Ram, Shri

Singh, Gulab

Singh, K P

IACS Institutional Repository

Ind ... " J. Phy •. 54B. lii7-562 (IQSO) Zero-relt-mass Icalar fields for certain space-times K P Singh, Gulab Smgh >l<lld Shl'i ltttm Depal'tment of M .. thoru"tlC •• n.H. U . V"",nlUil 221005. IlllJI~ Recoived 4 April 1970 Abstract: In thi8 papol' 1111 flhUd HolutuHl ofEin~tolll--l\1o.xwell·f:1callll.l· field oqUlt1,IOlll::l for an u.xI-AymI1lo1,r1O f>tRtic mot,rlC is olJtaiued whloh inoludos Mal. vin'a magnet,ic lmiv~rso as a Apocial CBF,le when fJralar flald iB a.bsent. Further an exo.ct solutlon of Eml:ltein's Held LH]uationa foJ;' zoro-roet-mBBS scala.r field hits 0.180 been 101Wd fOl' ,), sto.t:.lOno.l'Y cylinw.'ically symmetnc space-time. 1. latroductioa The study of 1'olo.tivi~tic field t'quat,io!lR ill tho pl'CH,meC of H(,alal' lllOoOl' field has di'IIiWIl t.hc aMent,ion of lllany work!'rs IIilld variOlls a"peds of tho prohlom havo been investigat.ed by .Bl'ahmachuIY (1960), Bergmann and Loipnik (1957), Buchd;,)'l (19,S9), Janis et al (1968), Penney (19G8), Gaut,roau (1969) and oth~rs. The oxistenco of tho solution of c(,uplcd electrnmagne1-ic and scalar fields in genem] rclativity is of considpmble physical interest and It may serve as a start-ing point fo1' furthor invostlgations. In Einstoin's field "quations R u' 1. I' V 8 '1' • ~ --2" "" .n = - 1r II' we take tho enorgy.momontum ton~or T#" IL~ T,/= E:+K;. (1.1) Horo E/ and K/ are tho pncl'gy-momontum tOll~ors corresponding t,o the eloo-tromagnotic and zero-t'(,st-mass scalar fiolds lospoet.ivoly. The tonsor E; is giv~n by wb,ere tho eleotromagnoj,ic fil,ld tonsor l!'., RlLtisfies Maxwell's oquatioIls (1.2) (1.3) 647 548 K P Singh Gulab Singh and Shri Ram Here a oomma and a semicolon sLand for ILll ordinary and" covariant differentia-tion respectively. The tensol' K,,' is givon by (1.4) whore the scalar potontial V slLti~fies y·'V .•• = o. (1.5) 2. Melvin magnetic universe Mclsin (1964) has given an exact axisymmotric solution of the Lllmbined s()W'ccloss Einstoin MlLxwdl field oquatiulls in ~t .. tic easo which J'PprOHontB a parallel ~lUlldlo of magnetic flux held together by its own gravitational field. It is de~cribed by tho metric \(2.1) with ( Br.) 1jr = 1jr(r) = log 1+--4-, where B is the magnetic intonsIty. His solution shows the persistent local energy-stress concentration wllich may be takon as the dofining characteriRti~ of a goon. Prasanna and Kumar (1973) studied tile spin prccission of a dlarg~d partiele in Melvin's magnetic universe including die effects of' generalrcla/.ivity. In this soction we have found I> solution of the Einstein-Maxwell field equa-tions corrosponding to 110 parallel bundle of magnetic flux coupled with zero-rest.-mass scalar field for tho axisymmetric stat,ic metric (2.1). The co-ordinat.cB r, tjJ, Z. t are levelled as x" x', x", x' respectively. Here the eleotroma.gnetie field is associated with an arbitrary constant and the scalar field is aasociated with anoiher arbitrary constani. When tho Bcalar field is absent we get tho solution of Melvin (1964). In ihe absence of magnetic field a solution of coupled Einstein-soalar field equations is obtained. Zero-re8t-mas8 8cnlar field8 for certa,in Bpace-timM 549 Now the oquations (1.2) and (l.4) give the electromagnetic and SCai8f energy mome.ntum tenAorR as 81TE: = B"e-'''' diag(+I, -1. +1, +1) 81TKu' = Vb-2'" diag(-I, +1, +1, +1) where the scalar potential V by symmet,ry is a function of r alono. (2.3) (2.4) FrolU oquations (2.1), (1.1), (2,3) and (2.4), the indl'pendent field equations afo (2./l) (2.6) Equatioll (1.5) giveR V V, 11 + -- = o. (2.7) The subs('ript o.enotes differentiMion with fOAfoet, to r r,hrnughom. 550 . K P 8ingh Gulab Singh ana 8hri Bam EquationH (2.9) am]. (2.10) lUll lCIUlld Hatisfiud by t,he value of if' in (2.12) equation provided that (2.13) Th~ relaLion (2.13) will determine (t, tllO conRt,ant IIH~ociaj,od with th" sealar field. Honce the soluLion of Einstoin-Maxwell·Rcalal' fipld l'qnationR for the metric (2.1) is dH2 = {r.,r 1- \/(I-a') I-c2r1 +V(I-IJ') };dt2-dr2-d;) _ s{ I-Vll_(2)+ I+ V(i::'ci')}-2 dA.2 r C,f C2f 'I' • (2.14) This Aolution has a singularity !tIling r = 0, which can he idontifb'd with the presenco of the SOUI'CC. The ~clLllLr denHit,y incrcaRllM as r incrC'l1ses. The space-time (2.14) roprosoutR a diR1,ribut,101l of zoro-rllH1,-mIlHR R«alar IJarticles in ~Ivin's magnetic univel'sc. Oase (i). When a = 0, tho scalltT' ficld dislLppearR lind tho Rolut.i(ln (2.14) reduclls to that (If Melvin's magnctie univt'I'A(,. ' Oase (ie). Whon B =- 0, tlw magnetic fidd is abRent lind tho relation (2.13) gives a2 = J if CI and C2 0.1'(\ non-Zl'r(l. III t.11iR On-RIl pquu.t.ioll (2 12) reducoR t,o (2.1li) and thl' line ulmnollt (2 14) t.U.kOB (,hl' fOl"1I1 (2.16) The apal·c-time (2.16), which iA not-fla.t, iA t,he solution of combin!'d Einstein-ScaJar field equatIOns for tho motric, (2.1). If llO\\,OVtlJ' cl 01' C2 bC' ZI'n), we> /,>'('1 n Aolutinn for Einstein-Scalar fiold for lIrbitrary valuc of a2 < 1. 4, Stationary eylilldrieally symmetric space-time In this seeMon an oxact, solution of F.inst"jn'R Hold f'qllations for coupled gra.vita.. tional and z('ro·rost-mas~ Mcalar field8 ha.s been f.oWld for tt (1 Htllt,iona,ry cylindri-cally symmctric spaoe.Linll' (8(1m and Raychaudhl1ry 1968) (3.1) where !Jr, Z and m are functions of r alone. For this stat.iona.ry metric the soala.r field V, by Hymmctry, is a function flf ,. only. From (1.4) o.nd (3.1), it followR t.hat 1\I+T2B = 0 (3.2) Zero-re,'·ma8B BCaZar fietda lor c:erlai~ Bpace.timeB 551 ~ hliAoe from (1.1) (3.3) We CMl now introduce Weyl-like clLIlonicaJ coordinate system in this stationary ClHle Buch that (Va.n Stookum 1937) (3.4) For the metric (3.1), in view of (1.1), (1.4) and (3.4), we Cim write the field equa. ~i~B oXl1lioitly &s -- ~ +2'" - -KV' 3 ( m )" 2 r 1"11- 1 • From (1.5), we got Vl1+.!L = O. r (3.5) (3.6) (3.7) .(3.8) (3.9) Here the subscripts 1 and 11 after !fr, m and V donote first and second derivative reIIplIotively with respeot to r throughout. The general solution of the equation (3.9) is given by V = adogr+p, where or; Mld P are arbitrary consta.nte aBBOciated with the soaJar field V. From equations (3.7) Mld (3.8), we aet Equations (3.6) and (3.7) give 11 (3.10) (3.11) (3.11) Combining oqulLtions (3.11) and (3.12) and integrating, we obt,(/'jn m'l =-(lj·~lii.ch implios that ! m=a, whore a is an ~rbitrary constlLnt. , , , Substituting for V and In in equation (3,5) and integrating, ~j) get 02'/1 = (o.l')b, (j boing ILn I>rbitmry consta!lt and b = krx2 (constant). Now equlltion (3.4) gives (3.15) Thu~ the ~olution of Einstein's fiold oquaj,ions for z~ro-re8t-maBS scalar fiold oorrosponding to tho ~tationary cylindrically symmetrio motric (3.1) is ~3.16) ",hero a and b are constants. Now the stationary solution (3.16), due t,o ~.l3) a.nd ,the Bubstitut,ion t = t+acp, reduces to the ~tatjc form (3.17) 1<'rom oquationR (3.10), (3.14) and (3.1~), it is obvious thai the solution obtained gives div(lrgellt gravitational and ma~blo~B scalar fields; and as r -. 00, V, Vr and l all the throe t,md to co. It. may be noted t,hat j,he abovo ~olutioll h0.8 a singularity I>long r = 0, which can be identified with the presenco of'tho sourco. Also along r = 0, we get a two dimensional fiat h~per8urfaco Oil which tho gravitaLional and zero-rust-mass scalar fiolds both become singular. Acknowledgment One of tho authors (GS) fecls ploo.Hure Lo -thank the refereo for his cqustructlve comments and U.G.C. (India) fO!' the award of Teacher-Fellowship to lIupPQrt the work. Referencn BrahmachlL1'Y H. L 1960 Pl'Og.'Theor. Phy •. 23 740 Borgmann 0 and Loipmk R 1957 Pity •. Rev. 107 1157 Buchdo.hl H A 1059 Pity •. Rev. 115 1325 ;rfl.ntS A, NewIllau E T and Wirucour J lU6M PhY8. Rev. Lett. 20878 Penney R 1968 Pity •. Rev. 174 1578 Gautr.ILu R 1969 Nt"'.o 0.111. BOB 360 Som M M and Raychaudhury A K 1968 Proc. Roy. Boo .\304, 81. Van Stockum W J 1937 Proo. Roy. Soc. Edinb. 57 135 M.lv4rI M A 1964 1'1'11';0 LeUe .. 8 65 PrBSanna A R and Kumar N 1973 Prog. T/ie~. pity •. 49 1653 '·,'1 

Zero-Rest-Mass Scalar Fields for Certain Space-Times

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Zero-Rest-Mass Scalar Fields for Certain Space-Times

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