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УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА
Физико-математические пауки
UDK 514.83
ON THE EINSTEIN EQUATION ON LORENTZIAN MANIFOLDS WITH PARALLEL DISTRIBUTIONS OF ISOTROPIC LINES
^4.5. Galae.v
Abstract
Some recent, results about Einstein Lorentzian manifolds that, admit, parallel distributions of isotropic lines are reviewed. We find all liolonomy algebras of such manifolds and describe special coordinates that, allow us to simplify the Einstein equation. Examples in dimension 4 are considered.
Key words: Lorent.zian manifold, Einstein equation. Walker metric, liolonomy algebra, recurrent, light-like vector field, Pet.rov classification.
Introduction
Let (M, g) be a Lorentzian manifold admitting a parallel distribution of isotropic lines. On any such manifold (of dimension n + 2, n > 0) there exist local coordinates v, x1,..., xn, u, the so-called Walker coordinates, such that the metric g has the form
g = 2dvdu + h + 2Adu + H ■ (du)2, (1)
where h = hj (x1,..., xn, u) dx® dxj is a u-dependent family of Riemannian metrics, .A == .A® (x ,. .., x , u ) dx® is a u-dependent family of one-forms, and H is a local func-
d
tion on M ill. The vector field dv = — defines the parallel distribution of isotropic
dv
lines. Lorentzian manifolds with this property are of interest both in differential geometry and theoretical physics (e.g. fl 6]). Recently in [5] G.W. Gibbons and C.N. Pope
(M, g)
physical interpretation for its solutions.
In Section 1 the Einstein equation for the metric (1) is rewritten as a system of
partial differential equations with respect to the components h, A and H defining the g
(M, g) A = 0
equation. In Section 3 we consider examples in dimension 4. In Section 4 all liolonomy
(M, g)
1. The form of the Einstein equation
(M, g) g
Ric = Ag, A e R,
where Ric is the Ricci tensor of the metric g. The number A e R is called the cos-mologtcal constant. If A = 0, i.e. Ric = 0, then (M,g) is called Ricci-flat or vacuum Einstein.
A special example of the metric (1) is the metric of a pp-wave
n
g = 2dvdu + ^(dx^2 + H • (du)2, dvH = 0. (2)
i= 1
If such metric is Einstein, then it is Ricci-flat. and it is Ricci-flat if and only if
n
Ed2H = 0.
i= 1
In [5] it is shown that the Einstein equation for a Lorentzian metric of the form (1) implies
H = Av2 + vHi + Ho, (3)
where H0 and H1 do not depend on v. Furthermore, in [5] it is proved that Eq. (2) is equivalent to Eq. (3) and the following system of equations:
AHo - - /■ '/:, - 2AidiH1 - H-iV'Aj + 2AA% - 2ViA; + 2
+ + + ^/''/';// = 0, (4)
VjFij + diHi - 2AAi + Vjhij - di(hjkhjk) = 0, (5)
AH1 - 2AViAi + Ahij hij = 0, (6)
Ricij = Ahij, (7)
where AH0 = hij (didj H0 — rj dk H0) is the Laplace - Beltrami operator of the metrics h(u) applied to H0, Fij = diAj — djA4 are the components of the differential of the one-form A = Aidxi. A dot denotes the derivative with respect to u.
2. Simplification of the Einstein equation
The Walker coordinates are not defined canonically and any other Walker coordinates v, x1,... ,xn, u such that d^ = dv are given by the following transformation (see [6. 7]):
v = v + f (x1,..., xn, u), xl = XV(x1,..., xn, u), V = u + c. Using this, in [7] the following theorem is proved:
Theorem 1. Let (M, g) be a Lorentzian manifold, of dimension n +2 (n > 2) admitting a parallel distribution of isotropic lines. If (M, g) is Einstein with the nonzero cosmological constant A, then there exist local coordinates (v, x1,..., xn, u) such g
g = 2dv du + hij dxi dxj + (Av2 + H0)(du)2
with dvhij = dvH0 = 0, hj defines a u-dependent family of Riemannian Einstein
A
All ■ '//:; II. (8)
Vj h ij = 0, (9)
hij h ij =0, (10)
Ricij = Ahij, (11)
where hij = duhij . Conversely, any such metric is Einstein.
A=0
a parallel distribution of isotropic lines to the stndy of families of Einstein Riemannian metrics satisfying Eqs. (9) and (10).
g
metric written with respect to some coordinates v, x1,..., xn, u % (1). Since g is an h
v ^ v + f (x1, .. ., xn, u), x® ^ x®, u ^ u.
Then Hi changes to Hi — 2A/. Taking / = l^Hi, w0 new coordinates such that H1 = 0
: X (xx , . . . , XX , XX ),
(12)
We get
Hence, if the equality
A® =
dxj dx®
dx
A, + h
dxk dx-
as =
(13)
holds, then A® = 0. Impose the conditions x® Then for each
set of numbers xk there exists a unique solution x®(X) of the above system of equations. Since the solution depends smoothly on the initial conditions, we may write the solution in the form x®(x1,..., xn, X). The obtained functions satisfy Eq. (13).
(dx® ~ \ (dx® N _ _
Since det ( J 0, we get that det ( ^-y J 0 f°r u rloar uo- Under this
transformation, H1 = H1 =0. We obtain the required transformation.
A=0
(M, g)
isotropic lines and assume that (M, g) ¿s Ricci-flat. Then there exist local coordinates (v, x1,..., xn, u) such that the metric is given as
g = 2dv du + h®j dx® dxj + vH1 (du)2,
where H1 and h®j are smooth functions with hkl = H1 = 0, satisfying the equations:
\hPhij + h:;h:, + V'/':;//
d®Hi + vjh®, - d®(hjkhjk)
AHi Ric®,
0, 0, 0.
(14)
(15)
(16) (17)
Conversely, any such metric is Ricci-flat.
To find the required coordinates it is enough to start with some Walker coordinates v, x1,..., xn, u and to solve the system of equations (13) for transformation (12).
x.
3. Examples in dimension 4
In f8] it is proved that metric (1) for n =2 satisfies the Einstein equation (2) if and only if after a proper choice of coordinates v, x, y, u the metric has the following form:
2
g = -j^dzdz+ (2di> + 2Wdz + 2W dz + (Av2 + H0) du) du, (18)
where
z = x + iy, 2P2 = |A|2P2 = |A| (l + A^ , W = idzL, the function L is R-valued depending on z, z, u and satisfying the equation
AL =
2 p2
2-dimensional sphere (or Lobachevsky space). All such functions are given by
2 2 _ where A = 2Pq dz d~ is the Laplace Beltrami operator of the metric — dzdz of the
L = 2Re^a(lnP0)-ia^, (19)
where ^ = ^(z, u) is an arbitrary function holomorphic in z and smooth in u. The function H0 = H0(z, z, u) can be expressed in a similar way in terms of ^ and another arbitrary function ^i(z, u) holomorphic in z and smooth in u.
In [9] it is shown that the Petrov type of any Einstein metric of the form (1) for n=2
II at generic points.
Example 1 [7]. Let ^>(z, u) be one of c(u), zc(u), z2c(u); then, using Theorem 1, the above metric can be rewritten as
2 ~
g = dz dz + (Av2 + H0) (du)2,
where H0 is a harmonic function, i.e. AH0 = 0.
Although formula (19) gives the complete solution to the Einstein equation, it is not useful for constructing examples of the form obtained in Theorems 1 and 2, since "simple" functions ^ define complicated functions L and metrics (18). For this reason, in [10] another method of finding partial solutions to the Einstein equation (2) is used. First, the following proposition is proved:
Proposition 1. Let (M, g) be a Lorentzian manifold, of dimension 4 admitting a parallel distribution of isotropic lines. If (M, g) is Einstein with the cosmological constant A, then in a neighborhood of each point of M there exist local coordinates v, x, y, u such that the metric g has one of the following forms: 1) if A > 0, then
g = 2di> du + ((dx)2 + sin2 x (dy)2 j +
+ 2 (--^-dx + sinxdxfdy] du + {Av2 + H0)(du)2, (20)
■where H0 and, f are functions depending on x, y, u and satisfying the equations:
As2 f = —2f, (21)
As*Ho = 4A/2 - 2A ( (dxf)2 + —2—(dyf)2 ) , (22)
1
sin2 x
■where AS2 = d2 -\--\—d2 -\-cotxdx is the Laplace Beltrami operator of the sphere
sin x
metric (dx)2 + sin2 x (dy)2 ; 2) if A < 0, then
g = 2dvdu + _1 ((dx)2 + (dy)2) + 2(-dyfdx + dxfdy)du + (Av2 + H0)(du)2, (23)
w/iere Ho and f are functions depending on x, y, u and satisfying the equations:
AL2 f = 2f, (24)
Al2 Hq = —4Af2 — 2Ax2((dx f )2 + (dy f )2), (25)
■where AL2 = x2(d2 + d2) is the Laplace Beltrami operator of the metric -^((dx)2 + +(dy)2) of the Lobachevsky space L2.
A
Partial solutions of Eqs. (21) and (24) can bo obtained by finding symmetries of these equations. This can be done using Maple 12. For example, partial solutions of (24) may be found in the following forms: /(x, y, u) = ^(x, u), /(x, y, u) = ^ u 0 x2 + y2 n
f(x,y, u) = ipy-—--,uj . Consider several examples from [10].
c(u) y x2 y2 Example 2. The functions / =-, c(u)— , c(u)'-- (where c(u) is a smooth
function) are partial solutions of (24). In each case the new coordinates can be chosen in such a way that the metric (23) takes the form:
g = 2dvdu + 2 ((dx)2 + (dy)2) + (Av2 + H0)(du)2,
where H0 satisfies AL2 H0 = 0.
Example 3. The functions f = x2^d H0 = — Ax4y are partial solutions of Eqs. (24) and (25). We get the following Einstein metric:
g = 2dvdu + x2 ((dx)2 + (dy)2) -
— 2x2 dx du + 4xy dy du + (Av2 — Ax4y2) (du)2. (26)
The Lie algebra of Killing vector fields is spanned by the vector fields 3vdv + xdx + ydy — 3ud„, d„.
Consider the transformation with the inverse one given by
v = V, x
x(1 + 3A V V3)-1/3, y = V(1 + 3A V V3)2/3, u = V.
With respect to the obtained coordinates, we get
5 = 2dvdM + ^A ((»«AW«3 + x2(1 + 3A.^)0 "
- 12A(1 + 3Aux3)yu dx dy + + (1 + 3W)2 + (dy)2) + (Av2 + 3AxV + , fK _ ) (dt()2. (27)
(1 + 3Aux3)2 g
Example 4. The function f = c(u) cos x (where c(u) is a smooth function) is a partial solution of (21). In each case the new coordinates can be chosen in such a way that the metric (20) takes the form
g = 2di' du + ^ ((dx)2 + sin2 x (dy)2) + (Av2 + H0)(du)2,
where H0 satisfies AS2H0 =0.
Example 5. The function / = In (tan cosx + 1 is a partial solution of (21). We get the following Einstein metric:
g = 2di' du + ((dx)2 + sin2 x (dy)2) +
+ 2 (cos x — In (cot sin2 x ) dy du + (Av2 + #«) (du)2, (28)
where H0 is a function satisfying (22). We will find the example of such function below. Consider the transformation
_ _ _ / x \ cos x \
v = v, x = x, y = y — Au In tan —--t,—
V ^ 2 J sin2 x J
With respect to the obtained coordinates, we get
'I 4Au2\ . A sin4 x J
i 1 4Au2 \ , N g = 2dvdu + - + —r- (dx)
4u sin2 x
dx dy + ^-(dy)2 + (Ai>2 + H0) (du)2, (29)
where Ho satisfies Ah Ho = -/''/'.;. where h is the Riemannian part of the
above metric. An example of such H0 is Ho = -A —— + In2 (cot - ) . Corii-
\sin2 x V 2))
ing back to the initial coordinates, we get Ho = A • (in (tan cosx + 1 j . The
Lie algebra of Killing vector fields of the metric (29) is spanned by the vector fields
dy,du + A ( --in (tan ) dy. The metric g is of Petrov type D on the set
\sin2 x V 2//
(0, x, y, u)| In (cot cosx — 1 = oj and it is of type II on the complement to this
set. The metric is indecomposable.
Ricci-flat Walker metrics in dimension 4 are found in [11, 12]. They are of the form g = 2dv du + (dx)2 + (dy)2 + 2A1dx du + (-(dxA1)v + H0)(du)2, (30)
u = u.
where A^d Ho satisfy Ai = H0 = 0,
+ dy2Ai = 0, (31)
d2Ho + d2Ho = 2d„dxAi - 2Aid2Ai - (d^Ai)2 + (dyAi)2. (32)
If dxAi = 0 and the metric g is indecomposable, then g is of Petrov type III at generic points [9, 11-13]. If dxAi = 0, then this is a pp-wave. If it is indecomposable, then it has Petrov type N at the point where the cnrvatnro is non-zero [9, 11 13].
Example 6. It is clear that A\ = xy and Ho = ~~ V4) aro t^10 solutions of
(31) and (32). We get the following Ricci-flat metric:
g = 2dvdu + (dx)2 + (dy)2 + 2xydxdu + ^-yv + y^'4 ~ (dw)2.
The Lie algebra of Killing vector fields of g is spanned by the vector field d„. Consider the transformation
v = v, x = xeyu, y = y, X = w. With respect to the obtained coordinates, we get
g = 2dv dw + e-2yu (dx)2 - 2xwe-2yu dxdy + (1 + xVe-2y") (dy)2 +
+ [~yv ~ x2y2e~2yu - ly4 + (d«)2. (33)
4. Holonomy algebras
Recall that any Riemannian manifold (N, h) can be locally decomposed into a product of a flat space and some Riemannian manifolds that can not be further decomposed [14]. In accordance to this, for the tangent space to (N, h) (that can be identified with M", n = dimN) and the holonomy algebra h C so(n) of (N, h), there exists a decomposition
M" = M"0 © M"1 © • • • © M"r (34)
and the corresponding decomposition into the direct sum of ideals
h = {0}© hi ©•••© hr (35)
such that each hi C 5o(n¿) is an irreducible Riemannian holonomy algebra, in particular
/ n ■ \ / n'
it coincides with one of the following snbalgobras of so(iii): so(ni), u 1 ^u
sp ffisp(l), sp , G2 C 50(7), spin7 C so(8) or it is an irreducible symmetric Berger algebra (i.e. it is the holonomy algebra of a symmetric Riemannian manifold and it is different from so(n~¿), u 1 5P ©5P(1))- It is known that if the manifold (N, h) is Ricd-lat, then each hi C so(ni) in the above decomposition is one of so(n.i), su , sp , G2 C so(7), spin7 C so(8). Conversely, if each f)¿ C so(n.¿)
is one of su , sp , G2 C so(7), spin7 C so(8), then (N, h) is Ricci-flat. Next, if (N, h) is an Einstein manifold with A = 0, then each hi C so(ni) coincides with one of so(n~¿), u (t^ i ^p © sp( 1) or with a symmetric Berger algebra, and it
holds no = 0. Conversely, if f) C so(n) is irreducible and f) = sp ffisp(l) or it is
a symmetric Berger algebra, then (N, h) is an Einstein manifold. Thus, Riemannian manifolds with some holonomy algebras are automatically Einstein or Ricci-flat.
( M, g)
be a Lorentzian manifold with a parallel distribution Z of isotropic lines. Without loss
( M, g)
( M, g)
can be identified with the Minkowski space M1'n+1 .Let p, e1,..., en, q be a Witt basis of M1'n+1 such that Rp corresponds to the distribution Z. The holonomy algebra g of (M, g) is contained in the maximal sub algebra of so(1, n + 1) preservi ng Rp,
g C sim(n) =
a G R, X G R", A G so(nU = (R © so(n)) x
The projection j of the holonomy algebra of (M, g) onto so(n) has to be a Riemannian holonomy algebra [16]. In [15] the following two theorems are proved:
( M, g)
1. The holonomy algebra of (M, g) coincides with (R © j) xR", and in the decomposition (35) of j C so(n) at least one subalgebra ( C so(nj) coincides with one of the
Lie algebras so('tii), u ? ^P ffisp(l) or with a symmetric Berger algebra.
2. The holonomy alge bra of (M, g) coincides w ith j xR", and in the decomposition (35) of j C so(n) esch aubalgebra j C so(nj) coincides with one of the Lie algebras so(m), su 5p (j) , G2 Cso(7), 5pin(7) C50(8).
( M, g)
of (M, g) coincides with (R © j) x R", and in the decomposition (35) of j C so(n) each suba,lgebras f)j C so(n.i) coincides with one of the Lie algebras so(n.i), u j ,
sp or with a symmetric Berger algebra. Moreover, in the decomposition (34)
it holds ns+1 = 0.
In [15] an example of a local Einstein (Ricci-flat) metric with each possible holonomy algebra from the above theorems is constructed.
The above two theorems show that if n = 2, i.e. dimM = 4 Mid (M, g) is Ricci-flat, then either g = (R © so(2)) x R2 or g = R2 (the last case corresponds to pp-waves). If (M, g) is Einstein with A = 0, then g = (R©so(2)) x R2. These statements are also proved in [9, 13, 17].
Unlike the case of Riemannian manifolds, it can not be stated that a Lorentzian manifold with some holonomy algebra is automatically an Einstein manifold, but there
( M, g) ( M, g)
a parallel spinor, then it is totally Ricci-isotropic (but not necessary Ricci-flat, unlike in the Riemannian case) [3, 4]. In [15] the following theorem is proved:
( M, g)
same as in Theorem 3. Conversely, if the holonomy algebra of (M, g) is j x R" and in the decomposition SS5) of j C so(n) each subalgebra j C so(nj) coincides with one of the Lie. algebras su j , sp j , G2 C so(7), spin(7) C so(8), then (M, g) is totally Ricci-isotropic.
This research is supported by the grant 201/09/P039 of the Grant Agency of Czech Republic and by the grant MSM 0021622409 of the Czech Ministry of Education.
Резюме
A.C. Галаеи, Уравпепия Эйнштейна па лорепцевых многообразиях с параллельным распределением изотропных прямых.
Приведен обзор недавних результатов исследования лорепцевых многообразий Эйнштейна, допускающих параллельные распределения изотропных прямых. Найдены алгебры голопомии таких многообразий. Описаны специальные координаты, позволяющие упростить уравнение Эйнштейна. Рассмотрены примеры в размерности 4.
Ключевые слова: лорепцево многообразие, уравнение Эйнштейна, метрика Уокера, алгебра голопомии. рекуррентное светоподобпое векторное поле, классификация Петрова.
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Поступила в редакцию 06.12.10
Galaev, Anton Sergeevich PliD in Physics and Mathematics, Scientific Researcher. Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Brno, Czech Republic.
Галаев Антон Сергеевич кандидат физико-математических паук, научный сотрудник отделения математики и статистики факультета естественных паук Университета им. Масарика, г. Брно, Чехия. E-mail: galaevQmath.muni.cz
