Chakraborty, S

Maity, K

Rahaman, F

IACS Institutional Repository

Indian J. Phys. 76B (1), 71-74 (2002)U P  ®an international journalNonstatic global monopole in higher dimensional space timeF Rahaman*^ S Chakraborty*,i;K Maity and M KalamKhodar Bazar, Baruipur-743 302, 24-Pargj^nas (South). West Bengal, India ‘Department of Mathematics, Jadavpur University, Kolkata-700 032, India B-mail )umathfi/)cal;vsnl net inReceived 10 August 2001, accepted 30 October 200JAbstract * We have obtained a class of nonstatic solutions around global monopole in higher dirnensioria! space lime We have used a field theoretic energy momentum tensor for monopole configuration This paper extends earlier work of ('hakraborty {Phystca Senptu 98 294 (1998)) to its five dimensional analogKeywords Nonstatic global monopole, higher dimension, deficit angles.PACSNos. . 98 80 cq, U4 501. IntroductionIn last few years, there are attempts to unify gravity with other fundamental forces in nature. Latest studies of super string and super gravity theories and unification of fundamental forces with gravity reveal that the space-time dimension should be different from four [1]. As a result, higher dimensional theory is receiving great attention both in cosmology and in particle physics. Moreover, solutions of Einstein field equations in higher dimensional space-time are believed to be of physical relevance, possibly at the extremely early times before the Universe underwent compactification phase transitions. Phase transitions in the early Universe, can give rise to various forms of topological defects. A defect is a discontinuity in the vacuum and depending on the topology of the vacua, the defects could be domain walls, cosmic strings, monopoles and textures [2].In particular, monopoles arc formed when the vacuum manifold contains surfaces which cannot be continuously shrunk to a point, Le. when ;r2(/i) ^  1. Global monopoles (which are formed when a global symmetry is broken) are important objects both for particle physicists and kosmologisls because it predicted to exist in Grand Unified theory.The existence of a (magnetic) monopole was first suggested by Dirac [31. L.ater, t' Hooft [4] and Polyakov [5] demonstrated monopole solutions in a gauge model possessing SO(3) symmetry with a Higgs field ^ in a triplet representation. Subsequently, monopole solutions of the realistic grand unified models based on the gauge groups SU(5) and SO(IO), have been considered by Dokos and Tomaras [6] and Li et al [7].In 1989, Barriola and Vilenkin [8] have found an approximate solution of a monopole resulting from breaking of the global SO(3) of a triplet scalar field in a schwarzchild background.An attractive feature of this topological defect is that it exerts no gravitational force on the matter around it (except for a tiny repulsive effect of the core). Some properties of the global monopole metric arc similar to gauge cosmic strings [2] but an important difference is that the monopole metric is not locally flat. So far, most of the analytical works in monopoles deal with their gravitational effects based on general relativistic static models [9].Recently, Chakraborty [10] described a non-static monopole solution. The consideration that the space time has more than four dimensions, has recently received muchCorresponding Author© 2002 lACS722[rH  2AH rB jF Rahaman, S Chakraborty, K Matty and M Kalamattention in its attempt to unify gravity with other gauge type interactions. As far our knowledge goes, there has not been any work in literature where nonstatic space-time has been considered for study of monopole in higher dimensional space-time. So, wc have thought it worth while to extend the Chakraborty's work in five dimensional space-time.2. The basic equationIn this section, we closely follow the formalism of Chakraborty and take the Lagrangion as flO](10)\  - j A  { r r  - i f f ,T! =~ T a 2B2A 2B+ 4I'O _  _  y ¥  _  ^ “h  -  2A 2B * 2T r = ^ r  <!>’".(4)r .  (5)- 772^ -^  (6)(6 . 1)Thus, the Einstein equations are 3 BH , 3 IF , 3 B' , 3: + -r—rrrr+-^-Trr-+-4 ABH 4 AH~ 2 B-r F H  BFA 23 H 3 AH 3 A' , 3 34 AH 4 h F  2 ABr F H  BF~~2A(b'<A 2fi ■*B i f , / / H^ A'B'2AB 4AB^ AH'^ AAH^ 4ABA" AB AH A'2AB 4BA^ 4BA^ 2HA^ rAB1 BH 1Br- ' 2 ABH1 . ^ r B ^  ^ F H(7)(8)(9)Here ‘ ’ stands for ^  and stands for( I )where is the triplet scalar field, a ~ I, 2, 3 and rj is the energy scale of the symmetry breaking and A is a constant.The metric ansatz describing monopole can be taken as-  ~ A ( r j ) d t -  + B{ r j )dt-  ^ r ^ U { ( )  ^ ^ ( d (P  t sin^ Od^ -  I sin^ ^sin^ <f>dy / ^ ). (2)The energy momentum tensor for the above Lagrangian is given by [lO)(3)Now for the above metric, the nonvanishing components of the energy momentum tensors are3. Solutions to the field equationsWe shall now solve the field equations assuming separable form of the metric coefficients as follows :A = A^(r)A2(0; B=^Bj(r)B2(0. (11)Also, we have taken the scalar field triplet in the separable form asr ( '* ,0  = «^f(r) + ^ 2(0. (12)Putting these separable forms (11) and (12) in (10), we getH A\H  1 A .rBi =  0 .rH 2 AiH\ /Also from (7) and (8), we get3 h j l  3 HB2 3 HBj 3 A^HB^ <^ ‘’ ^ 8 24 A2 l l '^  4 A2 H'^ 4 H A i A  HA2 A2(13)2 B}r 23 A\By P, (14)J, , ..where p is the separation constant.We shall now solve the.se equations with the following relations among the metric coefficients :Bi -  BqH” and Ay = (1*^ )where Bo^  n, Aq, d  are constants.From (13), we getI 2 J r f "  H (16)Hwhere k is the separation constant. Solving (16), we getr2= k\n{r^2H )and = AIn(^g,r),(17)(18)where ^oi ^02 Integrationconstants.From (14), using (15) and (18), we get for space part asB|' C _where C = rf+ |-*2.3 3A(j( 19)Nonstatic global munopole in higher dimensional space time 73Solving (19), we get 1Dr‘ + „2-</(2 0 )nwhere + vwhich has a frrst integral asif-(22)I w■3/?o. _ ~1----  //m-«+2 +m  +  2  ~ n - 2 l  *where />/ is an integration constant. The integral form of / /  isc U f(23)i, = + (/-/( ,)  (24)/ y / / ? /  --------  i f m - t H l ^  ^ ni \2  n ~ 2 r(/() ■ integration constant).I'or a different choice of the constants, the solution for H are :C a se  /  D i 02H oc {t provided m ^  n. (-5)C a se  11 . V -  0//yoc(/-/Q )/ I provided / I, (26)Case 111 ; / -  /(i) m -1, /? == -2 2/ /:(ii) w - 1, /? “2 f5"e-2>/7T('-»o)/y _ V V ________1+e-//>T('-'o)Caw IV : I ^ 2(i) n = 1, /w = 1N = J ^ s e c ^ / v ( t - t o ).(ii) « = -2, /w = 1.(27)(28)(29)Here, we getj2vH + D fc + d - 2 ‘ where D is an integration constant.For time part, we introduce an assumption without any loss of generality as^2=^7//'", (21)(The value of Ai different from unity only results in a transformation of time coordinate)where q, m are constants.from (19), we get by using (15), (17) and (21)D,//“ ±(/ “ A )• Case y : I ^ 0 (i) n ' -1, mDV: InV2V//+D, - , / A  J lv H + D i  +1/ /  = (ii)« - H -V1, nisin hyfv[i - /()).(30)(31)(32)Now, the expression for the metric (2) isds"- - ^ ^r^-H dn\. (33)If w^ e definem -n-  H i dtand R ^ J lh | 4 dr(34)then the above metric can be written asds2 = r ‘'W '[-dT -  + dR~ -» g( R )fi T) d n \  ](g (/? ) = r2 - ‘/,  / ( 7 - ) =  /y i " ) -It is to be noted that the metric describes a deficit solid angle which depends on radial and on time coordinates (except for a conformal factor) and hence, it represents a monopole [8,10] (since monopole exhibits some important properties, particularly in relation to the appearance of nontrivial spacetime topologies f8 ,10]).Again, if we consider /? =" 1, the explicit expression of the metric is (with a proper choice of radial and time coordinates).ds^  ^ -  -dV- + dR^ -^^{R )da\ (35)(except for conformal factor).fhis solution represents a static model and the solid angle of deficit is a function of radial coordinate only.Also for d ^ 2, we get the metric of the formds- = - d n  + dR- + f{T )  dQ\ (36)(except for the conformal factor).This solution represents a time-dependent model and hence, we get time dependent solid angle of deficit.However, for ^  = 2, 1, the above metric (34) isconformally flat and does not describe a monopole.But it is not possible to find a non-static deficit solid angle in self-similar form as shown by Cho and Vilenkin [11], because we have used a separable form of the metric coefficients.74 F Rahaman, S  Chakraborty, K  Maity and M  KalamFor future work, it will be interesting to obtain a nonseparable solution for the above field equations.AcknowledgmentWe are thankful to Dr. A A Sen and Dr. S Chatterji for their helpful discussions.Retcrencrs111 ,S Weinberg ?hysia m Higher Dimension (Singapore . World Scicntilic) (1986)|21 A Vilenkin and L P S Shcliard Cosmic String and other Topological Defects (Cambridge ■ Cambridge Univ. Prc-ss) (1994)[3] P A M  Dirac Proc. Roy. Soc. A133 60 (1931)[4] G t' Hooft N ud  Phys. 79 276 (1974)[$] A M Polyakov JEPT Lett. 43 1365 (1974)|6] C Dokos and T Tomaras Phys. Rev. D21 2940 (1980)[7] X I-i, K Wong and Zhang J. Phys U tt  HOB 209 (1984)[8] M Barriola and A Vilenkin Phys. Rev. Lett. 43 1365 (1989)[9] I) Harrari and C Luoslo Phys Rev. D42 2626 (1990); X Shi and X Li Class Q mn Grav. 8 761 (1991); A Baneiji and A Sen Class Quan Gray 13 3141 (1996)1101 S Chakraborty Phys Scripta 58 294 (1998)111] I Cho and A Vilenkin Phys Rev D56 7621 (1997)

Nonstatic Global Monopole in Higher Dimensional Space Time

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Nonstatic Global Monopole in Higher Dimensional Space Time

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