Small perturbations in general relativity: tensor harmonics of arbitrary symmetry

3 April 2000

Physics Letters A 268 Ž2000. 37–44 www.elsevier.nlrlocaterphysleta

Small perturbations in general relativity: tensor harmonics of arbitrary symmetry R.A. Konoplya Department of Physics, DnepropetroÕsk State UniÕersity, per. Nauchny 13, DnepropetroÕsk, 49050 Ukraine Received 26 January 2000; accepted 24 February 2000 Communicated by P.R. Holland

Abstract We develop a method for constructing of the basic functions with which to expand small perturbations of space–time in General Relativity. The method allows to obtain the tensor harmonics for perturbations of the background space–time admitting an arbitrary group of isometry, and to split the linearized Einstein equations into irreducible combinations. The essential point of the work is the construction of the generalized Casimir operator for the underlying group, which is defined not only on vector but also on tensor fields. As a quick illustration of the general method we consider construction of the basic functions for the case of the three-parameter group of isometry G 3 acting on the two-dimensional non-isotropic surface of transitivity. q 2000 Published by Elsevier Science B.V. All rights reserved.

1. Introduction There are the two widely accepted ways to construct tensor harmonics. The first is based on the construction of the tensor basis, corresponding to a given underlying symmetry of space w1x. The second way is to consider the action of the complete set of the invariant commutative operators on scalar harmonics of space. However each of these methods, being well developed in General Relativity for spaces of high symmetries, such as spherical or pseudospherical w2–4x, encounters serious difficulties when dealing with spaces of low symmetry. In the last few years the perturbation formalism in General Relativity has attracted some attention for a possibility of detecting gravitational waves from astrophysical

E-mail address: [email protected] ŽR.A. Konoplya..

sources by antennas. This stimulated the development of the second order perturbation formalism for Schwarzschild black holes and, in this connection, a review of the first one Žsee w5x and references therein.. In the present work we propose the method of the obtaining tensor harmonics that enormously simplifies calculations for the first order perturbations in General Relativity. In w6,7x, it was proposed the natural generalization of the Casimir operator defined so that the Lie derivatives are put instead of the corresponding infinitesimal operators Žsee formula Ž1. of the present work.. The metric used in the definition admits a group of isometry stipulated by the invariance condition for this operator. The generalized Casimir operator is invariant in the space of tensor functions. This made it possible to formulate correctly and solve the eigenvalue and tensor eigenfunction problems, and, thereby, to construct tensor representations of various groups. However, if we

0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 1 6 0 - 2

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follow this approach ‘directly’, the system of differential equations generated by a given generalized Casimir operator appears with ‘tangled’ components of tensor functions, and thus the direct solving of the tensor eigenfunction problem, in general case becomes impossible. An accurate and natural way to ‘disentangle’ such system of differential equations Ži.e. to separate variables. is to use invariant technique developed when considering decomposition of the tangent bundle of pseudo-Riemannian manifolds into the corresponding subbundles w8x. When at last the set of basic functions for a given symmetry of the background metric are obtained we can expand the small perturbations of a metric in terms of these basic functions, substitute such an expanded metric for that into the linearized Einstein equations, and try to separate variables into them. The paper is organized as follows. In Section 2 we consider the method for constructing of tensor harmonics and for obtaining of the linearized Einstein equations in the form of irreducible combinations. In Section 3 the tensor harmonics are obtained for the case of a three-parameter group on the two-dimensional non-isotropic surface of transitivity.

2. The method Let M n be an n-dimensional manifold and G r is an r-parameter transformation group on M n. Note, that unlike the Lie algebra of the tangent to oneparameter subgroups of the group G r vectors j i , defined only on functions, that of the Lie derivatives defined on tensors of an arbitrary type, are invariant operators Lj i under the general coordinate transformations. Hence it seems natural to define the generalized Casimir operator of the second order as G s g i k Lj i Lj k ,

Ž 1.

where g i k Ž i,k s 1,2, . . . r . are contravariant components of some unknown metric, which is subject to be determined. Since this operator commutes with all the operators Lj i of the representation we have

žL

jj

y1

g

ik

/

' j jg

ik

q C jli

g

lk

q C jlk

il

g s 0,

Ž 2.

where the tensor gy1 s g i k j i m j k

Ž rank < < gy1 < < s s .

Ž 3.

defines the metric on covectors belonging to the surfaces of transitivity M s ; M n Ž s F r, s F n.. Owing to Ž2., the group G r on the surfaces of transitivity is a group of isometries, where j i are the Killing’s vectors. Solutions of the Killing’s Eqs. Ž2. give us the metric for the definition of the generalized Casimir operator Ž1.. It turns out that for semisimple groups that will be enough to consider only constant solutions of Ž2.. In the general case the consideration of the constant solutions of Ž2. will be inadequate for construction of the generalized Casimir operator G, which is non-degenerate on M s. A tensor T of type Ž p,q . on M s will be the eigenfunction tensor of the generalized Casimir operator G, provided the equation GT ' g i k Lj i Lj kT s lT

Ž 4.

is satisfied. In the coordinate form it reads G Tba11...... baqp ' g i k Lj i Lj kTba11...... baqp s lTba11...... baqp ,

Ž 5.

where G Tba11...... baqp ' Ž GT . ba11 .. .. .. ab qp are representations of the generalized Casimir operator acting on the tensor T in an arbitrary vector basis e a Ž e aŽ a s 1,2, . . . , s . and e a, e a Ž e b . s d ba, are vector and co-vector bases on M s respectively., Tba11...... baqp are the components of the tensor T with respect to the basis e a . Since the Lie operators ‘tangle’ components of T, it is difficult to solve the Eq. Ž5. directly. In order to solve the system of Eqs. Ž5. it is necessary to ‘disentangle’ components of the tensor T in the equations, i.e. to diagonalize the operator G. The diagonalization of the generalized Casimir operator G can be realized invariantly, so that the tensor Eqs. Ž5. split into the system of scalar differential equations for irreducible components of the tensor T. Following w8x, we shall say that a split structure s H is introduced on M if the s linear symmetric operators Žprojectors. H a Ž a s 1,2, . . . s . of a constant rank with the properties s

H a P H b s d abH b ;

Ý H a s I,

Ž 6.

as1

where I is the unit operator Ž I P X s I, ; X g T Ž M .., are defined on T Ž M .. Here we consider a linear operator L on the tangent bundle T Ž M . as a tensor

R.A. Konoplyar Physics Letters A 268 (2000) 37–44

of type Ž1,1. for which L P X ' LŽ X . g T Ž M ., ; X g T Ž M .. The product of two linear operators L P H obeys the rule Ž L P H . P X s L P Ž H P X . g T Ž M ., ; X g T Ž M .. An operator H is called a symmetric one if Ž H P X,Y . s Ž X, H P Y ., ; X,Y g T Ž M .. Thus we can obtain the decomposition the tangent bundle T Ž M . and cotangent bundle T ) Ž M . into the Ž n1 q n 2 q . . . qn s . subbundles S a, Sa) , so that TŽ M.

s a s [as 1S ;

)

T ŽM.

s s [as1 Sa)

.

Ž 7.

Arbitrary vectors, covectors, and metrics are decomposed according to the scheme: s

s

X s Ý as1 X a ,

e a s Lba j b

Ž det < < Lba < < / 0 . ,

gy1 s Ý as1 gy1 a ,

Ž 8.

where X a s H a P X, H b P X a s 0, X a P X b s 0, v a s v P H a, v aŽ X b . s 0, Ž a / b .. Using this scheme we can obtain the decomposition of more complex tensors. We shall say that a split structure H s is compatible with a group of isometries if the conditions of invariance of H s are satisfied, i.e. if

Ž i s 1,2, . . . r ;a s 1,2, . . . s . .

Ž 9.

Eqs. Ž6., Ž9. define the invariant projection tensors. In order to construct the projectors we require existence of such a dual vector e a and co-vector e b bases on M s, that e a P e b s d ab ;

H a s ea m e a .

Ž 10 .

From now and on there is no summation on repeating indices a and b. The invariance condition of Ž9. yields

Ž Lj e a . P e b s 0 Ž a / b . . i

Lj i e a s ym ia e a ,

actually satisfies the Killing’s Eqs. Ž2., and thereby will be the inverse metric Ž3.. Now we shall return to the general case of arbitrary bases  e a ,e b 4 . The initial tensor T can be expanded in the series T s Ý A , B TˆBA s Ý A , B TBA eˆAB ,

Ž 16 .

where  eˆAB 4 s  e a1 m PPP m e a p m e b 1 m PPP m e b q 4 is the tensor basis, TˆBA s TBA e AB is the tensor monomial and TBA ' Tba11...... baqp is its component. A s  a1 , . . . ,a p 4 and B s  b 1 , . . . ,bq 4 are collective indices. The sum in Ž16. comprises the complete set of indices A, B. It is easy to show that since the projectors Ha are invariant, the eigenvalue Eqs. Ž4. and Ž5. split into the set of independent eigenvalue invariant equations for monomials GTˆBA ' g i k Lj i Lj kTˆBA s lTˆBA . Using this relation together with Ž12. and Ž13. we obtain

Ž 12 .

G TBA ' g i k Lj i Lj kTBA

where the factors of proportionality m ia are some functions, satisfying the equation

j i m ka y j k m ia s Cijk m aj ,

g a b s Lac Lbd d c d , Ž 15 .

Ž 11 .

Hence it follows Lj i e a s m ia e a ,

Ž 14 .

where the factors Lba satisfy the equations j b Lad q Cbaq Lqd s 0. The integrability conditions of these equations are satisfied owing to the Jacobi identity. Using the last equations it can easily be shown that the tensor gy1 , constructed by the formula

s

g s Ý as1 g a ,

Lj i H a s 0

of the factors m ia, or even all of them in some cases, can vanish. Then the projectors are constructed by means of the invariant basis  e a : Lj i e a s 04 . In particular, for simply transitive groups Ž r s s ., the invariant vector basis  e a4 can be expressed in the form

gy1 s d a b e a m e b s g a b j a m j b ;

v s Ý as1 v a ,

s

39

Ž 13 .

which follows from the integrability condition of Ž12.. Thus, the problem of the construction of the invariant projectors reduces to construction of the dual vector  e a4 and covector  e b 4 bases satisfying the corresponding systems of equations in Ž12.. Some

s g i k Ž j i y f iAB .Ž j k y f kAB . TBAs lTBA .

Ž 17 .

Here p

q

f iAB s Ý ks1 m ia k y Ý ns1 m ib n , A s  a1 , . . . ,a p 4 ,

B s  b 1 , . . . ,bq 4 .

Ž 18 .

In order that the tensor Eq. Ž4. could go over into the invariantly split Eqs. Ž17. for the irreducible compo-

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nents TBA , we must make a change T ™ TBA ; Lj i ™ Lj i s j i y f iAB . The Eqs. Ž17. can be rewritten in the form G TBA s K y 2g i kf iAB j k y g i k j i f kAB qg i kf iAB f kAB TBA s lTBA ,

Ž 19 .

ik

where K s g j i j k is the standard Casimir operator defined in the space of scalar functions. The solutions of the Eqs. Ž19. and Ž12., Ž13. give us the basic tensor functions TˆBA s TBA eˆAB in the space of a tensor representation of the group G r Žor, in other words, tensor harmonics.. Note that if there exists the invariant basis Ž14., then the generalized Casimir operator Ž4. with respect to this basis reduces to the standard Casimir operator K, and in order to construct the tensor basis of representation that will be enough to determine the basis of representation in the space of scalar functions. We proceed to apply this method to small perturbations in General Relativity. Let g mn be the background metric admitting some group of isometry generated by the set of arbitrary Killing vectors j i , Lj i g mn s 0, and hmn the perturbation. Small perturbations satisfy the linearized Einstein equations d Rmn s y8p Gd Smn , where Smn s Tmn y 12 g mn Tll. The perturbation of the Ricci tensor d Rmn satisfies the tensor relation w9x

d Rmn s 12 g lr hl r ; m ;n y hrm ;n ; l y hrn ; m ; l q hmn ; r ; l . Ž 20 . The perturbation of the metric and Ricci tensor, being tensors, can be written with respect to the basis constructed in accordance with Ž11., Ž12. and extended to the full manifold M 4 h s h i k e i m e k q h js Ž e r m e s q e s m e j . X

q h s sX e s m e s , i

Ž 21 .

k

r

s

s

j

d R s d R i k e m e q d R js Ž e m e q e m e . X

q d R s sX e s m e s ,

Ž 22 .

where i,k, j, . . . are the transversal indices, s, sX denote vectors on the surface of transitivity on which the underlying group of isometry acts. The basis Ž e i ,e s . s e a, Ž a s i, s s 0,1,2,3. we shall call a representation basis. The relations Ž21., Ž22. represent the expansion of these tensors in the terms of the

irreducible components d R i k , d R js , d R s sX , h i k , h js , h s sX . These components for the perturbation of the metric can be found by using the above procedure. To find them for the perturbation of the Ricci tensor we need to write the tensor relation Ž20. with respect to the representation basis. For this purpose one should substitute the components hmn for their ‘irreducible representatives’ h jk , h js , h s sX , and use the Christoffel symbols Gacb : Ž=e ae b s Gacb e c . with respect to the representation basis when dealing with the covariant derivatives in Ž20. or with the formula 2 d R ab s g a b ea eb h m n y en eb h m a ye m e b h n a q e b Ž Ž Gmln q Gn lm . h l a q Ž Galn y Gn la . h l m q Ž Galm y Gmla . h l n . qGalb Ž e n h m l q e m h n l y e l h m n . qGmlb Ž e n h l a q e l h n a y e a h l n . y Ž Gmkb Ž G kln q Gn lk . q Gn kb Ž Gmlk q G klm . . h l a qGn kb Ž e m h k a q e k h m a y e a h k m . q Ž Gmkb Ž G kla y Galk . q Gakb Ž G klm y Gmlk . . h l n yGakb Ž Gmln q Gn lm . q Ž Gakb Ž Gn lk y G kln . qGn kb Ž G kla y Galk . . h m l q e m e n h y Gmln e l hs0,

Ž 23 .

where a,b,m,n,k,l, . . . s 0,1,2,3 are indices associated with the basis vectors e 0 ,e1 , . . . , so that ga b , G bac , and h a b s h b a are the background metric, the Christoffel symbols, and the perturbations determined with respect to the basis e a Ž a s i, s s 0,1,2,3.. When considering space–times with matter one should express the source term Smn in the irreducible form as well. In order to simplify the form of the perturbations h a b one can use the gauge freedom, expanding the gauge vector in terms of the obtained basic functions belonging to the same eigenvalue as the corresponding perturbations of the metric: j s j c e c , where j c, being proportional to the basic functions, are adjusted to simplify h a b . Then the gauge transformations read hXa b s h a b q Lj c e ch a b .

Ž 24 .

R.A. Konoplyar Physics Letters A 268 (2000) 37–44

3. Tensor harmonics for G 3 groups on M 2 surfaces of transitivity with non-isotropic Killing vectors The method, being general, can easily be applied to construction of tensor harmonics for fields of gravity admitting an arbitrary group of isometry. Consider here as a quick illustration of the method one of the most physically interesting cases, including flat and spherical symmetries, when a threeparameter group acts on a two dimensional surface of transitivity. As is known the orbits M 2 of the group of isometry G 3 are the spaces of the constant Gauss curvature K Žand the space–time M 4 is of type either D or 0 by Petrov w11x.. Fields of gravity admitting the G 3 group of isometry acting on the two-dimensional non-isotropic surfaces of transitivity can be divided into the six canonical types: G 3VI, G 3VII ŽK s 0; the signatures on M 2 are qy and qq respectively., and G 3VIII and G 3 IX Žfour types. w10,11x. From now and on we shall denote an arbitrary set of coordinates as x, y, z. (a) G3 VI and G3 VII on M 2 . The Killing vectors have the form: j 1 s Ex , j 2 s E y , j 3 s x E y q ´ yEx , where ´ s "1 for the groups G 3VI and G 3VII respectively. They can be expressed in a more appropriate form in the cylindrical coordinates on Euclidean Žfor G 3VII . and pseudo-Euclidean Ž G 3VI . planes which we shall denote in both cases as r, w . y1

j 1 s ycos wEr q r

sin wEw ,

j 2 s ysin wEr y ry1 cos wEw ,

where c s 1 for G 3VI and c s i for G 3VII, can be chosen as the representation one on the surface of transitivity, for it satisfies Ž11., Ž12. at m13 s yc, m23 s c. In this case the construction of tensor harmonics reduces to that of scalar harmonics. The latter are the eigenfunctions of the operators:

j 3 tml s yEw tml s yi m tml ,

Ž 28 .

and Ktml s Ž j 12 y j 22 . tml s y Ž Er2 q ry1Er y ry2Ew2 . tmls y l2 tml .

Ž 29 .

Ktml s Ž j 12 q j 22 . tml s Ž Er2 q ry1Er q ry2Ew2 . tmls y l2 tml .

Ž 30 .

From Ž28 ., Ž29 ., Ž30 . it follows tml s Žyc . m e c mw f l Ž r . where f lŽ r . satisfies the corresponding equations

Ž Er2 q ry1Er y l2 y m2 ry2 . f l s 0, Ž Er2 q ry1Er q l2 y m2 ry2 . f l s 0.

Ž 31 .

The solution of the first equation in Ž31. is the McDonald function f l s KmŽ l r ., and of the second is the cylindrical function f l s ZmŽ l r . w12x. Thus the scalar harmonics can be expressed in the form m

tml s Ž yi . e mw Ž C1 Km Ž l r . q C2 Ky m Ž l r . . , m

tml s Ž yi . e i mw Ž C1 Zm Ž l r . q C2 Zy m Ž l r . . .

Ž 32 .

j 3 s yEw .

Ž 25 .

It is important for further application that the particular solutions of the second equation in Ž31. are the Hankel functions of the first and second orders. The tensor symmetric harmonic of weight l and type Ž0,2. are expressed in the form

j 3 s yEw .

Ž 26 .

T lm s TiŽklm . e i m e k q TjsŽ lm . Ž e j m e s q e s m e j .

j 1 s ycosh wEr q ry1 sinh wEw , j 2 s sinh wEr y ry1 cosh wEw ,

41

X

These groups are integrable and not semi-simple; the corresponding Cartan tensors are degenerate. The solution of the Killing equations in the cylindrical coordinates gives the metric for construction of the Casimir operator: g 11 s yg 22 s 1, for G 3VI and g 11 s g 22 s 1, g 33 s 0 for G 3VII. The basis e1 s e c w Ž Er q cry1Ew . ,

e 2 s eyc w Ž Er y cry1Ew . ,

Ž 27 .

. s X q TsŽslm e mes ,

Ž 33 .

where the basic vectors e i are invariant under the group transformations, and the basis e s Ž s, sX s 1,2., on the surface of transitivity dual to that defined by Ž27. is r r e 1 s 12 eyc w dr q d w , e 2 s 12 e c w dr y d w . c c Ž 34 .

ž

/

ž

/

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42

The irreducible components can be written in the form TiŽklm . s h i k t Ž lm . ; TjsŽ lm . s h js t Ž lm . ; . . Ž lm . X X TsŽslm s hŽslm . s t

Ž 35 .

From the above and the well-known recurrent relations for scalars w12x we obtain the corresponding relations for tensors. (b) G3 VIII and G3 IX on M 2 . Here we have the following sets of the Killing vectors:

j 1 s cosh yEx y sinh ycoth x E y , j 2 s sinh yEx y cosh ycoth x E y ,

j 3 s Ey

Ž 36 .

j 1 s ycos yEx q sin ycoth x E y , j 2 s sin yEx q cos ycoth x E y ,

j 3 s yE y ,

Ž 37 .

j 3 s yE y ,

Ž 38 .

j 1 s cosh yEx y sinh ycot x E y , j 2 s sinh yEx y cosh ycot x E y ,

j 3 s yE y .

Ž 39 .

The groups generated by these operators are semisimple and non-integrable. The Cartan tensor is non-degenerate in all these case, and g i k s diag Ž y 1,1,1 . , g i k s diag Ž 1,1,y 1 . , g i k s diagŽ1,1,1., g i k s diagŽy1,1,1.. Go over from j 1 , j 2 to the creation and annihilation operators Hs s e s y Ž sEx q a coth x E y . ,

Ž 40 .

where s s "1, a s y1 and s s "i, a s y1 for the groups generated by the operators Ž36. and Ž37. respectively, and Hs s e s y Ž sEx q a cot x E y . ,

Ž 41 .

where s s "ı, a s y1 and s s "1, a s y1 for the groups generated by the operators Ž38. and Ž39. respectively. The operators L H S are the creation and annihilation operators for tensor functions TˆBA , which, in their turn, are the eigenfunctions of the operator L H 3 . In the spirit of the book w12x we can show that there l. A is the set of tensor functions TŽŽm. B for which l. A Žl. A LH 3 TŽŽm. B s mTŽ m. B

(

Ž 43 .

where l is the weight of the representation. Herewith the tensor eigenfunction Eqs. Ž5. can be written in the form l. A Žl. A G TŽŽm. yG B s l Ž l q 1 . TŽ m. B ,

Ž 44 .

where G is the generalized Casimir operator. Suppose that we need to consider the tensor symmetric harmonics of type Ž2,0. and of weight l, which we shall denote T l. With respect to the initial differential basis e z , dx 1 s dx, dx 2 s dy they can be written in the form T l s TzŽzl . e z m e z q TrŽal . Ž e z m dx a q dx a m e z .

j 1 s ycos yEx q sin ycot x E y , j 2 s sin yEx q cos ycot x E y ,

l. A Žl. A LH S TŽŽm. B s l Ž l q 1 . y m Ž m q s . TŽ m. B ,

Ž m s yl,y l q 1, . . . ,l . ; Ž 42 .

q TaŽbl . dx a m dx b .

Ž 45 .

Note, that the covector e z s dz is invariant with respect to these groups. In order to split the system of Eqs. Ž44. for the tensor Ž45., into the irreducible components, one should go over into the basis of one-forms on the surfaces of transitivity M 2 satisfying the condition Ž12.. It turns out that the covectors e s s dx q s sin xdy ; e s s dx q s sinh xdy

Ž s g "1,ı .

Ž 46 .

are required for the groups generated by the operators Ž36., Ž37. and Ž38., Ž39. respectively. The condition of invariance of a split structure Ž9. for the Lie operators associated with the vectors Ž40. stipulates the corresponding relations

m ssX s

ssX e i s

m 3s s 0

X

sin x

y

;

m ssX s

ssX e i s

X

y

sinh x

;

Ž s, sX s "1. .

Ž 47 .

By using Ž46. the relation Ž45. can be rewritten in the form T l s TzŽzl . e z m e z q TzŽsl . Ž e r m e s q e s m e z . X

q TsŽslX. e s m e s ,

Ž 48 .

where the sum on s s "1 is implied. Then, if we suppose l X, TzŽzl . s h z z t l ; TzŽsl . s h z t sl ; TsŽX ls. s ht sqs

Ž 49 .

R.A. Konoplyar Physics Letters A 268 (2000) 37–44

where h z z , h z , h are functions of z, then for the function t nl Ž n s 0, . . . , s, . . . , s q sX . we obtain

½

E

1

E sinh x

sinh x E x 1 q

2

sinh x

ž 5

Ex

E2 Ey

E

2

y 2 in cosh x

Ey

y n2

/

ql Ž l q 1 . t nl s 0

Ž 50 .

for groups generated by the Killing vectors Ž37., and exactly the same equation with ordinary cos x,, sin x instead of the hyperbolic ones for Ž38.. For the other two cases we obtain

½

E

1

E sinh x

sinh x E x 1 y

2

sinh x

ž 5

Ex

E2 Ey

E

2

y 2 n cosh x

Ey

q n2

/

ql Ž l q 1 . t nl s 0

Ž 51 .

for Ž36. and exactly the same equation with the corresponding ordinary trigonometric functions for Ž39.. Owing to Ž42. we find the general solutions for groups generated by the Killing vectors Ž36., Ž37.,and Ž38., Ž39. respectively t nl s Sm Cm e cm y B ln m ,

Ž 52 .

and t nl s Sm Cm e cm y Pnlm ,

½

d

sinh x dx

sinh x

5

equations are the functions B ln m Žcosh x . and PnlmŽcos x . and, which are called the Legendre functions and Legendre polynomials respectively w12x. Recurrent relations for them follow from Ž42.,Ž43.. It is obvious that in a similar fashion we can treat all the other types of gravity in Petrov classification scheme. 4. Conclusion One of the question that remains is in which cases variables into the linearized Einstein equations will be separable after doing the above procedure? For an Abelian underlying group of symmetry owing to the Lie theorem it becomes obvious that the splitting of the linearized Einstein equations into irreducible combinations eventually leads to separations of the corresponding variables. However in the general case we cannot know beforehand whether variables will be separated or not. One more question for further investigation: for what classes of space, excepting trivial cases of the invariant basis, the generalized Casimir operator coincides with an ordinary one? Tensor harmonics obtained in the previous section include those of the spherical and a waste class of cylindrical symmetries. Following the present method we can easily obtain the linearized Einstein equations in terms of the irreducible combinations for these symmetries and thereby to separate the corresponding variables Žfor such consideration of the cylindrical waves as perturbations of the flat space– time w13x and of the Regge-Wheller spherical waves w2,14x see w15x..

Ž 53 .

where c s 1 for Ž36., Ž39. and c s i for Ž37., Ž38., Cm are some coefficients. Here the function B ln m satisfies the corresponding differential equation following from Ž50.: 1

43

m2 y 2 mn cosh x q n2

d y dx

ql Ž l q 1 . B ln m s0,

sinh2 x

Ž 54 .

and Pnlm satisfies exactly the same equation with ordinary cos x,, sin x. The solutions of the obtained

Acknowledgements I would like to acknowledge Professor V.D. Gladush for proposing the problem, much encouragement and reading the manuscript. I am also much indebted to Dr. S. Ulanov for computer help. References w1x E.P. Wigner, Group theory, New York, 1959. w2x T. Regge, J.A. Wheller, Phys. Rev. 108 Ž1957. 1063. w3x E.M. Lishitz, J. Phys. ŽUSSR. 10 Ž1946. 116.

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