Volume 145, number1 PHYSICSLETTERSA 26 March 1990

ON AXIAL VECTOR TORSIONIN VACUUM QUADRATIC POINCARE GAUGE FIELD THEORY

P. SINGHDepartmentofMathematicalSciences,TheUniversityofTechnology,Loughborough,LeicestershireLEJI 3TU, UK

Received27 November1989;revisedmanuscriptreceived29 January1990;acceptedfor publication30 January1990CommunicatedbyJ.P.Vigier

Thevacuumfield equationsof thequadraticPoincarégaugefield theory (qpgft) aresolvedfor a purelyaxial vectortorsion.Expressingthecontortiontensorfor this caseas~ we provethatthereareno solutionsof vacuumqpgftwhenS~isnon-null. However,when S~’is null, weprovethat anysolutionof vacuumqpgft,whoseV

4 part is alsoa solutionof Einstein’svacuumfield equationswith acosmologicalconstant,is necessarilyofalgebraictypeN. Thissolutionhasanexpansion-free,shear-freeandtwist-freeautoparallelrepeatedprincipalnull congruence.

1. Introduction ref. [1] mustbeusedinconjunctionwith thosegivenin ref. [81 in which the formalism of Newmanand

In a previouspaper [1], the vacuumfield equa- Penrose[9] had previously beenextendedto in-tionsoftheqpgft (seerefs. [2—51andreferencescited dudespace—timeswith torsionin any theory.therein)for the purelyquadraticLagrangiandensity The aim of this work is to usethetechniquegiven[61 in ref. [1] to obtain exactsolutions of the vacuum

field equationsof theqpgftwith axial vectortorsion.e ~ F ,~ e F’

4~ F ‘~ (1) Thistechniquecanproduceaclassof exactsolutions— 4/2 V~ 4k U VP which would be impossibleto obtain via the double

havebeenexpressedin the Newman—Penrosefor- duality ansatzof refs. [4,5,7,10]. Sucha classof so-malism involving spin coefficients.The two gauge lutions of algebraictype N with an expansion-free,field strengthsF,

4~1(torsiontensor)andF~~U(cur- shear-freeandtwist-free autoparallelrepeatedprin-

vaturetensor)alongwith the so-calledmodifiedtor- cipal null congruenceis presented.sion tensorT~~’andthe quantitiese, k and/are de- In orderto obtaina newclassof exactsolutionsoffined in ref. [11, asaretheconventionsandnotation the vacuumfield equationsof qpgft, it first seemsusedin thispaper.Argumentsin favourofthechoice necessaryto imposesomeconstraintson theform ofgivenby eq. (1) for the Lagrangiandensitymaybe the contortiontensor.Wefirst assumethat thecon-foundin refs. [3] and[6], andwill notbediscussed tortion tensoris axial i.e. totally anti-symmetric.Inhere.TheLagrangiandensityaboveisessentiallythe thiscasewe maydescribeit in termsof a vectorS~Lagrangiandensitygiven in ref. [71whentheweights definedbyof the irreduciblepiecesof thetorsionandcurvaturetakethefollowing values(in thenotationofref. [7]), S’4= � (3)

a1=—1, a2=2, a3=—l, thenx KPPK=�’4p,aSA. (4)

b~=b2=b3=b4=b5—b6=—l. (2)Thisrequiresthat the individual componentsof the

The vacuumfield equationsof the qpgft givenin contortiontensor,in the notationof ref. [8], satisfy

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Volume 145,number 1 PHYSICSLETTERSA 26 March 1990

andthe contortioncanbe expressedin termsof theK1 0, Pi Pi’ a1 =0, ri =—2ã1,

singleimaginarycomponentj~.= —2a1, ,~ =0, ~ii = —j~, v1 =0, Apart from the initial constraintsimposedupon

theform ofthecontortiontensor,in this sectionsome�‘=—~p~,fli=c~1, yi=—Lui. (5)

constraintson the form of the curvaturetensorareThe torsion is thus completelydescribedby the also imposed.We assumethat the V4 parts of the

complexcomponenta1 andby the purelyimaginary curvaturetensoraresolutionsof Einstein’svacuumcomponents/2~andPi• field equationswith a cosmologicalconstant,A~,i.e.

A short calculationwill show that S’4 is givenbyR°~~— ~R°g~~=A~g’4~, (11)

S’4=i/211’4+2ia1 m’4—2iã1i~’4—ip1n’4. (6)whereA~=—6A°=— ~R°.Of courseit is a foregone

WhenS’4 is null we musthave conclusion that the Ricci identities and Bianchiidentitiesfor curvaturein V4 mustalso hold.

~IpI —4a1c~1=0, (7a) With this additional assumptionit can easily beandwhenS’4 is non-null we musthave provedthat:

/1iPi —4a

1t~1~0. (7b) Theorem.Any solution of the vacuumfield equa-

tions of qpgft with S’4 null whoseV4 part is also asolutionof Einstein’svacuumfield equationswith a

2. S” non-null cosmologicalconstantisnecessarilyof algebraictypeN, with an expansion-free,shear-freeandtwist-free

In this caseS’4 is a continuouslydefinednon-null autoparallelrepeatedprincipal null congruence.vector.Thenwithout loss of generalitywe mayalign(1’4±n’4) with S’4. With alternativesignsthiscovers The class of exact solutionssatisfying these as-the caseswhenS’4 is time-like andspace-like.This sumptionsis givenby the metricchoice for S’4 imposesthe additional constraintson

dS22 du (— Udu+dv+ Wdz+Wd~)

the componentsof the contortion tensor,namely,

a1=0, u1=Rp1. (8) —dzdZ, (12)

Now, if we considerequations(30d,30eand30h) wherethe metric functionP is a real function of zin ref. [1] it is easily shown that

and~only satisfying(9) 4P

2(log~P) AO (13)

i.e. the torsionvanishes.Thus,we haveshown that andthemetric functions U and Ware givenbysolutionsof vacuumqpgft with S’4 non-null belongin V

4. U=(P2Ø,~O,

5+A°)v2—~O~v+U°,

W=O~v, (14)

3. S” null where 0=Ø( u, z, ~) = ~ and U°=U°(u, z, ~) = U0satisfy

In this caseS’4 is assumedto bea continuouslyde-fined null vector.Thisconditionis equivalentto as- 0,zz— ~ (0,z) 2 + O,z= 0,sumingthat the spin of the source field which gives

A°riseto thetorsionin space—timeis locallyalignedand ~ — — =null. It is then possibleto align the tetradvector l’4with S’4, andthis implies the conditions 0’U~

5+~p1=a1=0, /1~=—

2y~, /11=—U1, (10)

+(~Ø,~çb,5+A°/P2)U°=0. (15)

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Volume145, number1 PHYSICSLETTERSA 26March1990

Theonly non-zerocomponentsof thetorsiontensor The double duality ansatz (19), the secondfieldare equationof vacuumqpgft [1] andthe Bianchi iden-

tities for the curvature[1] give two possiblevalues/L~=im, y~= — him, (16) for y for non-vanishingtorsion, accompaniedbywhere m is an arbitraryreal function of u andthe constraintson the contortion tensorK,Jk, namelynon-zerospin-coefficientsare = ~, K1~kJ= 0 (21a)

a=1(PO,~+2P,2),fl=~(PØ~—2P5), and

— (P2O,~O,

5+A°)v+~ — kim,y=—l, KUk=K[Ukl. (2lb)

/2=im, v—P0~~v—2PU~—2PO,~U°, Furthermore,it is easily shown that thesetwo p05-

7t= ~P0,z, r=PO~. (17) sibleformsfor thegeneralizeddoubledualityansatz

The only non-zero componentsof the curvature areequivalentto the constraintstensorare WA 0, ~I~A 0, .�=0, A=k/81

2 (22a)

= 2P[2PU~2~+2(2P,~+P0,~)U~~+P(D.~)

2U°], and

= —m2, A=A°=—k/412, WA 0, ~A =0, E=0, A= —k/412, (22b)

612 = — mPØ~, respectively,in the notationof ref. [81.

622 = 2m (P2 O~O~+A°) v—~mØ,~— m~. (18) Theseconstraintsare considerablystrongerthanthoseconsideredin thepresentpaper.In particular,

The following specialcasesof this classof solu- the solutionsof section3, in general,have !P4 non-tions may also be noted. zeroandthegeneralizeddoubledualityansatzis not

(a) In thecasewhenW4 = 0, thesesolutionsreduce satisfied,exceptin the specialcasewhen W4 is zero.

to a generalizationof the conformallyflat solutions Recently,much attentionhasbeendevotedto thegivenby Lenzenin ref. [101: seealso section7, case HamiltonianstructureofPoincarégaugefield theory1 of ref. [lJ. [7,12,13]. Oneof the reasonsfor this is to help one

(b) Whenthetorsionvanishesthesesolutionsre- distinguishthe variablesin a theorywhich are dy-duceto the K(2) wavessolution of Diaz and Ple- namicalandphysically importantfrom thosewhichbañski [111. arenon-dynamicalandjustdescribegaugedegreesof

freedom.In the solutionsof section 3 m wasdeter-mined from the first field equationof the vacuum

4. Discussion qpgft (seeeqs.(30) in ref. [1]). Consequently,m

mustbe a dynamicalpieceof the Riemann—CartanIn view of eq. (11), onemay ask if the solutions connection,althoughit is arbitrary.

of the theoremobey the generalizeddoubleduality In conclusion,it canbe saidthat the advantageofansatzdiscussedat length in ref. [7]. The general- thespin coefficienttechniquegivenin ref. [1], overized doubleduality ansatzfor the purely quadratic the (generalized)double dualityansatz,in produc-Lagrangiandensityabovebecomes ing solutions of vacuumqpgft, lies in its ability to

F’4Mk/ = ~ *F*/4V,a — e

1,, ‘4e,1 ~, (19) producesolutionsof Petrovtypesotherthan0.

where ~ and ~‘ are dimensionlessconstants.This is Acknowledgementpreciselythe doubleduality ansatzof refs. [41and[5]. It canbeshown [41thatnon-vanishingtorsion Theauthorwishesto acknowledgetheawardof asolutionscanonly be obtainedfor researchscholarshipby the SERC, while this work

—1. (20) wascarriedout.He would alsolike to thankDr. J.B.

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Volume 145,number 1 PHYSICSLETTERSA 26 March 1990

Griffiths for his continuoussupportandencourage- [41P. Baelder,F.W. HehiandE.W. Mielke,in: Proc.2ndMarcel

ment andthe referee(s)for a numberof extremely GrossmannMeetingsin Generalrelativity, ed. R. Ruffini(North-Holland,Amsterdam,1982).helpful comments. [5] P. Baekler,F.W. HehIandH.-J.Lenzen,in: Proc.3rdMarcel

GrossmannMeeting in Generalrelativity, ed. Hu Ning(North-Holland,Amsteram,1983).

[6] F.W. Hehl, Y. Ne’eman,J. NitschandP. Von der Heyde,

References Phys.Lett.B78(1978)102.[71P. BaeklerandE.W. Mielke, Fortschr.Phys.36 (1988)549.[8] S. JogiaandJ.B. Griffiths, Gen.Rel. Gray. 12 (1980)597.

[1] P. SinghandJ.B. Griffiths, Gen. Rel. Gray. (1989),to be [9] E.T. NewmanandR.Penrose,J.Math.Phys.3 (1962) 566.published. [101H.-J.Lenzen,NuovoCimento82B (1984) 85.

[2] F.W. HehI, J. Nitsch and P. Von der Heyde,in: General [11] A.G. Diaz andJ.F. Plebañski,J. Math. Phys.22 (1981)relativityandgravitation.Onehundredyearsafterthebirth 2655.of AlbertEinstein,ed.A. Held (Plenum,NewYork, 1980). [121E.W. Mielke and R.P. Wallner, Nuovo Cimento 101 B

[3] F.W. HehI, in: Cosmologyand gravitation.Spin, torsion, (1988)607.rotationandsupergravity,eds.P.G. BergmannandV. De [13] E.W. Mielke and R.P. Wallner, Nuovo Cimento 1 02BSabbata(Plenum,NewYork, 1980). (1988)555.

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