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. Krivonosovв€—
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
П€ = в€’dt2 + a2 (t) dx2 + b2 (t)dy 2 + c2 (t)dz 2
were searched. If a = b = 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
(О±1 ) + (О±2 ) + (О±3 ) в€’ 1 =
1
2
(О±1 + О±2 + О±3 в€’ 1) .
2
в€— oxyzt2@ya.ru
†oxyzt@ya.ru
c Siberian Federal University. All rights reserved
– 472 –
Leonid N. Krivonosov, Vyacheslav A. Lukyanov
Solving Yang-Mills Equations for 4-metrics of Petrov...
When О±1 = О±2 = О±3 we obtain a metric of type I, when О±2 = О±3 = 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
g = dx2 + dy 2 +
2 3
y du в€’ dx
3
1
11
ydr + y 3 du + dx .
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
П€ = 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
П‰1
=
П‰2
=
1
в€’f
2
1
+f
2
dt в€’ gdx в€’ hdy,
П‰ 3 = dy,
dt + gdx + hdy,
П‰ 4 = dz.
In this case ψ = ηij ω i ω j , where ηij — Minkowski
components of conformal connection matrix
пЈ«
0 П‰1 П‰2
пЈ¬ П‰1 0
П‰12
пЈ¬
пЈ¬ 2
0
пЈ¬ П‰ П‰12
Ω= 3
3
пЈ¬ П‰ П‰1 П‰23
пЈ¬ 4
пЈ­ П‰ П‰14 П‰24
0 П‰ 1 в€’П‰ 2
tensor with signature (в€’ + ++) . We compute
П‰3
П‰13
в€’П‰23
0
П‰34
в€’П‰ 3
– 473 –
П‰4
П‰14
в€’П‰24
в€’П‰34
0
в€’П‰ 4
0
П‰1
в€’П‰2
в€’П‰3
в€’П‰4
0
пЈ¶
пЈ·
пЈ·
пЈ·
пЈ·
пЈ·
пЈ·
пЈ·
пЈё
(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
dω 1 = −dω 2 = fz ω 1 + ω 2 ∧ ω 4 + fy − u
1
в€’f
2
+ fy + u
− ht ω 2 ∧ ω 3 ,
1
+f
2
− ht ω 1 ∧ ω 3 +
dω 3 = dω 4 = 0,
def
u =
gy
.
g
(3)
Then we calculate Pfaffian forms of Christoffel for the metrics (1)
1 3
uω ,
2
1
ω13 = (−fy + uf + ht ) ω 1 + ω 2 + uω 1 ,
П‰34 = 0,
2
1
ω23 = (−fy + uf + ht ) ω 1 + ω 2 − uω 2 ,
П‰14 = П‰24 = в€’fz П‰ 1 + П‰ 2 .
2
Further we п¬Ѓnd 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
П‰12
=
k
We compute components of Ricci tensor Rij = Rijk
and its trace R = О· ij Rij
1
R12 = ∆f − (uf )y ,
R11 = R12 в€’
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
∆f − (uf )y ,
2
b12
=
b22
= b12 + K,
b44
1
1
= в€’ uy в€’ u2 ,
6
8
b11 = b12 в€’ K,
1
1
uy + u2 ,
3
8
1
1
K=
uy + u2 .
12
8
b33 =
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
пЈ«
0 О¦1
пЈ¬ 0 0
пЈ¬
пЈ¬
пЈ¬ 0 О¦21
О¦=пЈ¬
пЈ¬ 0 О¦31
пЈ¬
пЈ­ 0 О¦41
0 0
completely defined. Now we compute components of
О¦2
О¦21
0
О¦32
О¦42
0
О¦3
О¦31
в€’О¦32
0
О¦43
0
– 474 –
О¦4
О¦41
в€’О¦42
в€’О¦43
0
0
0
О¦1
в€’О¦2
в€’О¦3
в€’О¦4
0
пЈ¶
пЈ·
пЈ·
пЈ·
пЈ·
пЈ·
пЈ·
пЈ·
пЈё
(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
О¦21
О¦31
О¦41
О¦32
О¦42
= 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 в€’ fzz ) в€’ (uf )y , P = fyz в€’ ufz . It means, that
12
2
2
2
j
components of Weyl tensor Cimn
= О¦jimn are equal to О¦2112 = 2S, О¦3113 = T в€’ S, О¦3123 = T etc.
def
where we denote S =
О¦1
О¦2
= X ω 1 + ω 2 ∧ ω 3 + Zω 1 ∧ ω 3 + Y ω 1 + ω 2 ∧ ω 4 ,
1
= X П‰ +П‰
2
3
2
3
1
∧ ω − Zω ∧ ω + Y ω + ω
2
О¦3 = 0,
4
∧ ω , Φ4 = (b44 )y ω 3 ∧ ω 4 ,
1
1
def
def
where for brevity we denote X = в€’ (b12 )y + b12 u + uy (fy в€’ uf ) , Y = в€’ (b12 )z в€’
2
4
1
1
def 1
fz uy + u2 , Z =
uyy + uy u. Petrov matrix Q (О») [1, formula (18.14)], made with the
4
12
8
help of the components of Weyl tensor, looks like
пЈ¶
пЈ«
в€’2S + О»
0
0
пЈё.
Q (О») = пЈ­
0
в€’T + S + iP + О»
в€’P в€’ iT
0
в€’P в€’ iT
T + S в€’ iP + О»
It is a matrix of type II with 1-fold eigenvalue О»1 = 2S and double eigenvalue О»2 = в€’S. But if
S = 0 it is of type N.
To compose Yang-Mills equations we will write components of a dual matrix в€—О¦ (в€— is Hodge
star operator) [3, p. 352, item 6]
в€—О¦21
в€—О¦31
в€—О¦41
в€—О¦32
=
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 ,
в€—О¦42
= −P ω 1 + ω 2 ∧ ω 4 − T ω 1 + ω 2 ∧ ω 3 + Sω 1 ∧ ω 3 ,
в€—О¦2
= −X ω 1 + ω 2 ∧ ω 4 + Zω 1 ∧ ω 4 + Y ω 1 + ω 2 ∧ ω 3 , ∗ Φ3 = 0.
в€—О¦1
= −X ω 1 + ω 2 ∧ ω 4 − Zω 2 ∧ ω 4 + Y ω 1 + ω 2 ∧ ω 3 ,
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 в€’ K) + Zu = 0,
2S (K в€’ b33 ) в€’ (b44 )y u в€’ (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 в€’
– 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 в€—О¦1 produces
1
Xy + Z (fy в€’ uf в€’ ht ) + Yz + 2b12 S + (b33 в€’ b44 ) T в€’ uX = 0,
2
1
Xy + Zy + Z fy +
в€’ f u в€’ ht + Yz + 2b12 S+
2
1
+2SK в€’ (b33 + b44 ) S + (b33 в€’ b44 ) T в€’ 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 +
в€’
(uyy в€’ uy u) fy +
в€’2
3
2
6
2
uyy
5 (uy )
5uyy u
uy u
uyyy
f в€’ ht
= 0.
в€’
в€’
+
+
2
6
6
12
8
(8)
The Yang-Mills equation for the external form в€—О¦2 does not result new equations. Thus, the
whole system of Yang-Mills equations is reduced to (7) and (8). The equation (8) serves to п¬Ѓnd
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
+О±2 2 = 0.
4
∂z
∂z
Its general solution is f = О» cos О±z + Вµ sin О±z + Оґz + Оµ, where О», Вµ, Оґ, Оµ are arbitrary functions of
t.
∂4f
∂3f
∂2f
In the case, when f does not depend on z, equation (9) is reduced to
в€’2О± 3 +О±2 2 =
4
∂y
∂y
∂y
0, i.e. f = (О»y + Вµ) eО±y + Оґy + Оµ, where О», Вµ, Оґ, Оµ 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 = 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
П€ = 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
в€’ B dt в€’ dx,
2
П‰1
=
П‰12
= П‰14 = П‰24 = П‰34 = 0,
П‰1
= П‰2 =
П‰2 =
1
+ B dt + dx,
2
П‰ 3 = Ady,
П‰ 4 = dz,
.
def
where K =
B′
A
1
A
П‰13 = П‰23 = в€’
1
K П‰1 + П‰2 ,
2
A
B′ 1
П‰ + П‰2 + П‰3 ,
A
A
П‰3 = П‰4 = 0,
..
′
+
A
. The matrix of conformal curvature (5) has components
A
О¦31 = О¦32 =
1
K ω1 + ω2 ∧ ω3 ,
2
О¦21
=
О¦43 = 0,
О¦41
=
1
Φ42 = − K ω 1 + ω 2 ∧ ω 4 ,
2
Petrov matrix looks like
пЈ«
О»
пЈ¬
пЈ¬
Q (О») = пЈ¬ 0
пЈ­
0
О¦1 = О¦2 = в€’
0
1
в€’ K +О»
2
i
в€’ K
2
0
i
в€’ K
2
1
K +О»
2
О¦3 = О¦4 = 0,
K′ 1
ω + ω2 ∧ ω3 .
2A
пЈ¶
пЈ·
пЈ·
пЈ·.
пЈё
It is a matrix of type N.
Yang-Mills equation (6) for forms ∗Φ3 and ∗Φ4 are satisfied identically, and equations for
forms в€—О¦1 and в€—О¦2 result in the same equation
1
A
1
A
..
′
B′
A
+
A
A
K′
A
′
= 0, or in the unwrapped shape
′ ′
= 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 = О±y 3 + ОІy 2 + Оіy + Оґ, where О±, ОІ, Оі, Оґ are arbitrary functions of t.
3. For metric
2
2
П€ = 2dt (dx в€’ Оµzdy) + (a (t) dy + b (t) dz) + (c (t) dz) ,
where Оµ = const, the matrix of conformal connection (2) has components
1
1
dt в€’ dx + Оµzdy, П‰ 2 = dt + dx в€’ Оµzdy, П‰ 3 = ady + bdz, П‰ 4 = cdz,
2
2
.
1
a
Оµ
1
Оµ
= 0, П‰13 = П‰23 = П‰ 3 +
П‰1 + П‰2 ,
Pв€’
П‰ 4 , П‰34 =
P+
a
2
ac
2
ac
.
Оµ
1
c
P+
П‰ 3 + П‰ 4 , П‰1 = П‰2 = K П‰ 1 + П‰ 2 , П‰3 = П‰4 = 0,
= П‰24 =
2
ac
c
П‰1
=
П‰12
П‰14
.
.
..
..
b ab
a c 1 2 1 Оµ
def 1
where P = в€’ , K =
.
+ + P в€’
c ac
2 a c 2
2 a2 c2
Components of conformal curvature matrix (5) are
def
О¦21
=
О¦43 = О¦1 = О¦2 = О¦3 = О¦4 = 0,
О¦31
=
Φ32 = S ω 1 + ω 2 ∧ ω 3 + T ω 1 + ω 2 ∧ ω 4 ,
О¦41
=
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
Оµ
Оµa
Оµc
a
def 1 .
, T = P + P + 2 в€’ 2.
в€’ в€’ P2 в€’ 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 =
П€ = в€’dt2 + dx2 + dy 2 + dz 2 + f (t в€’ x, y, z) (dt в€’ dx)
after the substitution of variables t в€’ x = в€’u, t + x = 2v, F (u, y, z) =
2
1
f (t в€’ x, y, z) turns into
2
П€ = 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
П€ = в€’ (P + S) dt2 + 2Sdtdx + (P в€’ S) dx2 + Ady 2 + 2Bdydz + Cdz 2 ,
where A, B, C, P, S are functions of t в€’ x, after the substitution of variables u = t в€’ x, v = t + x
turns into
П€ = в€’P (u) dudv в€’ S (u) du2 + A (u) dy 2 + 2B (u) dydz + C (u) dz 2 .
Now, instead of v, we introduce a new variable w = в€’
new parameter П„ =
1
2
v+
S (u)
du , then we introduce a
P (u)
P (u) du. As a result,
П€ = 2dП„ dw + A (u (П„ )) dy 2 + 2B (u (П„ )) dydz + C (u (П„ )) 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
П€ = в€’ exp (2Вµ) dt2 в€’ dx2 + u2 exp (2ОЅ) dy 2 + exp (в€’2ОЅ) dz 2 ,
where Вµ and ОЅ are functions of u = t в€’ x, is a special case of the metric
П€1 = D (u) dt2 в€’ dx2 + A (u) dy 2 + 2B (u) dydz + C (u) dz 2 ,
and the latter is conformally equivalent to the metric (11) at Оµ = 0. Therefore, Yang-Mills
equations for the Rosen metric are satisfied identically.
7. Bondi-Piranha-Robinson metric
П€ = dt2 в€’ dx2 + О±dy 2 + 2ОІdydz + Оіdz 2 ,
where О±, ОІ and Оі are functions of t + x, is obviously isomorphic to the metric (11) at Оµ = 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
П€ = 2dt (dx + A (t, z) dt) + eО±t C (z) dy
2
+ dz 2 ,
О± = 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)
П‰1
=
П‰12
=
1
1
в€’ A dt в€’ dx, П‰ 2 =
+ A dt + dx, П‰ 3 = CeО±t dy, П‰ 4 = dz,
2
2
C′
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 + О±2 ,
R11 = R22 = R12 = A′′ +
C
C′
C ′′
C ′′
= R24 = О± ,
R33 = R44 =
,
R=2
C
C
C
(13)
The remaining 4 Pfaffian forms of the conformal connection matrix Ω are
П‰1
П‰2
П‰4
C ′′ 3
C ′′ 1 1 C ′ 4
П‰ + О± П‰ , П‰3 =
П‰ ,
6C
2 C
3C
C ′′ 2 1 C ′ 4
= b12 П‰ 1 + П‰ 2 в€’
П‰ + О± П‰ ,
6C
2 C
′
′′
C′
1
C
C 4
A′′ + A′
= О±
П‰ , b12 =
+ О±2 .
П‰1 + П‰2 +
2C
3C
2
C
= b12 П‰ 1 + П‰ 2 +
Now we write down the components of the conformal curvature matrix (5), where for brevity
C′
def
P = A′′ − A′
в€’ О±2
C
C′
C ′′ 1
ω ∧ ω2 + α
ω1 + ω2 ∧ ω4 ,
3C
2C
C ′′
C′ 3
− P ω1 ∧ 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 ,
Pв€’
6C
2C
C ′′ 3
C′
ω ∧ ω4 − α
ω1 + ω2 ∧ ω3 ,
3C
2C
′
A′ C ′′
C ′′
ω 1 ∧ ω 4 + −b′12 +
ω1 + ω2 ∧ ω4 ,
в€’
6C
2C
О¦21
= в€’
О¦31
=
О¦32
=
О¦41
=
О¦42
=
О¦43
=
О¦1
=
– 479 –
Leonid N. Krivonosov, Vyacheslav A. Lukyanov
C ′′
6C
О¦2
=
О¦3
= в€’
О¦4
= в€’
′
A′ C ′′
2C
ω 2 ∧ ω 4 + −b′12 +
C ′′
3C
′
ω3 ∧ ω4 +
′
C′
C
О±
2
ω1 + ω2 ∧ ω4 ,
ω1 + ω2 ∧ ω3 ,
′
C′
C
О±
2
Solving Yang-Mills Equations for 4-metrics of Petrov...
ω1 + ω2 ∧ ω4 .
Petrov matrix Q(О») is
пЈ«
пЈ¬
пЈ¬
пЈ¬
пЈ¬
пЈ¬
пЈ­
C ′′
+О»
3C
iαC ′
2C
αC ′
в€’
2C
iαC ′
2C
в€’
C ′′
в€’
+P +О»
6C
αC ′
2C
пЈ¶
iP
′′
iP
в€’
C
в€’P +О»
6C
пЈ·
пЈ·
пЈ·
пЈ·.
пЈ·
пЈё
2
2
P C ′′ 1 αC ′
P C ′′ 1 αC ′
= 0; type D if C ′′ = 0,
= 0;
+
+
C
2
C
C
2
C
type III if C ′′ = 0, αC ′ = 0; type N if C ′′ = 0, αC ′ = 0, P = 0; type O when C ′′ = 0, αC ′ = 0,
P = 0.
Yang-Mills equation for the external form в€—О¦3 gives
It has type II when C ′′ = 0,
′′
C ′′
C
в€’
C ′′
C
1
2
2
(14)
= 0.
Yang-Mills equation for the external form в€—О¦4 results
О±
C ′′
C
′
C ′′
C
′
C′
1
в€’
C
2
в€’
2
3
2
C ′′
C
C ′′ C ′
C2
=
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
C ′ ′′′
1 C′
A′′ +
в€’
в€’ A(4) в€’
A +
2
C
2 C
3C
(17)
3
2
2C ′′
C′
1 C′
C ′′ C ′
C ′′′
′
2
= 0.
A +О±
в€’
в€’
+
+ в€’
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
C ′ = βC 2 + γ
2
3
(18)
,
where ОІ and Оі are constants.
Put О± = 0. Then, eliminating the third derivative from (15) and (16), we obtain
C ′′
C
C ′′
4
в€’
C
3
C′
C
– 480 –
2
= 0.
(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=
О», Вµ = const.
(20)
3,
(О»z + Вµ)
This corresponds to (18) at γ = 0. Substituting (20) to (17), we obtain the differential equation
of Euler type
25О»3
О± 2 О»2
1
3О»
7О»2
′′
′
в€’ A(4) +
A′′′ −
A
в€’
A
+
2
3
2 = 0.
2
О»z + Вµ
2 (О»z + Вµ)
2 (О»z + Вµ)
(О»z + Вµ)
Its general solution
A = Оµ1 + Оµ2 ln z +
Вµ
Вµ
+ Оµ3 + Оµ4 ln z +
О»
О»
z+
Вµ
О»
6
+
О±2
Вµ
z+
32
О»
2
(21)
,
Вµ
where Оµ1 , Оµ2 , Оµ3 , Оµ4 are arbitrary functions of t. By making the change of variable z + в†’ z, we
О»
obtain the п¬Ѓnal solution
A = Оµ1 + Оµ2 ln |z| + (Оµ3 + Оµ4 ln |z|) z 6 +
О±2 2
z ,
32
C=
1
(22)
3,
(О»z)
depending on four arbitrary functions of the variable t. Since C ′′ = 0 and P
C ′′ 1
+ О±
C
2
C′
C
2
= 0,
this solution gives the metric of type II.
Although the solution (22) is obtained for О± = 0, it is also a solution in the case О± = 0, but
it is not a general solution, because equation (18) besides the solution (20) at Оі = 0, has other
non-elementary solutions at Оі = 0. But in the latter case we cannot п¬Ѓnd explicit solutions of the
equation (17). Solution (22) in the case of О± = 0 gives the metric of type II.
Now we consider the second possibility of the equality (19), C ′′ = 0, which is equivalent to
C = О»z + Вµ, where О», Вµ = const. Substituting this in (17), we again obtain the equation of Euler
type
1
О»
О»2
О»3
О± 2 О»2
в€’ A(4) в€’
A′′ −
A′ +
A′′′ +
2
3
2 = 0.
2
О»z + Вµ
2 (О»z + Вµ)
2 (О»z + Вµ)
(О»z + Вµ)
In its general solution, we replace z +
Вµ
О»
with z and get
A = Оµ1 + Оµ2 ln |z| + Оµ3 + Оµ4 ln |z| +
О±2 2
ln |z| z 2 ,
4
C = О»z.
(23)
At α = 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 = Оµ1 (t) + Оµ2 (t) z + Оµ3 (t) z 2 + Оµ4 (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) О±
=
2) О±
=
1
0, C = const, A = в€’ О±2 z 2 + Оµ1 (t) z + Оµ2 (t) ;
2
Вµ
+ Оµ2 (t) .
0, C = О»z + Вµ, A = Оµ1 (t) ln z +
О»
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
u2
7uy
в€’ fyyyy + ufyyy +
fyy +
в€’
2
3
2
2
uyyy
5 (uy )
5uyy u
+
f в€’ ht
в€’
в€’
2
6
6
11
(uyy в€’ uy u) fy +
6
uyy
uy u
= 0.
+
12
8
(24)
в€љ
Equation (7) allows reduction of order uy = в€’ 23 u2 + О» u, where О» = const. If О» = 0 we obtain
the solution in terms of elementary functions
3
,
2y + Вµ
u=
(25)
Вµ = const.
From (3) we obtain
3
g = Оі (2y + Вµ) 2 ,
(26)
Оі, Вµ = const.
Substituting (25) to (24), we obtain the equation of Euler type
fyyyy в€’
37
154
324
ht
6
fyyy +
2 fyy в€’
3 fy +
4f =
3.
2y + Вµ
(2y + Вµ)
(2y + Вµ)
(2y + Вµ)
2 (2y + Вµ)
Its general solution
f=
3
2y + Вµ
3
ht + (Оµ1 + Оµ2 ln |2y + Вµ|) (2y + Вµ) + (Оµ3 + Оµ4 ln |2y + Вµ|) (2y + Вµ) 2 ,
32
(27)
Оµi 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
П€ = 2dt (dx + A (x, z) dt) + C(z)eв€’О±t dy
2
+ dz 2 ,
О± = const
Yang-Mills equations are very complicated, but all the same they are solved.
Components of the conformal connection matrix Ω are
1
+ A dt + dx, П‰ 3 = Ceв€’О±t dy, П‰ 4 = dz,
2
1
в€’ A dt в€’ dx, П‰ 2 =
2
П‰1
=
П‰12
= в€’Ax П‰ 1 + П‰ 2 , П‰34 = в€’
Cz 3 3
П‰ , П‰1 = П‰23 = в€’О±П‰ 3 , П‰14 = П‰24 = в€’Az П‰ 1 + П‰ 2 .
C
The components of the Ricci tensor and its trace are
Cz
Az , R11 = R12 + Axx , R22 = R12 в€’ Axx ,
C
Cz
Czz
= R24 = в€’Axz в€’ О± , R = в€’2Axx + 2
.
C
C
R12
= Azz + О±2 в€’ О±Ax +
R33
= R44 =
Czz
, R14
C
The coefficients of Pfaffian forms ωi :
b12
=
1
2
Azz + О±2 в€’ О±Ax +
Cz
1
Az , b14 = b24 = в€’
C
2
– 482 –
Axz + О±
Cz
C
.
(28)
Leonid N. Krivonosov, Vyacheslav A. Lukyanov
Solving Yang-Mills Equations for 4-metrics of Petrov...
1
Czz
1
Czz
Czz
1
Axx +
+ b12 , b22 = в€’ Axx в€’
+ b12 , b33 = b44 =
+ Axx ,
3
6C
3
6C
3C
6
Pfaffian forms ωi :
b11
=
Cz
1
О±Cz
1
Czz
1
П‰1 +
Azz + О±2 в€’ О±Ax + Az
Axz +
П‰4 ,
П‰1 + П‰2 в€’
Axx +
3
6C
2
C
2
C
Cz
1
О±Cz
1
1
Czz
П‰2 +
Azz + О±2 в€’ О±Ax + Az
Axz +
П‰4 ,
= в€’
П‰1 + П‰2 в€’
Axx +
3
6C
2
C
2
C
1
О±Cz
Czz
1
1
Czz
=
Axz +
П‰1 + П‰2 +
+ Axx П‰ 3 , П‰4 = в€’
+ Axx П‰ 4 .
3C
6
2
C
3C
6
=
П‰1
П‰2
П‰3
Then we compute the components of the conformal curvature matrix О¦. For brevity we
Czz
Cz
Cz
def
def
def
denote S = Axx в€’
, T = Axz в€’ О± , L = Azz в€’ О±2 + О±Ax в€’ Az .
C
C
C
1
1
О¦21 =
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
О¦1 =
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 ,
О¦2 =
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
− (b12 )x ω 1 ∧ ω 2 + P − Axxz
4
1
− (b12 )x ω 1 ∧ ω 2 + P + Axxz
О¦2 =
4
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 .
О¦4 =
2
6
In the last four formulas we denoted
О¦1
=
1
1
Axxx A в€’
3
2
1
1
Axxx A +
3
2
1
ω 1 + ω 2 ∧ ω 4 + Sz ω 1 ∧ ω 4 ,
6
1
ω 1 + ω 2 ∧ ω 4 − Sz ω 1 ∧ ω 3 ,
6
1
Czz
1
в€’ (b12 )z + Axxz A + Ax b14 ,
Az Axx +
2
C
2
2
1
C
1
C
О±
Czz
1
def
z
z
+
в€’
в€’ Axx + Axxx ,
K = в€’ Axxx A + Axz
2
6
2
C
2 C
C
12
2
1
О±C
О±C
1
1
def
zz
M =
Axzz +
в€’ 2z в€’ Axxx A + Axxx .
2
C
C
3
12
def
P =
Petrov matrix Q(О») is
1
пЈ¬ в€’3S + О»
пЈ¬
i
пЈ¬
пЈ¬
T
пЈ¬
2
пЈ­
1
в€’ T
2
пЈ«
i
T
2
1
1
S+ L+О»
6
2
i
L
2
– 483 –
1
в€’ T
2
i
L
2
1
1
Sв€’ L+О»
6
2
пЈ¶
пЈ·
пЈ·
пЈ·
пЈ·.
пЈ·
пЈё
Leonid N. Krivonosov, Vyacheslav A. Lukyanov
Solving Yang-Mills Equations for 4-metrics of Petrov...
It has type II when S = 0, type III when S = 0, T = 0, subtype N when S = T = 0, L = 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 +О±)в€’
в€’
в€’
в€’
+
+
3
3
3C
6
6C
6C 2
3 C
2
в€’
Czz Cz2
= 0. (30)
6C 3
Equalities (29) mean that
(31)
Axxx = ОІ = const.
Taking into account these equations, Yang-Mills equations for the external form в€—О¦1 provide two
new conditions
1
Cz
Cz
5
в€’ Azzzz + Axxzz A в€’ О±ОІA + Axxz A
+ Az в€’ Axzz Ax + О±Axzz в€’ Azzz +
2
C
3
C
2
C
2Cz
О±
5 2
C
C
z
z
zz
+Axx
Azz + Az
+ Ax в€’ О± в€’ A2xz в€’ Axz Ax
+ О±Axz
в€’ О±Ax
+
3
3C
3
6
C
C
3C
2
2
3
Czz Cz
Cz
Cz
2Czz
Czzz
C
2Czz
+ Az
= 0,
+Azz
в€’
в€’
в€’ z3 + О±2
в€’
2C 2
3C
C2
6C
2C
C2
3C
(32)
1
Czz
2
1
О±
Axx Axz + Axz
в€’ ОІAz + Axzzz в€’ Axxz в€’
3 2
6C
3
2
2
(33)
Cz
Cz
Czzz
5Czz Cz
Cz
+ О±Axx
+О±
в€’О±
=
0.
в€’Axz 2 + Axzz
2C
2C
3C
2C
6C 2
Taking into account (31), Yang-Mills equations for the external form в€—О¦3 yield one new
equality
1
ОІ
Cz
О±ОІ
1
Axxzz в€’ Ax + Axxz
в€’
+
6
3
2C
2
3
1
6
Czz
C
1
Cz
1
ОІ
C2
Cz
в€’ Axxzz в€’ Axxz
в€’ A2xx + Ax + zz2 в€’
2
6C
6
3
6C
3C
Czz
C
Czz
C
zz
в€’
2
1
+ A2xx = 0,
6
(34)
О±ОІ
= 0.
6
(35)
and for the external form в€—О¦4 one new equation
+
z
Equation (31) is equivalent to
A=
1 3
ОІx + 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):
f ′′ −
C ′ ′ αβ
1
f +
в€’
C
2
2
C ′′
C
′′
– 484 –
+
1
2
C ′′
C
′
C′
= 0,
C
(37)
Leonid N. Krivonosov, Vyacheslav A. Lukyanov
Solving Yang-Mills Equations for 4-metrics of Petrov...
1 C ′ ′ 2 2 βg αβ
1
f + f в€’
в€’
в€’
3C
3
3
6
6
f ′′ +
C ′′
C
2
2
2βh′
C ′ ′′ 2f g ′ 2αf C ′
1 C′
C ′′
g ′′′
в€’
− αf ′ +
g +
+
+g ′
в€’
2
3
2C
3
3C
6C
2 C
1
C ′ ′′′
в€’ h(4) в€’
h + h′′
2
C
+h′
2C ′′
1
4f
в€’
+
3
3C
2
4f C ′
C ′′′
C ′′ C ′
1 C′
10f ′
+
в€’
+
в€’
2
3
3C
6C
C
2 C
− − g ′2 − g ′ g
αg ′ C ′
C′
+
+g
C
C
αC ′′
2О±f
в€’
3
3C
+
(38)
αC ′′′
5αC ′′ C ′
= 0, (39)
в€’
2C
6C 2
+
+ h 2f ′′ + 2f ′
5О±2 f
+ О±2
3
C′
= 0,
C
2
C′
C
3
в€’
′
C ′′
C
1
3
+
C′
в€’ О±ОІ
C
в€’
2C ′′
+
3C
− g ′′ (g − α)
(40)
2
C′
C
= 0.
Without additional constraints, the system of equations (37)–(40) can not be solved explicitly.
The п¬Ѓrst of these constraints is
C = eОіz ,
(41)
Оі = const = 0.
In this case system (37)–(40) is solved in terms of elementary functions, but the results are
different if β = 0 and β = 0.
At ОІ = 0 we obtain solution
f = Оµ1 +
О±ОІ
z,
2Оі
g=
2f 2
Оі4
в€’
,
ОІ
2ОІ
h=
4 3 О±Оі 3
f в€’
z + Оµ2 .
3ОІ 2
2ОІ
Directly through x and z formula (36) can be written as
A=
ОІ
6
x+
О±
z
Оі
3
+ Оµ1 x +
О±
z
Оі
2
+
2
ОІ
2
(Оµ1 ) в€’
Оі4
4
x+
О±
z
Оі
3
+
4 (Оµ1 )
+ Оµ2 ,
ОІ2
(42)
Оµ1 , Оµ2 , ОІ, Оі are arbitrary constants, but ОІ = 0 and Оі = 0. Formulas (42) and (41) together with
(28) give Yang-Mills metric of type II.
From (41) and ОІ = 0 it follows, that (by virtue of equations (37) and (38)) there are only two
possibilities for the function f (z):
1
f = В± Оі2.
(43)
2
If the sign is + the following solution to the equations (39) and (40) is obtained
g = Оµ1 + Оµ2 z + Оµ3 eв€’Оіz ,
Оµ2 Оµ 3
+ 2 zeв€’Оіz + z
Оі
2
(Оµ3 )
zeв€’2Оіz +
3Оі
2
2
О±Оµ2
Оµ 1 Оµ2
О±2
(Оµ2 )
(Оµ2 )
.
z
в€’
+
+
+
2Оі 2
Оі2
Оі2
2Оі
2Оі 3
h = Оµ4 + Оµ5 eв€’Оіz + Оµ6 eОіz + Оµ7 eв€’2Оіz +
(44)
Here Оµ1 , ..., Оµ7 , Оі are arbitrary constants, but Оі = 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 = Оµ1 + Оµ2 eО»1 Оіz + Оµ3 eО»2 Оіz в€’ О±Оіz,
1
h = eв€’ 2 Оіz
5
Оіz + Оµ7 sin
12
Оµ6 cos
5
Оіz
12
+
(45)
2
2
(Оµ3 )
(Оµ2 )
e2О»1 Оіz + 2
e2О»2 Оіz + Оµ4 + Оµ5 eв€’Оіz +
+ 2
2Оі (О»1 в€’ 3)
2Оі (О»2 в€’ 3)
Оµ2
Оµ3
1
О±Оµ1
+ 2 (О±zОі в€’ О±О»1 в€’ Оµ1 ) eО»1 Оіz + 2 (О±zОі в€’ О±О»2 в€’ Оµ1 ) eО»2 Оіz +
в€’ О±2 z z.
Оі
Оі
Оі
2
19
19
1
1
, О»2 = в€’ в€’
. Formulas
Here Оµ1 , ..., Оµ7 , Оі are arbitrary constants, but Оі = 0; О»1 = в€’ +
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
C = Оµ1 z + Оµ2 ,
h = Оµ9 ln z +
+Оµ8 +
1
8
Оµ2
Оµ1
в€’
z+
Оµ2
Оµ1
f = 0,
(Оµ3 )
36
g = Оµ3 z +
ОІ = 0,
2
z+
2
Оµ2
Оµ1
6
+
Оµ3
16
z+
Оµ2
Оµ1
4
Оµ2
Оµ1
2
+ Оµ4 ln z +
Оµ2
Оµ1
+ Оµ5 ,
2О± + Оµ4 в€’ 2Оµ5 в€’ 2Оµ4 ln z +
2
2
4Оµ6 в€’ 2Оµ7 + О±2 в€’ (Оµ4 ) + 4Оµ7 в€’ 2О±2 + 2 (Оµ4 )
ln z +
Оµ2
Оµ1
Оµ2
Оµ1
+
.
Here Оµ1 , ..., Оµ9 are arbitrary constants, but Оµ1 = 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
A = в€’ Оі 2 x2 + x (Оµ1 в€’ О±Оіz) + Оµ4 + Оµ5 eв€’Оіz + z
2
1
О±Оµ1
в€’ О±2 z ,
Оі
2
C = eОіz .
This gives Einstein metric of type II, which is a particular case of solution (45) at Оµ2 = Оµ3 =
Оµ6 = Оµ7 = 0.
3.
Metric of type III, satisfying Yang-Mills equations
Let’s investigate the metric
2
П€ = 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
в€’ A dt в€’ dx в€’ Bdy, П‰ 2 =
2
1
+ A dt + dx + Bdy, П‰ 3 = Cdy, П‰ 4 = dz,
2
П‰1
=
П‰13
C
B′ 4 2
B′ 3
B 1
П‰ + П‰2 + П‰3 +
ω , ω1 = 0, ω14 = ω24 = −A′ ω 1 + ω 2 −
П‰ ,
=
=
C
C
2C
2C
B′
B ′′ 1
B ′′ 3
= в€’
П‰ , П‰3 =
П‰ 1 + П‰ 2 , П‰1 = П‰ 2 = K П‰ 1 + П‰ 2 +
П‰ + П‰ 2 , П‰4 = 0,
2C
4C
4C
.
П‰34
.
П‰23
– 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 О¦ are:
О¦21
О¦31
О¦41
1
where S =
2
def
B ′′ 1
B ′′ 1
ω + ω2 ∧ ω3 ,
О¦43 =
ω + ω2 ∧ ω4 ,
4C
4C
B ′′ 1
ω ∧ ω2 + S ω1 + ω2 ∧ ω3 − T ω1 + ω2 ∧ ω4 ,
= О¦32 =
4C
B ′′ 3
= О¦42 =
ω ∧ ω4 − S ω1 + ω2 ∧ ω4 − T ω1 + ω2 ∧ ω3 ,
4C
=
..
1
, T =
2
C
− A′′
C
О¦1
def
=
1
О¦2 =
4
+
О¦3
= в€’
. ′
.
B
B′C
+ 2
C
C
.
B ′′
C
B ′′ B ′
− K′
8C 2
,
.
2B ′′ C
+
C2
ω1 + ω2 ∧ ω3 +
ω1 + ω2 ∧ ω4 −
B ′′′ 1
ω + ω2 ∧ ω4 ,
4C
B ′′′ 3
ω ∧ ω4 ,
4C
О¦4 = 0.
Petrov matrix Q(О») is
пЈ«
пЈ¬
пЈ¬
пЈ¬
пЈ¬
пЈ¬
пЈ­
О»
B ′′
в€’
4C
iB ′′
в€’
4C
B ′′
4C
в€’S в€’ iT + О»
iB ′′
4C
T в€’ iS
T в€’ iS
S + iT + О»
в€’
в€’
пЈ¶
пЈ·
пЈ·
пЈ·
пЈ·.
пЈ·
пЈё
..
C
This is a matrix of type III if B = 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 в€—О¦1 and в€—О¦2 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
′′
B
′′
′′
′′
= ОІ0 z 3 + ОІ1 z 2 + ОІ2 z + ОІ3 ,
A =
1
C2
2
9 (ОІ0 ) 6 9ОІ0 ОІ1 5
z +
z +
40
20
(ОІ1 )
3ОІ0 ОІ2
+
8
4
2
z 4 + О±0 z 3 + О±1 z 2 + О±2 z + О±3 ,
where ОІ0 , ОІ1 , ОІ2 , ОІ3 , О±0 , О±1 , О±2 , О±3 , C are arbitrary functions
of t.
..
1 B′
C
Mertic (46) is Einstein metric when B ′′ = 0, A′′ + −
C
2 C
2
= 0.
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 –

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