TRAVERSABLE SCHWARZSCHILD WORMHOLE SOLUTION

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1 JURNAL FIZIK MALAYSIA Volume 34, numbe 3 3 TRAVERSABLE SCHWARZSCHILD WORMHOLE SOLUTION Nu Izzati Ishak and Wan Ahmad Tajuddin Bin Wan Abdullah Theoetical Physics Laboatoy Depatment of Physics, Faculty of Science, Univesity Malaya 563 Kuala Lumpu,Malaysia Stating with spheically symmetic spacetime metic poposed by Mois-Thone; the equations of womhole stuctue ae deived to solve Einstein Field equation using Mois Thone famewok. The stuctue is obtained though embedding method whee it can be shown that to maintain the cuvatue at the thoat of womhole, mateial with adial tension that exceed the enegy density, c is essential to expand and hold the thoat fom collapse. Matte that possesses this chaacteistic is coined as exotic matte, violates all the enegy condition. Keywods: avity, geneal elativity, Einstein equation, enegy density. INTRODUCTION Pio to geneal elativity, special theoy of elativity states that the space and time ae combined to fom single entities in 4-dimensional Minkowski spacetime which the geomety of spacetime is kept intact o static. eneal elativity then descibes moe dynamics geometic elationship of spacetime. The Einstein field equation epesents this elationship and shows how the matte moves unde the influence of cuvatue of spacetime and how the spacetime cuve unde the influence of matte []. Womhole aises as one of the solution in Einstein field equation. Begin in 96, Kal Schwazschild discoveed the fist exact solution fo Einstein field equation, then in 935 Einstein-Rosen Bidge o Schwazschild womhole which is the Schwazschild geomety on hypespace consist of bidge that connects the two asymptotically flat univeses was mathematically discoveed by Einstein togethe with Nathan Rosen []. Following this discovey, in 988 K. S. Thone togethe with his student Mois successfully build a new class of womhole which possesses no event hoizon, making this womhole theoetically tavesable not only fo photon o subatomic paticles but also fo sizeable object as human [3]. We conside the Mois-Thone famewok in constucting a tavesable womhole [3] solution and deive the equation that descibes the exoticity of matte used as the souce to geneate spacetime cuvatue. The following conditions listed by Mois and Thone on chaacteistics of the tavesable womhole ae followed:. Metic is spheically symmetic and static. In evey point on the spacetime, the metic must fulfill Einstein field equation. 3. No event hoizon. 4. The solution must be consist shape of womhole; thoat between two mouths that joined the asymptotically flat space and flae-out condition at minimum adius. 5. Matte o field that poduces the cuvatue fo womhole must have physically acceptable enegy. One of ou motivations to study the subject is because the solution fo this type of womhole is simple that they can be used as guide to undestand basic geneal elativity, solve Einstein field equation, physically intepet the solution and exploe the popeties of the solution whee we will also discuss biefly about the enegy condition and phenomena that goven its violation. METHODOLOY I. The Riemann, Ricci and Einstein Tensos Stating fom static spheically symmetic fom of spacetime ( ct,,, ) coodinate is given as: ds e ( ) d c dt ( d sin d ) b/ () b and Whee ae abitay function of adial coodinates. Define b as shape function that detemine the shape of womhole and is the gavitational edshift. Radial coodinate is nonmonotonic which coves the ange, whee is minimum value at thoat. We wite () in metic tensos g fom: g diag g, g, g, g33 () And the conjugate metic tenso is Chistoffel symbols ae given as: g / g

2 JURNAL FIZIK MALAYSIA Volume 34 3 g g g g g, (3) We have calculated the nonzeo values fo ' 33 b sin b e c ' 33 sin cos ( b / ) ( b ) 3 3 b 3 3 cot (4) while othe components vanishes. The symbol pime denoting the diffeentiations with espect to adial coodinate. Riemann tenso R : R (5) Contaction of Riemann tenso yield Ricci tenso R : R (6) We obtain the nonzeo Ricci tensos by substitution of appopiate values in (4) into (6): b b ' R e c ' '' ' b b R ' " ' b (7) (8) b R b ' b (9) R33 Rsin () The final contaction of Ricci tenso yield cuvatue scala o Ricci scala R g R 33 R g R g R g R g R33 b b b R ' '' 4' ' () Einstein cuvatue tenso elates Ricci tenso, Ricci scala and metic tenso such as: R gr () Using the expession obtained fom the Ricci cuvatue tensos and Ricci scala, the nonzeo Einstein tensos () ae: e c b ' b b (3) (4) b ' b ' b b ' b " ' (5) 33 33sin (6) Relationship between the cuvatue tenso of space and stess-enegy tenso is given by Einstein field equation via the Bianchi identity: 8 4 T c (7) II. Einstein Field Equation Mois-Thone [3] and Lemos et al.[4] simplified mathematical analysis and physical intepetation by intoducing othonomal basis vecto that epesents the pope efeence fame fo an obseve who emains at est in the coodinate system ct,,, with,, fixed [3,4]. Basis vectos in this coodinate system ae denoted as e e, e, e e t, e, e, e, 3 with tansfomation e e whee ˆ ˆ ˆ ˆ / b /,, sin diag e, (8) Thus, we obtained othonomal tems fo metic tensos, Ricci tensos and Einstein tensos in basis vectos such as: g g (9) ˆ ˆ R ˆ ˆ ˆ ˆ ˆ ˆ ˆ R ˆ () ˆ ˆ () The Einstein tenso in othonomal efeence fame is R g R with the nonzeo components: ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ' b () ˆ ˆ 3 (3) ˆ ˆ b b b " b b b ' b ' ' (4) (5) 3ˆ 3ˆ ˆˆ Bikhoff theoem testifies the existence of spheical (nontavesible) Schwazschild womhole as the only solution that can exists in vacuum Einstein field equation. Hence, the tavesable womhole must be intetwined by matte of field with nonzeo stessenegy tenso. Einstein field equation demands the popotionality between the stess-enegy tenso with Einstein cuvatue tenso ˆ ˆ T ˆ ˆ giving us the

3 JURNAL FIZIK MALAYSIA Volume 34 3 same algebaic stuctue between both tensos. The non-zeo stess-enegy tensos ae T, T, T and T cay intepetation of measuement made by the 3ˆ3ˆ static obseve: T t ˆ t ˆ ( ) c (6) T ˆ ˆ ( ) (7) T ˆ ˆ T ˆ ˆ p( ) (8) in which () is the total enegy density, () is the adial tension and p () is the pessue measued in diections of othogonal to adial diection [3,4]. Using elationship (7) with (6-8) to obtain: c 8 4 ˆˆ ˆˆ ˆˆ (9) c b b ' 8 3 (3) c b b p( ) '' ' 8 b / (3) b ' b / Diffeentiate (3) with espect to adial coodinate then eliminate b ' and ' ' in (9) and (3) to obtain elationship: p (3) ' c ' which is Eule equation o equation of hydostatic equilibium fo mateial theading the womhole [4].Fom (4),ewite (3) as: p ( ) c ' ' (33) Thus we obtain thee diffeential equations that coelate with five unknown functions of, p,, and b. Fom (9-3) function is detemined by b() while is detemined by b () and '( ). Nomal way of solving this type of Einstein field equations is by assuming specific matte o field that contibutes to the souce of stess-enegy tenso. Fom the natue of this souce, we deived the equation of state of adial tension and tangential pessue as a function of enegy density. Then we obtained two equations of state and thee field equations that yield five equations with five unknown functions that is sufficient to geneate geomety of spacetime given in metic g. By following Mois and Thone famewok, we fist geneate suitable geomety fo womhole solution by contolling shape function b and edshift function. Then we detemine the favoable stess-enegy tenso fom the geomety though the field equations of, p and [5]. III. Mathematical of Embedding We epesent the shape of womhole via the embedding diagam and obtain the infomation of ou chosen shape function though it. An equatoial slice, / though spheical symmety geomety at the fixed t consttime is consideed. Then the line element epesents this slice is e-paameteized by using (): d (34) ds d b / Ou stategies ae to constuct a mental image of stuctue fo this slice by embedding this metic into 3-D Euclidean space. We wite metic in this suface in cylindical coodinate,, z as: ds dz d d (35) Due to the symmety aound the axis, equation fo the embedded suface is assigned by single equation z z(), which esults to the metic of this suface: dz ds d d (36) d Equating (34) with (36), we obtained equation of embedding suface: dz d b / (37) FIURE. Embedding -dimensional cuve pofile of womhole nea thoat egion in 3-dimensional Euclidean space, Womhole shape is geneated by otating cuve about π though z-axis Since adial coodinate behave pooly nea the thoat, we intoduce the pope adial distance l to descibe the divegence of thoat i.e.: uppe pat of womhole z a d l( ) (38) / b( )/ and lowe pat of womhole z a d l( ) (39) / b( ) /

4 JURNAL FIZIK MALAYSIA Volume 34 3 Lowe limit of the integation is the adius of thoat, and the maximum uppe limit is set up as the adius of womhole mouth, a as pesented in Fig.Shape function b () is positive so that / b is eal. Solution fo womhole equied the flaing-out condition at the thoat. Fom (37) we obtain invese embedding function (z) as: / d ( ) dz b (4) The geomety of flae- out condition is applied by stict minimality at the womhole thoat which is: d b dz b (4) FIURE. Embedding diagam fo geomety of womhole that sepaating two pats of univese. IV. Absence of Hoizon Setting edshift, () eveywhee finite as l to eliminate event hoizon means that the pope time lapse vanished at the finite coodinate time. V. Exotic Matte At the thoat of a womhole, b, the enegy density is finite with ' b (9) and b / '.Fom (3) we obtain tension in womhole thoat. Futhe calculation [3] eveals the huge 9magnitude of tension appoximately equal to pessue inside the most massive neuton sta fo b 3km As the esult, the dimensionless function is intoduced to define the configuation equied to suppot the womhole i.e. ) / ( c c. Fom the substitution (9, 3, and 33) into this dimensionless function, we obtain: c b / ( b) ' c (4) Compaing (44) with (43): b d b ' (43) dz Again at the thoat of womhole, is finite, lead to the finiteness of elationship of: c c and b/ ' give us (44) which is defined as the exoticity function of mateial. RESULTS AND ANALYSIS The constaint in c is vey clamoous since it states that the adial tension at thoat is lage as it must supassed the magnitude of the enegy density. The mateial that exhibits popety of c is called exotic matte. This popety means the measuement is made by the obseve who moves though thoat with adial velocity appoaching speed of light,. Let the basis vectos fo this static obseve e, e, e, e applied to the standad Loentz ˆ ˆ ˆ 3ˆ tansfomation. We ae inteested with the obsevation of enegy density measued by the obseve s time. It is epesented by the pojection of stess-enegy tenso (3) with the basis vecto on the obseve s time obtained by e e ( v/ c) e : ˆ' v / c T ( v/ c T ˆ T T ) ˆ 'ˆ' ˆˆ ˆˆ c ( v/ c) whee / v / c c (45). As the obseve move with vey fast adial velocity ( vey lage) he will obseved negative enegy density. This is due to null geodesic that ente womhole at uppe mouth and emege at lowe having nonmonotonic coss section aea. This situation only happens if the matte in this egion exhibit gavitational epulsion popeties, which epulse the passing light ay. A epulsion equied negative enegy density which only possible if the null enegy condition, NEC [3, 4, 6] is violated. The negative mass enegy acts like divegent lens that display negative angle of deflection [7]. Amount of exotic matte used in womhole solution is quantified via the ( ) c c Due to the poblematic natue of the exotic matte we must minimize ou usage of this matte in womhole solution. DISCUSSION VI. Enegy Condition The clamoous point of exotic matte leads us to minimize usage of this matte in womhole solution ˆ ˆ ˆ

5 JURNAL FIZIK MALAYSIA Volume 34 3 by applying some constaint on enegy density. The enegy condition can be genealized into thee types; namely weak enegy condition (WEC), stong enegy condition (SEC) and null enegy condition (NEC). NEC is the weakest enegy condition that can be satisfied by any classical matte. Many mathematical theoems in geneal elativity ae built based on the assumption that classical matte must satisfy all enegy condition mentioned. Howeve, we also can show the violation of this enegy condition [] by consideing the aveage weak enegy condition (AWEC) that is taking the aveage point wise enegy condition along the obseve s geodesic to obtain appopiate solution fo Einstein field equation. Instead of measuing the enegy density at one point in obseve s geodesics, we aveaged all the enegy density measued acoss the geodesics. This allows the existence of negative enegy at some potion of the geodesics as long as the aveage sum of this enegy density is positive. We also can use the same definition fo aveage null enegy condition (ANEC) and aveage stong enegy condition (ASEC). By quantum inequalities (QI), the inetial obseve at Minkowski spacetime will obseve that the amount and duation of negative enegy density is limited [9]; this obseve can only see the lage negative enegy density duing a shot peiod of time which is the suitable condition in maintaining womhole shape. Howeve, QI is invalid fo the cuve spacetime. If we conside small spacetime volume at the thoat of womhole, then the spacetime can be appoximated as flat suface, so that QI can be applied. Fom [] analysis, the thoat possesses by womhole is slightly lage than Planck size. VII. Violation The violation of the enegy condition has been obseved in both classical and quantum level [] by phenomenon such as Casimi effect, Hawking evapoation, squeezed vacuum state in non -linea optics and many othes. Cosmologically it has been obseved [] that the cosmological fluid violates SEC which lead to the possibility that NEC is also violated in this egime. These esults play a vital ole in the constuction of womholes since the quantum effect has influences on the stess-enegy tenso and semiclassical gavity of womhole solution satisfied QI. Hence the at the quantum egime, these exotic spacetime aise natually. CONCLUSION In conclusion, by consideing a spheical symmety spacetime metic we obtained impotant equation of spacetime stuctue in solving Einstein field equation.following Mois and Thone appoach in solving these equations yield the definition of dimensionless function known as exoticity function c c This implies toublesome constaints on popeties of mateial at the thoat of womhole whee it must exhibit negative enegy density. The exoticity of this mateial epesents the measue of enegy conditions violation. ACKNOWLEDEMENT We thank M. Anua Alias and D Bijan Nikouavan. fo the poductive comments, discussion, citicism and idea. REFERENCES [] C. W. Misne, K. S. Thone & J. A. Wheele. (973). avitation. New Yok: W.H.Feeman and Company. [] A.Einstein & N.Rosen. (935). The paticle poblem in the geneal elativity. Physics Review, [3] M.S.Mois & K.S.Thone. (988). Womholes in spacetime and thei use fo intestella tavel: A tool fo teaching eneal Relativity. Ameica Jounal of Physics, 5(56), [4] Lemos, J.P.S., Lobo, F.S.N. & Oliveia, S.Q. (3). Mois-Thone womholes with a cosmological constant. Physics Review D, 6(68), 644. [5] A D.Benedictis & A Das (). On a geneal class of womhole geometies. Classical and Quantum avity, 8(7), 8 87 [6] M. Visse (989). Tavesable womholes: Some simple examples. Physics Review D, 39(), 38 [7] M. Safanova, D.F. Toes &.E. Romeo, (6). avitational lensing by womholes. Physics Review D, 74(), 4. [8] Fank J. Tiple, (978). Enegy conditions and spacetime singulaities. Physics Review D, 7(), [9] L.H Fod & T.A Roman (995). Aveaged Enegy Conditions and Quantum Inequalities.Physics Review D, 5(8), []M.Visse.(995).Loentzian Womholes: Fom Einstein to Hawking. New Yok: Ameican Institute of Physics. []L.H Fod &T. A. Roman (999).The quantum inteest conjectue. Physics Review D, 6(), 48. []A.. Rises et al., (4). Type IA Supenova Discoveies at z > Fom the Hubble Space Telescope: Evidence fo Past Deceleation and Constaints on Dak Enegy Evolution.Astophysics Jounal 67,