International Journal of Scientific & Engineering Research, Volume 4, Issue 10, October ISSN

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1 Intentionl Jounl of Scientific & Engineeing Resech, Volume 4, Issue, Octobe-3 4 ISSN MORRIS-THORNE TRAVERSABLE WORMHOLE WITH A GENERIC COSMOLOGICAL CONSTANT N M Emn*, M S Alm, S M Khushed Alm, Q M R Nizm Deptment of Physics, Univesity of Chittgong, Chittgong-433, Bngldesh. *Coesponding utho: numemn@yhoo.com Abstuct The sttic nd spheiclly symmetic Mois-Thone tvesble womhole solutions in the pesence of cosmologicl constnt e nlyzed. We mtched n eio solution of spheiclly symmetic tvesble womhole to unique eio vcuum solution t junction sufce. The sufce tngentil pessue on the thin lye of shell is deduced. The specific womhole solutions e constucted with geneic cosmologicl constnt. I INTRODUCTION Womholes e hndles o tunnels in the spcetime topology connecting two septe nd distinct egions of spcetime. These egions my be pt of ou Univese o of diffeent Univeses. The sttic nd spheiclly symmetic tvesble womhole ws fist oduced by Mois nd Thone in thei clssic ppe []. Fom the stnd po of cosmology,the cosmologicl constnt Λ, seved to cete kind of epulsive pessue to yield sttiony Univese. Zel dovich [] identified Λ with the vcuum enegy density due to quntum fluctutions. Mois-Thone womholes with cosmologicl constnt Λ hve been studied ensively, even llowing Λ to be eplced by spce vible scl field. These womholes cnnot exist, howeve, if Λ e both spce nd time dependent. Such Λ will theefoe ct s topologicl censo. In this ticle, we oduce n exct blck hole solution of the Einstein field equtions in fou dimensions with positive cosmologicl constnt to electomgnetic nd confomlly coupled scl fields. This solution is often clled Mtinez-Toncoso-Znelli (MTZ) blck hole solution. In geement with ecent obsevtions [3], this blck hole only exists fo positive cosmologicl constnt Λ, nd if qutic self-ection coupling is consideed. Sttic scl field configutions such s those pesented hee, which e egul both t the hoizon s well s outside, e unexpected in view of the no-hi conjectue [4]. The confoml coupling fo the scl field is the unique pesciption tht guntees the vlidity of the equivlence pinciple in cuved spcetime [5]. In the litetue, numbe of tvesble womhole solutions with cosmologicl constnt e vilble [6-]. A genel clss of womhole geometies with cosmologicl constnt nd junction conditions ws nlyzed by De Benedictis nd Ds [9], nd futhe exploed in highe dimensions []. It is of eest to study positive cosmologicl constnt, s the infltiony phse of the ult-ely univese demnds it, nd in ddition, ecent stonomicl obsevtions po to Λ >. Lobo [], with the ension of minimizing the exotic mtte used, mtched sttic nd spheiclly symmetic womhole solution to n eio vcuum solution with cosmologicl constnt, nd he clculte the sufce stesses of the esulting shell nd the totl mount of exotic mtte using volume egl quntifie [3]. The constuction of tvesble womhole solutions by mtching n eio womhole spcetime to n eio solution, t junction sufce, ws nlyzed in [3-5]. A thin-shell tvesble womhole, with zeo sufce enegy density ws nlyzed in [5], nd with geneic sufce stesses in [4]. A genel clss of womhole geometies with cosmologicl constnt nd junction conditions ws exploed in [9], nd lineized stbility nlysis fo the plne symmetic cse with negtive cosmologicl constnt is done in [7]. Mois-Thone womholes, with Λ, hve two symptoticlly flt egions spcetime. By dding positive cosmologicl constnt Λ >, the womholes hve two symptoticlly de-sitte egions, nd by dding negtive cosmologicl constnt, Λ <, the womholes hve two symptoticlly nti-de Sitte egions. We nlyze symptoticlly flt nd sttic tvesble Mois-Thone womholes in the pesence of cosmologicl constuct. An eqution connecting the dil tension t the mouth with the tngentil sufce pessue of the thinshell is deived. The stuctue s well s sevel physicl popeties nd chcteistics of tvesble womholes due to the effects of the cosmologicl tem e studied. This ticle is ognized s follows: In Sec. II we studied Einstein s field equtions nd totl stess-enegy with cosmologicl constnt Λ. In Sec. III, we oduce n exct blck hole solution with electomgnetic nd confomlly coupled scl fields. The junction conditions nd the sufce tngentil pessue e discussed in Sec. 3

2 Intentionl Jounl of Scientific & Engineeing Resech, Volume 4, Issue, Octobe-3 5 ISSN IV. Specific constuction of womhole with geneic cosmologicl constnt is discussed in Sec. V. Finlly, conclusion of the esults is given in Sec. VI. II EINSTEIN S FIELD EQUATIONS AND SURFACE STRESSES WITH A COSMOLOGICAL CONSTANT Λ ) Fom of the Spcetime Metic The eio spcetime metic fo the womhole in the sttic nd spheiclly symmetic isotopic coodinte ( t,,θ, φ ), is given by [] Φ( ) d ds e dt ( dθ sin θdφ ), () b( ) Φ( is denoted s the edshift function, fo it is elted to the gvittionl edshift nd () whee ) b is clled the fom function, s it detemines the shpe of the womhole; both e functions of the dil coodinte. Fo the tvesble womhole, one must demnd tht thee e no hoizons pesent, which e identified s the sufces with Φ e, so the Φ () must be finite eveywhee. The dil coodinte hs nge tht inceses fom minimum vlue t, coesponding to the womhole thot to. Mximum vlue of coesponding to the mouth t one hs to join smoothly this spheicl volume to nothe one copy with nging gin fom to. In ddition, one hs then to join ech copy to the enl spcetime fom to, s will be done. The detils of subsequent mthemtics nd of physicl epettions will be simplified using set of othonoml bsis vectos s the pope efeence fme, the obseves emin t est in this coodinte system ( t,,θ, φ ), with (,θ, φ ) constnt. The bsis vectos in this coodinte system e denoted by et, e, e θ, eφ. The tnsfomtion of these bsis vectos fom the pope efeence fme to boosted fme is s follows: φ et e e t, eˆ ( b ) e e ˆ e θ θ nd e ˆ ( sinθ ) e φ φ. () In this bsis the metic coefficients ssume on thei stndd, specil eltivity foms e given by g ˆ µ ˆ ν e ˆ µ. eˆ ν η ˆ µ ˆ ν dig (,,, ). (3) In the othonoml efeence fme, the Einstein field eqution with geneic cosmologicl constnt cn be witten s G ˆ µ ˆ ν Λη ˆ µνˆ T ˆ µνˆ. (4) b) The Totl Stess-Enegy Tenso with Cosmologicl Constnt One my wite the Einstein field eqution with cosmologicl constnt in the following mnne; ( vc) G G T T, (5) whee ( ) µ ˆ ν π ˆ µνˆ ˆ µ ˆ ν ˆ 8 ( ) g ( Λ ( πg) ) T vc ˆ µ ˆ ν ˆ µνˆ 8, is the stess-enegy tenso ssocites with the vcuum, nd in the othonoml efeence fme is given by T ( vc) dig [ ( G ) ( G ) ( G ) ( G ) ˆ µ ˆ ν Λ Λ Λ Λ ]. (6) Fo the metic (), the non-zeo components of the Einstein tenso in the othonoml efeence fme cn be witten s [] b G t ˆˆ t, (7) b b Φ 3 G ˆ, (8) 3

3 Intentionl Jounl of Scientific & Engineeing Resech, Volume 4, Issue, Octobe-3 6 ISSN Φ Φ Φ Φ b b b b b G ˆ ˆ G ˆ ˆ ( ) ( b) ( b ). (9) θθ φφ Using the Einstein field equtions with non-zeo cosmologicl constnt in n othonoml efeence fme, we obtin the following stess-enegy scenio b b Φ τ ( ) Λ 8 G, () 3 π b Φ b b b b p( ) Φ ( Φ ) Φ Λ G ( ) ( ) b b, () whee ρ () is the enegy density, τ () is the dil tension, p () is the pessue mesued in the ltel diections, othonoml to the dil diection. We obtin the eqution fo τ by tking the deivtive of Eq. () with espect to the dil coodinte nd eliminting b nd Φ, given in Eqs. () nd (), espectively, τ ( ρc τ ) Φ ( ρ τ ). (3) Eqution (3) is known s the eltivistic Eule eqution o the hydosttic equilibium eqution fo the mteil theding the womhole. This eqution cn lso be obtined using the consevtion of the stess-enegy tenso ˆ µνˆ T, putting µ. The consevtion of the stess-enegy tenso cn lso be deduced fom the Binchi ; identities, which e equivlent to µνˆ G ; ˆ ν. III EXTERIOR SOLUTION WITH GENERIC Λ The eio vcuum solution of Einstein field equtions is given by Λ GM Λ GM ds dt d ( dθ sin θdϕ ) (4) 3 3 whee <. This is the solution of de-sitte blck hole with confomlly coupled scl field nd lso known s MTZ solution. The scl field is given by φ( ) 3 GM. 4π GM (5) The MTZ solution exists only fo dimensionless constnt, α πλg, nd descibe sttic nd 9 spheiclly symmetic blck hole with positive cosmologicl constnt Λ. The mss of the blck hole stisfies > GM > l / 4, whee l is the cosmologicl dius nd is given by l 3 / Λ. The inne, event nd cosmologicl hoizon stisfies < GM < l / < c l, whee l ( 4GM / l ), (6) l ( 4GM / l ), (7) l ( 4GM / l ). (8) l The solution (4) hve singulities t the dii ± ( ± 4GM / l ). b cn be consideed s the event hoizon of the vcuum blck hole solution, but since the womhole mtte will fill egion up 3

4 Intentionl Jounl of Scientific & Engineeing Resech, Volume 4, Issue, Octobe-3 7 ISSN to womhole dius supeio thn b. This dius does not ente o the poblem. Fo the sme eson,, the inne event hoizon of the blck hole is not consideed in the pesent poblem. So c cn be egded s the position of the cosmologicl event hoizon of the de-sitte spcetime. Keeping Λ fixed, if one inceses M, will incese nd will decese. Fo the mximum llowed vlue of the mss, M l(4g), the blck hole event hoizon nd cosmologicl hoizon e sme i.e., l /. In the cse of vnishing cosmologicl constnt Λ, the geomety of the eme Reissne-Nodstöm metic ( dθ sin θ d ) GM GM φ ds dt d, (9) which hs colesced inne nd event hoizons t GM. Fo the mssless cse, M, the blck hole geomety in de-sitte spcetime nd the metic tkes simple fom Λ d ds dt ( dθ sin θ dφ ). () 3 Λ 3 Fo Λ, the de-sitte metic tends to the Minkowskin spcetime. IV JUNCTION CONDITIONS In ode to mtch the eio nd eio mtices, one needs the boundy sufce S tht connects them. The fist condition is tht the metic must be continuous t S, i.e., g µν S gµν S. This condition is not sufficient to join diffeent spcetimes. The second condition fo mking the mtch cn be done diectly with the field eqution, due to the spheiclly symmetic. We cn use the Einstein field equtions, Eqs. (7), (8) nd (9), to detemine the enegy density nd stesses of the sufce necessy to hve mtch between the eio nd eio solutions. When thee is null stess-enegy tems t S, we cn sy tht the junction is boundy sufce. On the othe hnd, if sufce stess-enegy tems e pesent, the junction is clled the thin-shell. Since both the inside nd outside mtices e spheiclly symmetic, the components G θθ nd G φφ e ledy continuous, nd theefoe one is left with imposing the continuity G tt nd G, these cn be witten s g nd g. At, with g nd g being the metic components fo the g tt tt g eio egion t, nd g tt nd g the eio metic components fo the vcuum solution t. We e consideing the eio solution Eq. () nd the MTZ eio solution Eq. (4) mtched t sufce, S. The continuity of the mtices then give geneiclly Φ Φ nd b b. Now comping Eqs. () nd (4), the ed shift nd shpe functions cn be witten s Λ GM Φ( ) ln, () 3 G M Λ 3 b GM. () 3 We conside pticul choice in which the sttic eio obseve mesues zeo tidl foces, i.e., Φ const. nd Φ. Since the shell is infinitesimlly thin in the dil diection, so thee is no dil sufce pessue. Theefoe we e left with sufce enegy density σ nd sufce tngentil pessue P. At the boundy S, the stess-enegy tenso T ˆ µ ˆ ν is popotionl to Dic delt function, so one cn wite T ˆ ˆ t ˆ ˆ ( ˆ ˆ µ ν µν δ ). To find t ˆ µ ˆ ν we then use tt 3

5 Intentionl Jounl of Scientific & Engineeing Resech, Volume 4, Issue, Octobe-3 8 ISSN whee ( ˆ ˆ ) dˆ G ˆ dˆ µ ˆ ν t ˆ µ ˆ νδ, (3) mens n infinitesiml egl though the shell. Now using the popety of the δ function [ ] ( ) ( f ( x) ) / f ( x) δ x, nd g ( x) δ ( x x ) g( ) δ t ˆ µνˆ b G t ˆˆ t x, we find G dˆ. (4) ˆ µ ˆ ν We see tht only depends on the fist deivtive of the metic, which e continuous fo eio nd eio solutions. Thus, since the egl gives the vlue of the metic on the eio side ( b, sy) minus the vlue of the metic on the eio side ( b, sy), it gives zeo, nd one finds σ. In Eq. (9) we see tht b /, the othe tems in this eqution G hs n impotnt tem ( ) ˆ θ ˆ θ [ ]Φ depend t most on the fist deivtive nd do not contibute to the egl. Fom Eq. (3), we obtin G / p b / Φ. Now Φ is tken befoe nd GM G M Λ Φ ( b / ) 3 3. Theefoe, the sufce tngentil pessue cn be obtined s GM G M Λ P 3. (5) 8 π G b / This eqution cn be witten moe explicitly s GM G M Λ P 3. (6) 8 π G GM Λ 3 One cn obtin the mtching eqution of the dil pessue coss the junction boundy of the thin-shell. This is done by consideing two genel solutions of Eq. (), nd n eio nd eio solutions mtched t the junction sufce. The dil component of the Einstein Eq. (), povides us b b Φ 8 G ( ), 3 τ Λ b b Φ 8 G τ 3 ( ) Λ Φ Φ ( nd b b π (7) π. (8) At the junction boundy, one hs obtined ) consideing Φ ( ). Fom Eq. () we hve. Fo simplicity, we e 3

6 Intentionl Jounl of Scientific & Engineeing Resech, Volume 4, Issue, Octobe-3 9 ISSN GM G M Λ 3 ( ) 3 Φ. (9) Λ GM 3 Using Eqs. (9) nd (6), we veify tht Eqs. (7) nd (8) povide us with n eqution which govens the behvio of the dil tension t the boundy, nmely Φ( ) τ Λ τ Λ Pe, (3) 8 G 8 G π Φ( ) GM Λ whee we hve put e 3 the tngentil pessue of the thin-shell. π. This eqution eltes the dil tension t the sufce with V SPECIFIC CONSTRUCTION OF WORMHOLE WITH GENERIC Λ To constuct specific womhole solutions with geneic cosmologicl constnt Λ, we biefly discuss the two cses Λ, Λ >. The specific womhole solutions e given below. ) Specific Tvesble Womhole Solution with Λ.) Junction with P With the junction hving the tngentil pessue, P, we conside mtching of n eio solution to n eio MTZ solution, so we hve τ nd Λ 3 l. In the cse of the boundy sufce, i.e. P, we obtin Λ. Thus thee is no womhole solution with P..) Junction with P Agin we conside mtching of the eio solution to n eio MTZ solution with the tngentil pessue of the junction, P, we hve τ nd Λ. At the junction of the shell, the behvio of the dil tension is given by Eq. (3) nd consideing Eq. () we find the shpe function t the junction simply G M educes to b GM. Fo diffeent womhole solutions, we shll conside vious choice of the shpe function b ().. Fist we conside the womhole solution fo the functions ( ) b( ) o ; Φ( ) Φ. (3) whee is the thot dius of the womhole. The Einstein field equtions e given by / ρ ( ) Λ, (3) 5 / 6π G / τ ( ) Λ, (33) 5 / G / p( ) Λ (34) 5 / 3π G In this cse the enegy density ρ cn be positive o zeo, depending on the vlue of the enl cosmologicl G M constnt Λ. The thot dius of the womhole fte mtching the shpe functions b GM nd 3

7 Intentionl Jounl of Scientific & Engineeing Resech, Volume 4, Issue, Octobe-3 3 ISSN ( ) b ( ) must be gete thn the blck hole dius. The constnt φ must stisfy the escling eio metic,, is given by ( dθ sin θd ) GM d ds dt φ, (35) while the eio metic,, is the MTZ solution (4).. Second specific womhole solution is b( ) ; Φ( ) Φ, (36) whee is the thot dius of the womhole. The Einstein field equtions e given by ρ( ) Λ, (37) 4 G G Φ e nd τ ( ) Λ, (38) 4 G G p( ) Λ. (39) 4 G 3π G In this cse the enegy density ρ cn be positive o zeo, depending on the vlue of the enl cosmologicl constnt Λ. G M The dius of the womhole thot fte mtching the two shpe functions b GM nd b( ) Φ GM must be gete thn the blck hole dius. The constnt φ must stisfy e. To find the eio metic of the womhole, we must impose the condition,, nd this is given by ( dθ sin θd ) GM d ds dt φ. (4) The eio metic,, is the MTZ metic (4). b) Specific Tvesble Womhole Solution with Λ > b.) Junction with P Now we shll conside the mtching of n eio solution to n eio MTZ solution, τ nd Λ >, t boundy sufce, P. One my obtin Eq. (3) tht holds the following condition τ Λ Λ, (4) 3 t the boundy sufce. Now in view of Eq. (4), we hve b Λ. We shll conside identicl shpe functions s in the pevious section.. Fist specific womhole solution fo the following functions: GM 3

8 Intentionl Jounl of Scientific & Engineeing Resech, Volume 4, Issue, Octobe-3 3 ISSN b ( ) ( ) ; Φ( ) Φ. (4) Fom mtching the shpe functions b b ( ) (, one cn find the dius of the womhole 3 Λ nd ) nd this dius must be gete thn the blck hole dius. Moeove, the constnt φ must stisfy the ed shift Φ GM e. The escling eio metic of the womhole t function, is l given by GM d ds dt ( dθ sin θdφ ). (43) l The escling eio metic,, is the sme s eio solution of MTZ blck hole (4).. The second specific womhole solution is b( ) ; Φ( ) Φ. (44) 3 The dius of the womhole fte mtching the shpe functions b Λ nd b( ) must be gete thn Φ GM the blck hole dius. In the cse, the ed shift function tkes the fom e nd l the escling eio metic is GM d ds dt ( dθ sin θdφ ). (45) l The eio metic,,, is sme s the MTZ solution (4). IV CONCLUSIONS We hve studied Mois-Thone sttic tvesble womhole with geneic positive cosmologicl constnt Λ by mtching the enl nd enl geometies of two blck solutions. In the enl egion we impose ppopite geomety to obtin spheiclly symmetic tvesble womhole, while, in eio egion we use MTZ blck hole solution. The sufce tngentil pessue with the sufce enegy density of the exotic mtte is locted t the thot of the womhole. To mtch vcuum eio solution with eio solution, we hve deduced n eqution fo the tngentil sufce pessue nd nothe one which influences the behvio of the dil tension t the boundy. We see tht thee is no womhole solution with zeo tngentil pessue t p, it fom boundy sufce. The womhole solutions e obtined with non-zeo tngentil pessue, i.e., p. We biefly we epesent some specific solutions of the tvesble womholes fo diffeent choices of the shpe functions of the womhole. REFERENCES. M. S. Mois nd K. S. Thone, Am. J. Phys. 56, 395 (988).. Y. B. Zel dovic Sov. Phys. Uspekhi, Vol., No. 3, pp. 38 (968). 3. S. Pelmutte et l., Ntue (London) 39, 5 (998); Astophysics. J. 57, 565 (999). 4. M. Heusle, Blck Hole Uniqueness Theoems (Cmbidge Univesity Pess, Cmbidge, Englnd, (996). 5. V. Foni nd S. Sonego, Clss. Qunt. Gv., 85 (993). 6. S. Kim, Phys. Lett. A 66, 3 (99). 3

9 Intentionl Jounl of Scientific & Engineeing Resech, Volume 4, Issue, Octobe-3 3 ISSN T. A. Romn, Phys. Rev. D 47, 37 (993). 8. M. S. R. Delgty nd R. B. Mnn, Int. J. Mod. Phys. D 4, 3 (995). 9. A. DeBenedicts nd A. Ds, Clss. Qunt. Gv. 8, 87 ().. A. DeBenedicts nd A. Ds, Nucl. Phys. Rev. B 653, 79 (3).. S. V. Sushkov nd Y. Z. Zhng, Phys. Rev. D 77, 44 (8).. F. S. N. Lobo, Phys. Rev. D 7, 4 (5). 3. F. S. N. Lobo,Xiv:g-qc/ F. S. N. Lobo, Clss. Qunt. Gv., 48 (4). 5. J. P. S. Lemos, F. S. N. Lobo nd S. Q. de Olivei, Phys. Rev. D 68, 644 (3). 6. F. S. N. Lobo nd P. Cwfod, Clss. Qunt. Gv., 39 (4). 7. J. P. S. Lemos nd F. S. N. Lobo, Phys. Rev. D 69, 47 (4). 8. J. P. S. Lemos, Phys. Lett. B 35, 46 (995). 9. J. P. S. Lemos nd V. T. Znchin, Phys. Rev. D 54, 384 (996).. J. P. S. Lemos, Xiv:g-qc/9.. J. P. S. Lemos nd F. S. N. Lobo, Phys. Rev. D 59, 44 (999). 3