + ε3 eλ2 γz − αγz, ! r r 5 5 − 21 γz h=e γz + ε7 sin γz + ε6 cos 12 12 (45) 2 2 (ε3 ) (ε2 ) 2λ1 γz 2λ2 γz −γz e + 2 e + ε 4 + ε5 e + + 2 2γ (λ1 − 3) 2γ (λ2 − 3) ε2 ε3 1 2 αε1 λ1 γz λ2 γz + 2 (αzγ − αλ1 − ε1 ) e + 2 (αzγ − αλ2 − ε1 ) e + − α z z. γ γ γ 2 r r 19 19 1 1 , λ2 = − − . Formulas Here ε1 , ..., ε7 , γ are arbitrary constants, but γ 6= 0; λ1 = − + 2 12 2 12 (44), (43) (with minus in the right side), (41), (36) and (28) give Yang-Mills metric of type II. Consider now instead of (41) other constraints: C ′′ = 0, S = 0. Then we obtain the following solution 2 ε2 ε2 + ε5 , + ε4 ln z + C = ε1 z + ε2 , f = 0, β = 0, g = ε3 z + ε ε1 6 4 1 2 ε2 ε2 ε2 (ε3 ) ε2 ε3 h = ε9 ln z + z+ z+ − 2α + ε4 − 2ε5 − 2ε4 ln z + + + ε 36 ε1 16 ε1 ε1 1 2 1 ε2 ε2 2 2 +ε8 + z+ 4ε6 − 2ε7 + α2 − (ε4 ) + 4ε7 − 2α2 + 2 (ε4 ) ln z + . 8 ε1 ε1 Here ε1 , ..., ε9 are arbitrary constants, but ε1 6= 0. We have received Yang-Mills metric of type III. Thus, the metric (28) has provided us with two series of Yang-Mills metric of types II and III. Another remarkable feature of the metric (28) is that at suitable function A(x, z) and C(z) it gives Einstein metric of type II, which is a rare phenomenon. We do not give the full solution, but only one particular case: 1 1 αε1 A = − γ 2 x2 + x (ε1 − αγz) + ε4 + ε5 e−γz + z − α2 z , C = eγz . 2 γ 2 This gives Einstein metric of type II, which is a particular case of solution (45) at ε2 = ε3 = ε6 = ε7 = 0. 3. Metric of type III, satisfying Yang-Mills equations Let’s investigate the metric 2 ψ = 2dt (dx + B (t, z) dy + A (t, z) dt) + (C (t) dy) + dz 2 , (46) which, as we shall see, may be of types III, N or O. The components of conformal connection matrix Ω are 1 1 1 2 ω = − A dt − dx − Bdy, ω = + A dt + dx + Bdy, ω 3 = Cdy, ω 4 = dz, 2 2 . ω13 ω34 . C B′ 3 B′ 4 2 B 1 ω + ω2 + ω3 + ω , ω1 = 0, ω14 = ω24 = −A′ ω 1 + ω 2 − ω , = = C C 2C 2C B ′′ 3 B′ B ′′ 1 = − ω , ω3 = ω 1 + ω 2 , ω1 = ω 2 = K ω 1 + ω 2 + ω + ω 2 , ω4 = 0, 2C 4C 4C ω23 – 486 – Leonid N. Krivonosov, Vyacheslav A. Lukyanov Solving Yang-Mills Equations for 4-metrics of Petrov... where dot denotes the derivative with respect to t, stroke ′ denotes derivative with respect to z, .. ′ 2 ! C 1 B def 1 A′′ + − . K = 2 C 2 C The components of the conformal curvature matrix Φ are: B ′′ 1 B ′′ 1 ω + ω2 ∧ ω3 , Φ43 = ω + ω2 ∧ ω4 , 4C 4C B ′′ 1 ω ∧ ω2 + S ω1 + ω2 ∧ ω3 − T ω1 + ω2 ∧ ω4 , Φ31 = Φ32 = 4C B ′′ 3 4 4 Φ1 = Φ2 = ω ∧ ω4 − S ω1 + ω2 ∧ ω4 − T ω1 + ω2 ∧ ω3 , 4C ! .. . ′ . ! ′ C B B C 1 def def 1 − A′′ , T = + 2 , where S = 2 C 2 C C . ! ′′ . B 1 2B ′′ C Φ1 = Φ2 = + ω1 + ω2 ∧ ω3 + 2 4 C C ′′ ′ B B B ′′′ 3 ′ 1 2 4 + − K ω + ω ∧ ω − ω ∧ ω4 , 8C 2 4C B ′′′ 1 ω + ω2 ∧ ω4 , Φ4 = 0. Φ3 = − 4C Φ21 = Petrov matrix Q(λ) is λ B ′′ − 4C iB ′′ − 4C B ′′ 4C −S − iT + λ iB ′′ 4C T − iS T − iS S + iT + λ − − . .. C This is a matrix of type III if B 6= 0; of type N if B = 0 and of type O if B = 0, A = , C B ′ C = ε (z) , where ε (z) is arbitrary function of z. Yang-Mills equations for the external form ∗Φ4 are satisﬁed identically, for the form ∗Φ3 ∂4B = 0. External forms ∗Φ1 and ∗Φ2 also yield only one they are reduced to a single equation ∂z 4 ′′ 2 ′′′ ′ 4 3B B 3 B ∂ A + = . The system of these two equations has a solution equation 4 ∂z 2 C 2C 2 ′′ = β0 z 3 + β1 z 2 + β2 z + β3 , " 2 1 9 (β0 ) 6 9β0 β1 5 A = z + z + 2 C 40 20 ′′ ′′ ′′ B (β1 ) 3β0 β2 + 8 4 2 ! z 4 # + α0 z 3 + α1 z 2 + α2 z + α3 , where β0 , β1 , β2 , β3 , α0 , α1 , α2 , α3 , C are arbitrary functions of t. .. ′ 2 1 B C = 0. Mertic (46) is Einstein metric when B ′′ = 0, A′′ + − C 2 C References [1] A.Z.Petrov, New methods in General Relativity, 1966, Mosscow, Nauka (in Russian). – 487 – Leonid N. Krivonosov, Vyacheslav A. Lukyanov Solving Yang-Mills Equations for 4-metrics of Petrov... [2] Yu.S.Vladimirov, Geometrophysics, BINOM, Moscow, 2010 (in Russian). [3] L.N.Krivonosov, V.A.Luk’yanov, The full solution of Yang-Mills equations for the centralsymmetric metrics, Journal of Siberian Federal University. Mathematics & Physics, 4(2011), no. 3, 350–362 (in Russian). [4] L.N.Krivonosov, V.A.Luk’yanov, Purely time-dependent solutions to the Yang–Mills equations on a 4-dimensional manifold with conformal torsion-free connection, Journal of Siberian Federal University. Mathematics & Physics, 6(2013), no. 1, 40–52. [5] M.Korzyjnski, J.Levandowski, The Normal Conformal Cartan Connection and the Bach Tensor, arXiv:gr-qc/0301096v3, 2003. Решение уравнений Янга-Миллса для 4-метрик типов II, N, III Петрова Леонид Н. Кривоносов Вячеслав А. Лукьянов Приводятся по 4 серии 4-метрик для каждого из видов II, N, III, удовлетворяющих уравнениям Янга-Миллса. Ключевые слова: уравнения Эйнштейна, уравнения Янга-Миллса, многообразие конформной связности с кручением и без кручения. – 488 –