Reductive decompositions and Einstein–Yang–Mills equations associatedto the oscillator groupRaúl Durán Díaz, Pedro M. Gadea, and José A. Oubiña Citation: J. Math. Phys. 40, 3490 (1999); doi: 10.1063/1.532902 View online: http://dx.doi.org/10.1063/1.532902 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v40/i7 Published by the AIP Publishing LLC. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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JOURNAL OF MATHEMATICAL PHYSICS VOLUME 40, NUMBER 7 JULY 1999

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Reductive decompositions and Einstein–Yang–Millsequations associated to the oscillator group

Raul Duran DıazInstituto de Fı´sica Aplicada, CSIC, Serrano 144, 28006-Madrid, Spain

Pedro M. GadeaInstituto de Matema´ticas y Fısica Fundamental, CSIC, Serrano 123, 28006-Madrid, Spain

Jose A. OubinaDepartamento de Xeometrı´a e Topoloxı´a, Facultade de Matema´ticas,Universidade de Santiago de Compostela, 15706-Santiago de Compostela, Spain

~Received 22 February 1999; accepted for publication 10 March 1999!

All of the homogeneous Lorentzian structures on the oscillator group equipped witha bi-invariant Lorentzian metric, and then the associated reductive pairs, are ob-tained. Some of them are solutions of the Einstein–Yang–Mills equations.© 1999 American Institute of Physics.@S0022-2488~99!00107-3#

I. INTRODUCTION

The oscillator group has interesting features from the viewpoints of both differential geomand physics~see, for instance, Refs. 1–12!. It was introduced by Streater,1 who named it sobecause its Lie algebra can be identified to that generated by the differential operators asswith the harmonic oscillator problem, acting on functions of a variable. The oscillator group ionly simply connected four-dimensional non-Abelian solvable Lie group which admits ainvariant Lorentzian metric.3,4 Moreover, it is8 an example of homogeneous space–time, whichcausal space satisfies the so-called causal continuity.

Levichev studied in Ref. 5 the oscillator group with the biinvariant Lorentzian metricgiven in Refs. 6 and 8, proving that this group~which geometrically is a Lorentzian symmetrspace and physically is related to an isotropic electromagnetic field! provides a solution of thesourceless Einstein–Yang–Mills equations.

On the other hand, generalizing the classical characterization by Cartan13 of Riemanniansymmetric spaces as the spaces of parallel curvature, Ambrose and Singer14 gave a characterization of Riemannian homogeneous spaces in terms of a~1,2! tensor fieldS, called homogeneousRiemannian structure in Ref. 15, satisfying certain differential equations~see~1! below!. In Refs.16 and 17, we have extended this concept to the pseudo-Riemannian case.

In Ref. 12, all of the homogeneous Lorentzian structures corresponding to a family oinvariant metrics on the general oscillator groups have been obtained, and the reductive dpositions for the non-bi-invariant metrics in that family have been determined.

In this paper we consider the four-dimensional oscillator group equipped with its uLorentzian bi-invariant metric and we determine all the homogeneous Lorentzian structureand all the associated reductive pairs. There appear six types of such pairs, and we prove tof them are solutions of the Einstein–Yang–Mills equations. They have sources exceptparticular case, where the symmetric pair appears as a solution of the sourceless Einstein–Mills equations.

II. PRELIMINARIES

Ambrose and Singer14 gave a characterization for a connected, simply connected andplete Riemannian manifold to be homogeneous, in terms of a~1,2! tensor fieldS, called in Ref. 15a homogeneous Riemannian structure. In Ref. 16 it is defined ahomogeneous pseudo-Riemanni

34900022-2488/99/40(7)/3490/9/$15.00 © 1999 American Institute of Physics

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structureon a pseudo-Riemannian manifold~M,g! as a tensor fieldSof type ~1,2! such that being¹ the Levi-Civita connection andR its curvature tensor, the connection¹5¹2S satisfies theAmbrose–Singer equations

¹g50, ¹R50, ¹S50. ~1!

In Ref. 16 it is proved thatif ~M,g! is connected, simply connected and geodesically completeit admits a homogeneous pseudo-Riemannian structure if and only if it is a reductive homogepseudo-Riemannian manifold, which means thatM5G/H, whereG is a connected Lie groupacting transitively and effectively onM as a group of isometries,H is the isotropy group at a poinoPM , and the Lie algebrag of G may be decomposed into a vector space direct sum of thealgebrah of H and an Ad(H)-invariant subspacem, that is g5h% m, Ad(H)m,m. ~If G isconnected andM is simply connected thenH is connected, and the latter condition is equivalen@h,m#,m!.

Let ~M,g! be a connected, simply connected and geodesically complete pseudo-Riemmanifold, and suppose thatS is a homogeneous pseudo-Riemannian structure on~M,g!. We fix apoint oPM and putm5To(M ). If R is the curvature tensor of the connection¹5¹2S, we canconsider the holonomy algebrah of ¹ as the Lie subalgebra of antisymmetric endomorphisms(m,go) generated by the operatorsRZW , whereZ,WPm. Then, according to the Ambrose–Singconstruction,14,15 a Lie bracket is defined in the vector space direct sumg5 h% m by

@U,V#5UV2VU, U,VP h,

@U,Z#5U~Z!, UP h, ZPm, ~2!

@Z,W#5RZW1SZW2SWZ, Z,WPm,

and we say that (g,h) is thereductive pairassociated with the homogeneous pseudo-RiemanstructureS. The connected and simply connected Lie groupG whose Lie algebra isg acts tran-sitively on M as a group of isometries andM[G/H, whereH is the connected Lie subgroup oG whose Lie algebra ish. The setK of the elements ofG which act trivially onM is a discretenormal subgroup ofG, and the Lie groupG5G/K acts transitively and effectively onM as agroup of isometries, with isotropy groupH5H/K. ThenM is ~diffeomorphic to! the reductivehomogeneous pseudo-Riemannian manifoldG/H.

On the other hand, we consider~see e.g., Ref. 18, p. 134! the Einstein–Yang–Mills equationon a Lorentzian manifold~M,g!,

r 2 12sg1Lg5kT, ~3!

whereg denotes the metric tensor,r the Ricci tensor, ands the scalar curvature ofg, L andk thecosmological and gravitational constants, respectively, andT the stress-energy tensor of the gaufield with field strengthF, given by

T~X,Y!5g21~ i XF,i YF !2 14g~X,Y!iFi2, ~4!

i denoting the interior product.Let g be the Lie algebra with generatorsP,X,Y,Q, and brackets

@X,Y#5P, @Q,X#5Y, @Q,Y#52X.

The corresponding simply connected Lie groupG is called theoscillator group.

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III. LORENTZIAN HOMOGENEOUS STRUCTURES AND REDUCTIVE DECOMPOSITIONS

Consider the bi-invariant Lorentzian metricg on the oscillator groupG given in the basis$P,X,Y,Q%, by

g5S 1

1

1

1

D . ~5!

Let $h,a,b,j% be the dual basis to$P,X,Y,Q%. The Levi-Civita connection¹ of g is determinedby 2g(¹WU,V)5g(@W,U#,V)2g(@U,V#,W)1g(@V,W#,U) for all U,V,WPg. So, we obtainthat ¹ is given by

¹WP50,

¹WX52 12b~W!P1 1

2j~W!Y,

¹WY5 12a~W!P2 1

2j~W!X,

¹WQ5 12b~W!X2 1

2a~W!Y,

for everyWPg. The nonvanishing components of the curvature tensor field, for which we athe conventionRWUV5¹@W,U#V2¹W¹UV1¹U¹WV, are given by

RQXX52 14P, RQXQ5 1

4X, RQYY52 14P, RQYQ5 1

4Y.

Thus, the only nonvanishing component of the Ricci tensor isr (Q,Q)5 12 and the scalar curvatur

is s50.In Ref. 12 all the homogeneous Lorentzian structures on the general oscillator grou

determined by solving the Ambrose–Singer equations~1!. As a consequence, we have all thhomogeneous Lorentzian structures on the four-dimensional oscillator group.

Proposition: All the homogeneous Lorentzian structures on~G,g! are given by

S5u ^ ~a∧b!1r ^ ~a∧j!1s ^ ~b∧j!, ~6!

wherer, s, andu are left invariant 1-forms on G satisfying the conditions

¹u50,

¹r5s∧u2 12a ^ u1 1

2j ^ s, ~7!

¹s52r∧u2 12b ^ u2 1

2j ^ r,

with ¹5¹2S.Proof: Expression~6! follows from the equations¹g50 and¹R50 in ~1!. Conditions~7! are

equivalent to the equation¹S50. ~See Theorem 4.1 in Ref. 12.! h

By ~6!, we can write the following general expression for the homogeneous Lorentziantures on~G,g!:

S5~a1h1a2a1a3b1a4j! ^ ~a∧b!1~b1h1b2a1b3b1b4j! ^ ~a∧j!

1~c1h1c2a1c3b1c4j! ^ ~b∧j!, a1 ,...,c4PR.

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Equations~7! give us, after some long but straightforward calculations, the relations to be satby the coefficients. Ifa1Þ0 ~we denote this case as~i!!, we get the expression ofS in terms ofa1 ,a2 ,a3 ,a4 ,

S5~a1h1a2a1a3b1a4j! ^ ~a∧b!

1S 2a3h2a2a3

a1a2S a3

2

a11

1

2Db1S a3

2a12

a3a4

a1D j D ^ ~a∧j!

1S a2h1S a22

a11

1

2Da1a2a3

a1b1S a2a4

a12

a2

2a1D j D ^ ~b∧j!.

If a150, some tedious computations lead to another five cases, and the coefficients in the si~i!–~vi! are as follows:

~i! ~ii ! ~iii ! ~iv! ~v! ~vi!

a1 Þ0 0 0 0 0 0

a2 PR 0 0 0 0 0

a3 PR 0 0 0 0 0

a4 PR 12

12

12 Þ 1

2 Þ 12

b1 2a3 0 0 0 0 0

b2 2a2a3

a1

0 0 PR 0 0

b3 2a3

2

a12

12

2 12 2 1

2 Þ2 12 2 1

2 2a4

b4a3

2a12

a3a4

a1

0 PR PR PR 0

c1 a2 0 0 0 0 0

c2a2

2

a11

12

Þ 12

12

12

22b2

2

2b311

12 a4

c3a2a3

a1

0 0 2b2 0 0

c4 2a2

2a11

a2a4

a1

PR PR 22b2b4

2b311PR 0

In case~i!, making the changex5a1Þ0, y5a2 /a1 , z5a3 /a1 , w5a42 12, we can write the

family of homogeneous Lorentzian structures as

S~x,y,z,w!5~xh1xya1xzb1~w1 12!j! ^ ~a∧b!

2~xzh1xyza1~xz21 12!b1zwj! ^ ~a∧j!

1~ xyh1~xy21 12!a1xyzb1ywj! ^ ~b∧j!,

x,y,z,wPR, xÞ0.

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In cases~ii !–~vi! one arrives~after some changes of notations for the coefficients! to the followingfamilies:~ii !

S~q,c!512j ^ ~a∧b!2 1

2b ^ ~a∧j!1~qa1cj! ^ ~b∧j!, q,cPR, qÞ 12.

~iii !

S~b,c!512j ^ ~a∧b!1~2 1

2b1bj! ^ ~a∧j!1~ 12a1cj! ^ ~b∧j!, b,cPR.

~iv!

S~k,t,b!512j ^ ~a∧b!1~ka1~ t2 1

2!b1bj! ^ ~a∧j!

1~~ 122~k2/t !!a2kb2~kb/t !j! ^ ~b∧j!, k,t,bPR, tÞ0.

~v!

S~a,b,c!5aj ^ ~a∧b!1~2 12b1bj! ^ ~a∧j!1~ 1

2a1cj! ^ ~b∧j!,

a,b,cPR, aÞ 12.

~vi!

Sa5a~j ^ ~a∧b!2b ^ ~a∧j!1a ^ ~b∧j!!, aPR, aÞ 12.

Next, we determine¹5¹2S. We however omit the expression of¹ for the sake of brevity.Then, in each case we calculate the curvatureR, and so we obtain generators of the holonomalgebrah of ¹, which will be expressed in terms of the basis$P,X,Y,Q% of g.

In case~i!, we have that the only not always null curvature operators areRXY5wU, RQX

5xzU and RQY52xyU, where

U5S 0 z 2y 0

0 0 1 2z

0 21 0 y

0 0 0 0

D .

If y5z5w50 then R50, but if eithery, z, or w is not zero then the holonomy algebrah isone-dimensional and generated byU.

In case~ii !, the holonomy algebrah is generated by the vector

V5S 0 0 1 0

0 0 0 0

0 0 0 21

0 0 0 0

D ,

and the only nonvanishing curvature operator isRQY5(q2 12)V.

In case~iii ! ~and ~v!! we deduceR50.In case~iv!, the holonomy algebrah is generated by

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U5S 0 2t k 0

0 0 0 t

0 0 0 2k

0 0 0 0

D ,

and the curvature of¹ is determined byRQX5U and RQY52(k/t)U.In case~vi!, the holonomy algebrah has two generators

U5S 0 1 0 0

0 0 0 21

0 0 0 0

0 0 0 0

D , V5S 0 0 1 0

0 0 0 0

0 0 0 21

0 0 0 0

D ,

and the curvature is determined by the operatorsRQX5(a22 14)U, RQY5(a22 1

4)V. If a52 12

then R50 and if aÞ2 12, then h is two-dimensional and Abelian.

Through the usual identification ofg with the tangent space at the identity ofG, we considerm5g and use~2! in order to obtain the Lie bracket ofg5 h% g in each case. In particular, we wihave the reductive pair (g,h) associated with each homogeneous structure.

We next explicitly give those brackets. Some cases are broken in two subcases accorwhether the dimension of the holonomy algebra depends on the particular values of thecients in the corresponding homogeneous structure.

Case (i1). ~x,y,z,wPR; xÞ0; y,z,wnot all null!. The reductive decomposition associatwith S(x,y,z,w) is g(x,y,z,w)5 h% g5^$U,P,X,Y,Q%&, with not always null brackets

@U,X#5zP2Y, @U,Y#52yP1X, @U,Q#52zX1yY,

@P,X#52xzP1xY, @P,Y#5xyP2xX, @P,Q#5xzX2xyY,

@X,Y#5wU1~x~y21z2!11!P2xyX2xzY,

@Q,X#5xzU2zwP2xyzX1~w1xy211!Y,

@Q,Y#52xyU1ywP2~w1xz211!X1xyzY.

Case (i2) ~xÞ0; y5z5w50). The reductive decomposition associated withS(x,0,0,0) is gx

50% g5^$P,X,Y,Q%&, with nonvanishing brackets

@P,X#5xY, @P,Y#52xX, @X,Y#5P, @Q,X#5Y, @Q,Y#52X.

Case (ii) ~q,cPR; qÞ 12!. The reductive decomposition associated toS(q,c) is g(q,c)5 h% g

5^$V,P,X,Y,Q%&, with nonvanishing brackets

@V,Y#5P, @V,Q#52Y, @X,Y#5~q1 12!P,

@Q,X#5~q1 12!Y, @Q,Y#5~q2 1

2!V1cP2X.

Case (iii) (b,c,PR). The reductive decomposition associated withS(b,c) is g(b,c)50% g

5^$P,X,Y,Q%&, with nonvanishing brackets

@X,Y#5P, @Q,X#5bP1Y, @Q,Y#5cP2X.

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In this case, one obtains the Lie algebra of the oscillator group. In fact, puttingX5X2cP, Y5Y1bP, we have@X,Y#5P, @Q,X#5Y, @Q,Y#52X.

Case (iv)~k,t,bPR; tÞ0!. The reductive decomposition associated withS(k,t,b) is g(k,t,b)

5 h% g5^$U,P,X,Y,Q%&, with Lie brackets given by

@U,X#52tP, @U,Y#5kP, @U,Q#5tX2kY, @X,Y#5~12~k21t2!/t !P,

@Q,X#5U1bP1kX1S 12k2

t DY, @Q,Y#52k

tU2

kb

tP1~ t21!X2kY.

Case (v)~a,b,cPR; aÞ 12!. The reductive decomposition associated toS(a,b,c) is g(a,b,c)50

% g5^$P,X,Y,Q%&, with nonvanishing brackets

@X,Y#5P, @Q,X#5bP1~a1 12!Y, @Q,Y#5cP2~a1 1

2!X.

Case (v i 1) ~aPR; aÞ 12,2

12!. The reductive decomposition associated withSa is ga5 h% g

5^$U,V,P,X,Y,Q%&, with nonvanishing brackets

@U,X#5P, @U,Q#52X, @V,Y#5P, @V,Q#52Y, @X,Y#52aP,

@Q,X#5~a22 14!U12aY, @Q,Y#5~a22 1

4!V22aX.

In particular, if a50, one hasS50, and the associated reductive pair (g,h) is a symmetricpair, which defines the oscillator group as a symmetric Lorentzian space.

Case (v i 2) (a52 12). The reductive decomposition associated toS2(1/2) is g2(1/2)50% g

5^$P,X,Y,Q%&, with nonvanishing brackets

@X,Y#52P, @Q,X#52Y, @Q,Y#5X,

which also gives the Lie algebra of the oscillator group.

IV. CURRENTS, AND EINSTEIN–YANG–MILLS EQUATIONS

We consider the connection formv on g, which is theh-component of the canonical 1-formon g, and we obtain, in each case, the curvature formF5dv and then!F, where! denotes theHodge operator with respect to the Lorentzian metricg. The following table provides the values oF and!F for the cases with nontrivial holonomy,

F5dv !F

(i1) (2wa∧b1xza∧j2xyb∧j) ^ U 2(wh∧j1xzh∧b1xyh∧a) ^ U

~ii ! (q2 12)(b∧j) ^ V (q2 1

2)(h∧a) ^ V

~iv! (a∧j2(k/t)b∧j) ^ U 2(h∧b1(k/t)h∧a) ^ U

(vi1) (a22 14)((a∧j) ^ U1(b∧j) ^ V) (a22 1

4)(2(h∧b) ^ U1(h∧a) ^ V)

Moreover, as the second Yang-Mills equation isD!F5J, whereJ denotes the current, applying the usual formula19 one obtains for all the cases

J5D!F5d!F1v∧!F2!F∧v5d!F. ~8!

Thus, the currents are given, respectively, byJ

(i1) (xzh∧a∧j2xyh∧b∧j1wa∧b∧j) ^ U

~ii ! (q2 12)(h∧b∧j1ca∧b∧j) ^ V

~iv! (h∧a∧j2(k/t)h∧b∧j1b(11(k/t)2)a∧b∧j) ^ U

(vi1) 2a(a22 14)((h∧a∧j) ^ U1(h∧b∧j) ^ V)

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On the other hand, we consider the metric ong which is the direct product of the metricg0 ong and the metric onh for which the basic vectors~U and/orV! are unitary.

Now we haveiFi252w2 in the case (i1) and iFi250 in the remaining cases. We necompute the stress-energy tensorT, given by~4!, and we express it in terms of the basis$P,X,Y,Q%of g. In case (i1) we have

T5S 0 0 0 2 12w

2

0 12w

2 0 2xyw

0 0 12w

2 2xzw

2 12w

2 2xyw 2xzw x2~y21z2!

D .

In cases~ii !, ~iv!, and (vi1), the only nonvanishing component isT(Q,Q), which is, respectively,(q2 1

2)2, 11(k/t)2, and 2(a22 1

4)2.

It is then a matter of calculation to see that the field equations~3! are satisfied in cases~ii !,~iv!, and (vi1), and in the case (i1) if w50 and y21z2Þ0, and to obtain the correspondincosmological and gravitational constants.

Theorem: The Lorentzian reductive pairs( g,h) determined by the homogeneous Lorentzstructures on the four-dimensional oscillator group, in cases( i 1) (with w50 and y21z2Þ0), (ii),(iv) and (vi1) above, are solutions of the Einstein–Yang–Mills equations for an electromagnetigauge field with cosmological constantL50, and gravitational constantk given, respectively, by

1

2x2~y21z2!,

1

2~q2 12!

2,

t2

2~k21t2!,

1

4~a22 14!

2.

The gauge algebrah is Abelian and two-dimensional in the last case and one-dimensional inother three cases. The currents are given in the table following (8). They are nonvanishingin the case (vi1) for a50.

If a50 in case (vi1), the associated symmetric pair (g,h) appears as a solution of thEinstein–Yang–Mills equations for a sourceless gauge field with cosmological constantL50 andgravitational constantk54 ~see Levichev5!.

ACKNOWLEDGMENTS

This work has been partially supported by DGES~Spain! under Project No. PB95-0124, anby Xunta de Galicia, under Project XUGA No. 20703B98.

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