Chapter 2 Wormhole Basics

Transcription

1 Chpte 2 Womhole Bsics Fncisco S.N. Lobo 2.1 Sttic nd Spheiclly Symmetic Tvesble Womholes Spcetime Metic Thoughout this book, unless stted othewise, we will conside the following spheiclly symmetic nd sttic womhole solution [1 ds 2 = e 2Φ) dt 2 + d 2 1 b)/ + 2 dθ 2 + sin 2 θ dφ 2 ). 2.1) The metic functions Φ) nd b) e bity functions of the dil coodinte. AsΦ) is elted to the gvittionl edshift, it hs been denoted the edshift function, nd b) is clled the shpe function, s it detemines the shpe of the womhole [1 3, which will be shown below using embedding digms. The dil coodinte is non-monotonic in tht it deceses fom + to minimum vlue 0, epesenting the loction of the thot of the womhole, whee b 0 ) = 0, nd then inceses fom 0 to +. Although the metic coefficient g becomes divegent t the thot, which is signlled by the coodinte singulity, the pope dil distnce l) =± 0 [1 b)/ 1/2 d is equied to be finite eveywhee. The pope distnce deceses fom l =+, in the uppe univese, to l = 0 t the thot, nd then fom zeo to in the lowe univese. One must veify the bsence of hoizons, F.S.N. Lobo B) Fculdde de Ciêncis, Instituto de Astofísic e Ciêncis do Espço, Univesidde de Lisbo, Cmpo Gnde, Lisbo, Potugl e-mil: fslobo@fc.ul.pt Spinge Intentionl Publishing AG 2017 F.S.N. Lobo ed.), Womholes, Wp Dives nd Enegy Conditions, Fundmentl Theoies of Physics 189, DOI / _2 11

2 12 F.S.N. Lobo in ode fo the womhole to be tvesble. This condition must imply tht g tt = e 2Φ) = 0, so tht Φ) must be finite eveywhee. 1 Anothe inteesting fetue of the edshift function is tht its deivtive with espect to the dil coodinte lso detemines the ttctive o epulsive ntue of the geomety. In ode to veify this, conside the fou-velocity of sttic obseve given by U μ = dx μ /dτ = e Φ), 0, 0, 0). The obseve s fou-cceletion is μ = U μ ;ν U ν, which hs the following components: t = 0, = Φ 1 b ), 2.2) whee the pime denotes deivtive with espect to the dil coodinte. Now, note tht fom the geodesic eqution, dilly moving test pticle which stts fom est initilly hs the eqution of motion d 2 dτ = Γ 2 tt ) dt 2 =. 2.3) dτ Hee, is the dil component of pope cceletion tht n obseve must mintin in ode to emin t est t constnt, θ,φ, so tht fom Eq. 2.2), sttic obseve t the thot fo geneic Φ) is geodesic obseve. In pticul, fo constnt edshift function, Φ ) = 0, sttic obseves e lso geodesic. Thus, womhole is ttctive if > 0, i.e. obseves must mintin n outwd-diected dil cceletion to keep fom being pulled into the womhole. If < 0, the geomety is epulsive, i.e. obseves must mintin n inwd-diected dil cceletion to void being pushed wy fom the womhole. Indeed, this distinction depends on the sign of Φ, s is tnspent fom Eq. 2.2) The Mthemtics of Embedding We cn use embedding digms to epesent womhole nd extct some useful infomtion fo the choice of the shpe function, b). Due to the spheiclly symmetic ntue of the poblem, one my conside n equtoil slice, θ = π/2, without loss of genelity. The espective line element, consideing fixed moment of time, t = const, is given by ds 2 d 2 = 1 b)/ + 2 dφ ) 1 This follows fom esult oiginlly due to C.V. Vishveshw stted s follows: In ny symptoticlly flt spcetime with Killing vecto ξ ξ = e 0 fo the metic 2.1)) which i) is the odiny timetnsltion Killing vecto t sptil infinity nd ii) is othogonl to fmily of thee-dimensionl sufces, the 3-sufce ξ ξ = 0, i.e. e 0 e 0 = g tt = 0, is null sufce tht cnnot be cossed by ny outgoing, futue-diected timelike cuves, i.e. hoizon.

3 2 Womhole Bsics 13 Fig. 2.1 The embedding digm of two-dimensionl section long the equtoil plne t = const, θ = π/2) of tvesble womhole. Fo full visuliztion of the sufce sweep though 2π ottion ound the z xis, s cn be seen fom the gphic on the ight To visulize this slice, one embeds this metic into thee-dimensionl Eucliden spce, in which the metic cn be witten in cylindicl coodintes,,φ,z), s ds 2 = dz 2 + d dφ ) In the thee-dimensionl Eucliden spce the embedded sufce hs eqution z = z), so tht the metic of the sufce cn be witten s [ ds 2 = 1 + ) dz 2 d dφ ) d Comping Eq. 2.4) with 2.6), one deduces the eqution fo the embedding sufce, which is given by ) dz 1/2 d =± b) ) To be solution of womhole, the geomety hs minimum dius, = b) = 0, denoted s the thot, t which the embedded sufce is veticl, i.e. dz/d. F fom the thot, one my conside tht spce is symptoticlly flt, dz/d 0 s. To be solution of womhole, one lso needs to impose tht the thot fles out see Fig. 2.1 fo detils). This fling-out condition entils tht the invese of the embedding function z) must stisfy d 2 /dz 2 > 0 t o ne the thot 0.Diffeentiting d/dz =±/b) 1) 1/2 with espect to z, wehve

4 14 F.S.N. Lobo d 2 dz = b b > ) 2 2b 2 This fling-out condition is fundmentl ingedient of womhole physics, nd plys fundmentl ole in the nlysis of the violtion of the enegy conditions. At the thot we veify tht the fom function stisfies the condition b 0 )<1. Note, howeve, tht this tetment hs the dwbck of being coodinte dependent, nd we efe the ede to Refs. [4, 5 fo covint tetment Equtions of Stuctue fo the Womhole Fom the metic expessed in the fom ds 2 = g μν dx μ dx ν, one my detemine the Chistoffel symbols connection coefficients), Γ μ αβ, defined s Γ μ αβ = 1 2 gμν g να,β + g νβ,α g αβ,ν ), 2.9) which fo the metic 2.1) hve the following nonzeo components: Γ t t = Φ, Γ tt = 1 b ) Φ e 2Φ, Γ = b b 2 b), Γ θθ = + b, Γ φφ = b) sin 2 θ, Γ θ θ = Γ φ φ = 1, Γ θ φφ = sin θ cos θ, Γ φ θφ = tn θ. 2.10) The Riemnn tenso is defined s R α βγδ = Γ α βδ,γ Γ α βγ,δ + Γ α λγ Γ λ βδ Γ α λδγ λ βγ. 2.11) Howeve, the mthemticl nlysis nd the physicl intepettion is simplified using set of othonoml bsis vectos. These my be intepeted s the pope efeence fme of set of obseves who emin t est in the coodinte system t,,θ,φ), with,θ,φ)fixed. Denote the bsis vectos in the coodinte system s e t, e, e θ, e φ ). Thus, the othonoml bsis vectos e given by eˆt = e Φ e t eˆ = 1 b/) 1/2 e e ˆθ = ) e θ e ˆφ = sin θ) 1 e φ The nontivil Riemnn tenso components, given in the othonoml efeence fme, tke the following fom:

5 2 Womhole Bsics 15 R ˆt ˆ ˆt ˆ = R ˆt ˆ ˆ ˆt = R ˆ ˆt ˆt ˆ = R ˆ ˆt ˆ ˆt = 1 b )[ Φ Φ ) 2 + R ˆt ˆθ ˆt ˆθ = R ˆt ˆθ ˆθ ˆt = R ˆθ ˆt ˆt ˆθ = R ˆθ ˆt ˆθ ˆt = 1 b ) Φ ) Φ R ˆt ˆφ ˆt ˆφ = R ˆt ˆφ ˆφ ˆt = R ˆφ ˆt ˆt ˆφ = R ˆφ ˆt ˆφ ˆt = 1 b b b 2 b) Φ, 2.13), 2.14), 2.15) R ˆ ˆθ ˆ ˆθ = R ˆ ˆθ ˆθ ˆ = R ˆθ = R ˆθ ˆ ˆθ ˆ ˆ ˆ ˆθ = b b 2 3, 2.16) R ˆ ˆφ ˆ ˆφ = R ˆ ˆφ ˆφ ˆ = R ˆφ = R ˆφ ˆ ˆφ ˆ ˆ ˆ ˆφ = b b 2 3, 2.17) R ˆθ ˆφ ˆθ ˆφ = R ˆθ ˆφ ˆφ ˆθ = R ˆφ ˆθ ˆφ ˆθ = R ˆφ ˆθ ˆθ ˆφ = b 3, 2.18) whee, s befoe, pime denotes deivtive with espect to the dil coodinte. The Ricci tenso, R ˆμˆν, is given by the contction R ˆμˆν = R ˆα ˆμ ˆα ˆν, nd the nonzeo components e the following: Rˆt ˆt = 1 b )[ Φ + Φ ) 2 b 3b + 4 Φ, 2.19) 2 b) Rˆ ˆ = 1 b )[ Φ + Φ ) 2 + b b 2 b) Φ + b b, 2.20) 2 b) R ˆθ ˆθ = R ˆφ ˆφ = 1 b )[ b + b 2 2 b) Φ. 2.21) The cuvtue scl o Ricci scl, defined by R = g ˆμˆν R ˆμˆν, is given by R = 2 1 b )[ Φ + Φ ) 2 b b) b + 3b 4 Φ. 2.22) 2 b) Thus, the Einstein tenso, given in the othonoml efeence fme by G ˆμˆν = R ˆμˆν 1 2 Rgˆμˆν, yields fo the metic 2.1), the following nonzeo components: G ˆt ˆt = b, 2.23) 2 G ˆ ˆ = b b ) Φ 3, 2.24) G ˆθ ˆθ = 1 b )[ Φ + Φ ) 2 b b 2 b) Φ b b 2 2 b) + Φ, 2.25) G ˆφ ˆφ = G ˆθ ˆθ, 2.26) espectively.

6 16 F.S.N. Lobo Stess Enegy Tenso Though the Einstein field eqution, G ˆμˆν = 8πT ˆμˆν, one veifies tht the stess enegy tenso T ˆμˆν hs the sme lgebic stuctue s G ˆμˆν,Eqs.2.23) 2.26), nd the only nonzeo components e pecisely the digonl tems Tˆt ˆt, Tˆ ˆ, Tˆθ ˆθ nd T ˆφ ˆφ.Using the othonoml bsis, these components cy simple physicl intepettion, i.e. Tˆt ˆt = ρ), Tˆ ˆ = τ), Tˆθ ˆθ = T ˆφ ˆφ = p), 2.27) whee ρ) is the enegy density, τ) is the dil tension, with τ) = p ), i.e. it is the negtive of the dil pessue, p) is the pessue mesued in the tngentil diections, othogonl to the dil diection. Thus, the Einstein field eqution povides the following stess enegy scenio: ρ) = 1 8π τ) = 1 8π p) = 1 8π b, 2.28) [ 2 b 2 1 b ) Φ, 2.29) 3 1 b )[ Φ + Φ ) 2 b b b/) Φ b b b/) + Φ. 2.30) Note tht one now hs thee equtions with five unknown functions of the dil coodinte. Sevel sttegies to solve these equtions e vilble, fo instnce, one cn impose n eqution of stte [6 10 nd conside specific choice of the shpe function o of the edshift function. Note tht the sign of the enegy density depends on the sign of b ). One often comes coss the misleding sttement, in the litetue, tht womholes should necessily be theded by negtive enegy densities, o negtive mtte; howeve, this is not necessily the cse. Note, howeve, tht due to the fling-out condition, obseves tvesing the womhole with sufficiently high velocities, v 1, will mesue negtive enegy density. This will be shown below. Futhemoe, one should pehps coectly stte tht it is the dil pessue tht is necessily negtive t the thot, which is tnspent fo the dil tension t the thot, which is given by p ) = τ 0 ) = 8π 2 0 ) 1. By tking the deivtive with espect to the dil coodinte,of Eq.2.29), nd eliminting b nd Φ,giveninEqs.2.28) nd 2.30), espectively, we obtin the following eqution: τ = ρ τ)φ 2 p + τ). 2.31)

7 2 Womhole Bsics 17 Eqution 2.31) is the eltivistic Eule eqution, o the hydosttic eqution fo equilibium fo the mteil theding the womhole, nd cn lso be obtined using the consevtion of the stess enegy tenso, T ˆμˆν ;ˆν = 0, inseting ˆμ =. The effective mss, m) = b)/2 contined in the inteio of sphee of dius, cn be obtined by integting Eq. 2.28), which yields m) = πρ ) 2 d. 2.32) 0 Theefoe, the fom function hs n intepettion which depends on the mss distibution of the womhole Exotic Mtte nd Modified Gvity Exoticity Function To gin some insight into the mtte theding the womhole, Mois nd Thone defined the dimensionless function ξ = τ ρ)/ ρ [1, which tking into ccount Eqs. 2.28) nd 2.29) yields ξ = τ ρ ρ = b/ b 21 b/)φ b. 2.33) Combining the fling-out condition, given by Eq. 2.8), with Eq. 2.33), the exoticity function tkes the fom ξ = 2b2 b d 2 dz 2 1 b ) Φ 2 b. 2.34) Now, tking into ccount the finite chcte of ρ, nd consequently of b, nd the fct tht 1 b/)φ 0 t the thot, we hve the following eltionship: ξ 0 ) = τ 0 ρ 0 ρ 0 > ) The estiction τ 0 >ρ 0 is somewht toublesome condition, depending on one s point of view, s it sttes tht the dil tension t the thot should exceed the enegy density. Thus, Mois nd Thone coined mtte constined by this condition exotic mtte [1. We shll veify below tht this is defined s mtte tht violtes the null enegy condition in fct, it violtes ll the enegy conditions) [1, 2. Exotic mtte is pticully toublesome fo mesuements mde by obseves tvesing though the thot with dil velocity close to the speed of light. Conside Loentz tnsfomtion, x ˆμ = Λ ˆμ ˆν x ˆν, with Λ ˆμ ˆα Λ ˆα ˆν = δ ˆμ ˆν nd Λ ˆμ ˆν defined s

8 18 F.S.N. Lobo γ 00γ v Λ ˆμ ˆν ) = γ v 00 γ. 2.36) The enegy density mesued by these obseves is given by Tˆ0 ˆ0 = Λ ˆμ ˆ0 Λˆν ˆ0 T ˆμˆν, i.e. Tˆ0 ˆ0 = γ 2 ρ 0 v 2 τ 0 ), 2.37) with γ = 1 v 2 ) 1/2. Fo sufficiently high velocities, v 1, the obseve will mesue negtive enegy density, Tˆ0 ˆ0 < 0. This fetue lso holds fo ny tvesble, nonspheicl nd nonsttic womhole. To see this, one veifies tht bundle of null geodesics tht entes the womhole t one mouth nd emeges fom the othe must hve coss-sectionl e tht initilly inceses, nd then deceses. This convesion of decesing to incesing is due to the gvittionl epulsion of mtte though which the bundle of null geodesics tveses The Violtion of the Enegy Conditions The exoticity function 2.33) is closely elted to the null enegy condition NEC), which ssets tht fo ny null vecto k μ,wehvet μν k μ k ν 0. Fo digonl stess enegy tenso, this implies ρ τ 0 nd ρ + p 0. Using the Einstein field equtions 2.28) nd 2.29), evluted t the thot 0, nd tking into ccount the finite chcte of the edshift function so tht 1 b/)φ 0 0, we veify the condition ρ τ) 0 < 0. This violtes the NEC. In fct, it implies the violtion of ll the pointwise enegy condition. Although clssicl foms of mtte e believed to obey the enegy conditions, it is well-known fct tht they e violted by cetin quntum fields, mongst which we my efe to the Csimi effect. Thus, the flingout condition 2.8) entils the violtion of the NEC, t the thot. Note tht negtive enegy densities e not essentil, but negtive pessues e necessy to sustin the womhole thot. It is inteesting to note tht the violtions of the pointwise enegy conditions led to the veging of the enegy conditions ove timelike o null geodesics [11. The veged enegy conditions pemit loclized violtions of the enegy conditions, s long on vege the enegy conditions hold when integted long timelike o null geodesics. Now, s the veged enegy conditions involve veging ove line integl, with dimensions mss)/e), not volume integl, they do not povide useful infomtion egding the totl mount of enegy condition violting mtte. In ode to ovecome this shotcoming, the volume integl quntifie ws poposed [12. Thus, the mount of enegy condition violtions is then the extent tht these integls become negtive.

9 2 Womhole Bsics Womholes in Modified Theoies of Gvity Genelly, the NEC ises when one efes bck to the Rychudhui eqution, which is puely geometic sttement, without the need to efe to ny gvittionl field equtions. Now, in ode fo gvity to be ttctive, the positivity condition R μν k μ k ν 0 is imposed in the Rychudhui eqution. In genel eltivity, contcting both sides of the Einstein field eqution G μν = κ 2 T μν whee κ 2 = 8π) with ny null vecto k μ, one cn wite the bove condition in tems of the stess enegy tenso given by T μν k μ k ν 0, which is the sttement of the NEC. In modified theoies of gvity the gvittionl field equtions cn be ewitten s n effective Einstein eqution, given by G μν = κ 2 Tμν eff eff, whee Tμν is n effective stess enegy tenso contining the mtte stess enegy tenso T μν nd the cuvtue quntities, ising fom the specific modified theoy of gvity consideed [13.Now, the positivity condition R μν k μ k ν 0 in the Rychudhui eqution povides the genelized NEC, T eff μν kμ k ν 0, though the modified gvittionl field eqution. Theefoe, the necessy condition to hve womhole geomety is the violtion of the genelized NEC, i.e. T eff μν kμ k ν < 0. In clssicl genel eltivity this simply educes to the violtion of the usul NEC, i.e. T μν k μ k ν < 0. Howeve, in modified theoies of gvity, one my in pinciple impose tht the mtte stess enegy tenso stisfies the stndd NEC, T μν k μ k ν 0, while the espective genelized NEC is necessily violted, T eff μν kμ k ν < 0, in ode to ensue the fling-out condition. Moe specificlly, conside the genelized gvittionl field equtions fo lge clss of modified theoies of gvity, given by the following field eqution: [13 g 1 Ψ i )G μν + H μν ) g 2 Ψ j ) T μν = κ 2 T μν, 2.38) whee H μν is n dditionl geometic tem tht includes the geometicl modifictions inheent in the modified gvittionl theoy unde considetion; g i Ψ j ) i = 1, 2) e multiplictive fctos tht modify the geometicl secto of the field equtions, nd Ψ j denote geneiclly cuvtue invints o gvittionl fields such s scl fields; the tem g 2 Ψ i ) coves the coupling of the cuvtue invints o the scl fields with the mtte stess enegy tenso, T μν. It is useful to ewite this field eqution s n effective Einstein field eqution, s mentioned bove, with the effective stess enegy tenso, Tμν eff, given by Tμν eff 1 +ḡ 2Ψ j ) T g 1 Ψ i μν H μν, 2.39) ) whee ḡ 2 Ψ j ) = g 2 Ψ j )/κ 2 nd H μν = H μν /κ 2 e defined fo nottionl convenience. In modified gvity, the violtion of the genelized NEC, Tμν effkμ k ν < 0, implies the following estiction:

10 20 F.S.N. Lobo 1 +ḡ 2 Ψ j ) g 1 Ψ i ) T μν k μ k ν < H μν k μ k ν. 2.40) Fo genel eltivity, with g 1 Ψ j ) = 1, g 2 Ψ j ) = 0, nd H μν = 0, we ecove the stndd violtion of the NEC fo the mtte theding the womhole, i.e. T μν k μ k ν < 0. If the dditionl condition [1 +ḡ 2 Ψ j )/g 1 Ψ i )>0 is met, then one obtins genel bound fo the noml mtte theding the womhole, in the context of modified theoies of gvity, given by 0 T μν k μ k ν < g 1Ψ i ) 1 +ḡ 2 Ψ j ) H μν k μ k ν. 2.41) Tvesbility Conditions In constucting tvesble womhole geometies, we will be inteested in specific solutions by imposing specific tvesbility conditions. Assume tht tvelle of n bsudly dvnced civiliztion begins the tip in spce sttion in the lowe univese, t pope distnce l = l 1, nd ends up in the uppe univese, t l = l 2. Futhemoe, conside tht the tvelle hs dil velocity v), s mesued by sttic obseve positioned t. One my elte the pope distnce tvelled dl, dius tvelled d, coodinte time lpse dt, nd pope time lpse s mesued by the obseve dτ,by the following eltions: Φ dl v = e dt = e Φ v γ = dl dτ = 1 b 1 b ) 1/2 d ) 1/2 d dt, 2.42) dτ. 2.43) It is lso impotnt to impose cetin conditions t the spce sttions [1. Fist, conside tht spce is symptoticlly flt t the sttions, i.e. b/ 1. Second, the gvittionl edshift of signls sent fom the sttions to infinity should be smll, i.e. Δλ/λ = e Φ 1 Φ, so tht Φ 1. The condition Φ 1 imposes tht the pope time t the sttion equls the coodinte time. Thid, the gvittionl cceletion mesued t the sttions, given by g = 1 b/) 1/2 Φ Φ, should be less thn o equl to the Eth s gvittionl cceletion, g g, so tht the condition Φ g is met. Fo convenient tip though the womhole, cetin conditions should lso be imposed [1. Fist, the entie jouney should be done in eltively shot time s mesued both by the tvelle nd by obseves who emin t est t the sttions. Second, the cceletion felt by the tvelle should not exceed the Eth s

11 2 Womhole Bsics 21 gvittionl cceletion, g. Finlly, the tidl cceletions between diffeent pts of the tvelle s body should not exceed, once gin, Eth s gvity Totl Time in Tvesl The tip should tke eltively shot time, fo instnce, Mois nd Thone consideed 1 ye, s mesued by the tvelle nd fo obseves tht sty t est t the spce sttions, l = l 1 nd l = l 2, i.e. espectively. Δτ tvelle = Δt spce sttion = +l2 l 1 +l2 l 1 dl vγ 1 ye, 2.44) dl 1 ye, 2.45) veφ Acceletion Felt by Tvelle An impotnt tvesbility condition equied is tht the cceletion felt by the tvelle should not exceed Eth s gvity [1. Conside n othonoml bsis of the tvelle s pope efeence fme, eˆ0, eˆ1, eˆ2, eˆ3 ), given in tems of the othonoml bsis vectos of Eq. 2.12) of the sttic obseves, by Loentz tnsfomtion, i.e. eˆ0 = γ eˆt γ v eˆ, eˆ1 = γ eˆ + γ v eˆt, eˆ2 = e ˆθ, eˆ3 = e ˆφ, 2.46) whee γ = 1 v 2 ) 1/2, nd v) being the velocity of the tvelle s he psses,s mesued by sttic obseve positioned thee. Thus, the tvelle s fou-cceletion expessed in his pope efeence fme, ˆμ = U ˆν U ˆμ ;ˆν, yields the following estiction: = 1 b ) 1/2 e Φ γ e Φ) g. 2.47) Tidl Acceletion Felt by Tvelle It is lso convenient tht n obseve tvesing though the womhole should not be ipped pt by enomous tidl foces. Thus, nothe of the tvesbility conditions equied is tht the tidl cceletions felt by the tvelle should not exceed, fo instnce, the Eth s gvittionl cceletion [1. The tidl cceletion felt by the tvelle is given by

12 22 F.S.N. Lobo Δ ˆμ = R ˆμ ˆν ˆα ˆβ U ˆν η ˆα U ˆβ, 2.48) whee U ˆμ = δ ˆμ ˆ0 is the tvelle s fou-velocity nd η ˆα is the seption between two bity pts of his body. Note tht η ˆα is puely sptil in the tvelle s efeence fme, s U ˆμ η ˆμ = 0, so tht ηˆ0 = 0. Fo simplicity, ssume tht ηî 2m long ny sptil diection in the tvelle s efeence fme. Tking into ccount the ntisymmetic ntue of R ˆμ ˆν ˆα ˆβ in its fist two indices, we veify tht Δ ˆμ is puely sptil with the components Δî = Rî ˆ0 ĵ ˆ0 η ĵ = Rî ˆ0 ĵ ˆ0 η ĵ. 2.49) Using Loentz tnsfomtion of the Riemnn tenso components in the sttic obseve s fme, eˆt, eˆ, e ˆθ, e ˆφ ), to the tvelle s fme, eˆ0, eˆ1, eˆ2, eˆ3 ), the nonzeo components of Rî ˆ0 ĵ ˆ0 e given by Rˆ1 ˆ0 ˆ1 ˆ0 = Rˆ ˆt ˆ ˆt = 1 b )[ Φ Φ ) 2 + b b 2 b) Φ, 2.50) Rˆ2 ˆ0 ˆ2 ˆ0 = Rˆ3 ˆ0 ˆ3 ˆ0 = γ 2 R ˆθ ˆt ˆθ ˆt + γ 2 v 2 R ˆθ ˆ ˆθ ˆ = γ 2 [v 2 b b ) + 2 b)φ. 2.51) 2 2 Thus, Eq. 2.49) tkes the fom Δ ˆ1 = Rˆ1 ˆ0 ˆ1 ˆ0 ηˆ1, Δ ˆ2 = Rˆ2 ˆ0 ˆ2 ˆ0 ηˆ2, Δ ˆ3 = Rˆ3 ˆ0 ˆ3 ˆ0 ηˆ ) The constint Δ ˆμ g povides the tidl cceletion estictions s mesued by tvelle moving dilly though the womhole, given by the following inequlities: 1 b γ )[ Φ + Φ ) 2 b b ηˆ1 2 b) Φ g, 2.53) [v 2 b b ) + 2 b)φ ηˆ2 g. 2.54) The dil tidl constint, Eq. 2.53), constins the edshift function, nd the ltel tidl constint, Eq. 2.54), constins the velocity with which obseves tvese the womhole. These inequlities e pticully simple t the thot, 0,

13 2 Womhole Bsics 23 Φ 0 ) γ 2 v 2 2g 0 1 b ) ηˆ1, 2.55) 2g b ) ηˆ2, 2.56) Fo the pticul cse of constnt edshift function, Φ = 0, the dil tidl cceletion is zeo, nd Eq. 2.54) educes to γ 2 v b b ) ηˆ2 g. 2.57) Fo this specific cse one veifies tht sttiony obseves with v = 0 mesue null tidl foces. 2.2 Dynmic Spheiclly Symmetic Thin-Shell Tvesble Womholes An inteesting nd efficient mnne to minimize the violtion of the null enegy condition is to constuct thin-shell womholes using the thin-shell fomlism [2, 14 nd the cut-nd-pste pocedue s descibed in [2, Motivted in minimizing the usge of exotic mtte, the thin-shell constuction ws genelized to nonspheiclly symmetic cses [2, 15, nd in pticul, it ws found tht tvelle my tvese though such womhole without encounteing egions of exotic mtte. In the context of lineized stbility nlysis [16, two Schwzschild spcetimes wee sugiclly gfted togethe in such wy tht no event hoizon is pemitted to fom. This sugey concenttes nonzeo stess enegy on the boundy lye between the two symptoticlly flt egions nd dynmicl stbility nlysis with espect to spheiclly symmetic petubtions) ws exploed. In the ltte stbility nlysis, constints wee found on the eqution of stte of the exotic mtte tht compises the thot of the womhole. Indeed, the stbility of the ltte thin-shell womholes ws consideed fo cetin specilly chosen equtions of stte [2, 16, whee the nlysis ddessed the issue of stbility in the sense of poving bounded motion fo the womhole thot. This dynmicl nlysis ws genelized to the stbility of spheiclly symmetic thin-shell womholes by consideing lineized dil petubtions ound some ssumed sttic solution of the Einstein field equtions, without the need to specify n eqution of stte [18. This lineized stbility nlysis ound sttic solution ws soon genelized to the pesence of chge [19, nd of cosmologicl constnt [20, nd ws subsequently extended to pletho of individul scenios see [21 nd efeences theein). The key point of the pesent section is to develop n extemely genel, flexible, nd obust fmewok tht cn quickly be dpted to genel spheiclly symmetic tvesble womholes in dimensions see [21.

14 24 F.S.N. Lobo We shll conside stndd genel eltivity, with tvesble womholes tht e spheiclly symmetic, with ll of the exotic mteil confined to thin shell Geneic Sttic Spheiclly Symmetic Spcetimes To set the stge, conside two distinct spcetime mnifolds, M + nd M, with metics given by g + μν x μ +) nd g μν x μ ), in tems of independently defined coodinte systems x μ + nd x μ. A single mnifold M is obtined by gluing togethe the two distinct mnifolds, M + nd M, i.e. M = M + M, t thei boundies. The ltte e given by Σ + nd Σ, espectively, with the ntul identifiction of the boundies Σ = Σ + = Σ. Conside two geneic sttic spheiclly symmetic spcetimes given by the following line elements: [ ds 2 = e 2Φ ± ± ) 1 b [ ± ± ) dt 2± + 1 b ± ± ) 1 d± 2 ± + ± 2 dω2 ±, 2.58) ± on M ±, espectively. Using the Einstein field eqution, G μν = 8π T μν with c = G = 1), the othonoml) stess enegy tenso components e given by ρ) = 1 8π 2 b, 2.59) τ) = 1 [ 2Φ b ) + b, 8π ) p t ) = 1 [ b + 3b 2)Φ 16π 2 + 2b )Φ ) 2 + 2b )Φ + b, 2.61) whee we hve denoted the quntity τ) hee s the dil tension the vible τ in this section denotes the pope time, s mesued by comoving obseve on the thin shell). The ± subscipts wee tempoily) dopped so s not to ovelod the nottion Extinsic Cuvtue The mnifolds e bounded by hypesufces Σ + nd Σ, espectively, with induced metics g + ij nd g ij. The hypesufces e isometic, i.e. g+ ij ξ) = g ij ξ) = g ijξ), in tems of the intinsic coodintes, invint unde the isomety. As mentioned bove, single mnifold M is obtined by gluing togethe M + nd M t thei boundies, i.e. M = M + M, with the ntul identifiction of the boundies

15 2 Womhole Bsics 25 Σ = Σ + = Σ. The thee holonomic bsis vectos e i) = / ξ i tngent to Σ hve the following components e μ i) ± = x μ ±/ ξ i, which povide the induced metic on the junction sufce by the following scl poduct g ij = e i) e j) = g μν e μ i) eν j) ±. The intinsic metic to Σ is thus povided by ds 2 Σ = dτ τ) dθ 2 + sin 2 θ dφ 2 ), 2.62) whee τ is the pope time of n obseve comoving with the junction sufce, s mentioned bove. Thus, fo the sttic nd spheiclly symmetic spcetime consideed in this section, the single mnifold, M, is obtined by gluing M + nd M t Σ, i.e. t f,τ)= τ) = 0. The position of the junction sufce is given by x μ τ,θ,φ)= tτ), τ), θ, φ), 2.63) nd the espective 4-velocities s mesued in the sttic coodinte systems on the two sides of the junction) e U μ ± = e Φ ±) 1 b ±) 1 b ±) +ȧ 2, ȧ, 0, 0, 2.64) whee the ovedot denotes deivtive with espect to τ. We shll conside timelike junction sufce Σ, defined by the pmetic eqution of the fom f x μ ξ i )) = 0. The unit noml 4 vecto, n μ,toσ is defined s n μ =± gαβ f x α f x β 1/2 f x μ, 2.65) with n μ n μ =+1nd n μ e μ i) = 0. The Isel fomlism equies tht the nomls point fom M to M + [14. Thus, the unit nomls to the junction sufce, detemined by Eq. 2.65), e given by n μ ± =± e Φ ±) 1 b ±) ȧ, 1 b ±) ) +ȧ 2, 0, ) Note tht the bove expessions cn lso be deduced fom the contctions U μ n μ = 0 nd n μ n μ =+1. The extinsic cuvtue, o the second fundmentl fom, is defined s K ij = n μ;ν e μ i) eν j). Tking into ccount the diffeentition of n μe μ i) = 0 with espect to ξ j, the extinsic cuvtue is given by

16 26 F.S.N. Lobo K ± 2 x μ ij = n μ + Γ μ± ξ i ξ j αβ x α ξ i x β ). 2.67) ξ j Note tht fo the cse of thin shell K ij is not continuous coss Σ, so tht fo nottionl convenience, the discontinuity in the second fundmentl fom is defined s κ ij = K + ij K ij. Thus, using Eq. 2.67), the nontivil components of the extinsic cuvtue cn esily be computed to be K θ ± θ =± 1 K τ ± τ =± 1 b ±) ä + b ±) b ± ) 2 2 +ȧ 2, 2.68) 1 b ±) +ȧ 2, 2.69) + Φ 1 b ±) +ȧ 2 ± ) whee the pime now denotes deivtive with espect to the coodinte Lnczos Equtions: Sufce Stess Enegy The Lnczos equtions follow fom the Einstein equtions pplied to the hypesufce joining the fou-dimensionl spcetimes, nd e given by S i j = 1 8π κi j δi j κk k ), 2.70) whee S i j is the sufce stess enegy tenso on Σ. In pticul, becuse of spheicl symmety consideble simplifictions occu, nmely κ i j = dig κ τ τ,κθ θ θ),κθ.the sufce stess enegy tenso my be witten in tems of the sufce enegy density, σ, nd the sufce pessue, P, ss i j = dig σ, P, P). The Lnczos equtions then educe to σ = 1 4π κθ θ, 2.71) P = 1 8π κτ τ + κθ θ ). 2.72) Tking into ccount the computed extinsic cuvtues, Eqs. 2.68) nd 2.69), we see tht Eqs. 2.71) nd 2.72) povide us with the following expessions fo the sufce stesses:

17 2 Womhole Bsics 27 [ σ = 1 1 b +) 4π P = 1 8π +ȧ b ) 1 +ȧ2 + ä b +)+b + ) b +) +ȧ ȧ2 + ä b )+b ) b ) +ȧ 2 +ȧ 2, 2.73) 1 b +) 1 b ) +ȧ 2 Φ + ) +ȧ 2 Φ ). 2.74) Note tht the sufce enegy density σ is lwys negtive, consequently violting the enegy conditions. The sufce mss of the thin shell is given by m s = 4π 2 σ, which will be used below Consevtion Identity The fist contcted Guss Codzzi eqution is given by G μν n μ n ν = 1 2 K 2 K ij K ij 3 R), 2.75) which combined with the Einstein equtions povides the evolution identity S ij K ij = [ T μν n μ n ν ) The convention [X + X + Σ X Σ nd X 1 2 X + Σ + X Σ ) is used. The second contcted Guss Codzzi eqution is G μν e μ i) nν = K j i j K, i, 2.77) which togethe with the Lnczos equtions povides the consevtion identity S i j i = [T μν e μ j) nν ) When intepeting the consevtion identity Eq. 2.78), conside the momentum flux defined by [T μν e μτ) nν + = [ T μν U μ n ν + = ± ) ȧ Tˆt ˆt + Tˆ ˆ 1 b) +ȧ 2 1 b) +, 2.79)

18 28 F.S.N. Lobo whee Tˆt ˆt nd Tˆ ˆ e the fou-dimensionl stess enegy tenso components given in n othonoml bsis. This flux tem coesponds to the net discontinuity in the momentum flux F μ = T μν U ν which impinges on the shell. Applying the Einstein equtions, we hve [T μν e μτ) nν + = ȧ 4π [Φ + ) It is useful to define the quntity Ξ = 1 4π [Φ + ) 1 b +) 1 b +) +ȧ 2 + Φ ) +ȧ 2 + Φ ) 1 b ) 1 b ) +ȧ ) +ȧ ) nd to let A = 4π 2 be the sufce e of the thin shell. Then in the genel cse, the consevtion identity povides the following eltionship: dσ A) dτ + P da dτ = Ξ A ȧ. 2.82) The fist tem epesents the vition of the intenl enegy of the shell, the second tem is the wok done by the shell s intenl foce, nd the thid tem epesents the wok done by the extenl foces. If we ssume tht the equtions of motion cn be integted to detemine the sufce enegy density s function of dius, tht is, ssuming the existence of suitble function σ), then the consevtion eqution cn be witten s σ = 2 σ + P) + Ξ, 2.83) whee σ = dσ/d. Note tht the flux tem Ξ is zeo wheneve Φ ± = 0, which is ctully quite common occuence, fo instnce in eithe Schwzschild o Reissne Nodstöm geometies, o moe genelly wheneve ρ + p = 0. In pticul, fo vnishing flux Ξ = 0 one obtins the so-clled tnspency condition, [ Gμν U μ n ν + = 0[22. The consevtion identity, Eq. 2.78), then educes to the simple eltionship σ = 2 σ + P)ȧ/, which is extensively used in the litetue Eqution of Motion To qulittively nlyse the stbility of the womhole, it is useful to enge Eq. 2.73) into the thin-shell eqution of motion given by 2ȧ2 1 + V ) = 0, 2.84)

19 2 Womhole Bsics 29 whee the potentil V ) is given by { V ) = 1 1 b) [ ms ) 2 [ } Δ) ) 2 2 m s ) Hee m s ) = 4π 2 σ) is the mss of the thin shell. The quntities b) nd Δ) e defined, fo simplicity, s b) = b +) + b ) 2, Δ) = b +) b ), 2.86) 2 espectively. This gives the potentil V ) s function of the sufce mss m s ). By diffeentiting with espect to, we see tht the eqution of motion implies ä = V ). It is useful to evese the logic flow nd detemine the sufce mss s function of the potentil. Following the techniques used in [23, suitbly modified fo the pesent womhole context, we hve [ m 2 s ) = 22 1 b) 2V ) + 1 b +) nd in fct m s ) = [ 1 b +) 2V ) + 2V ) 1 b ) 1 b ) 2V ), 2.87) 2V ), 2.88) with the negtive oot now being necessy fo comptibility with the Lnczos equtions. Note the novel ppoch used hee, nmely, by specifying V ) dicttes the mount of sufce mss tht needs to be inseted on the womhole thot. This implicitly mkes demnds on the eqution of stte of the exotic mtte esiding on the womhole thot. In completely nlogous mnne, fte imposing the eqution of motion fo the shell one hs σ) = 1 4π [ 1 b +) 2V ) + 1 b ) 2V ), 2.89) P) = 1 8π 1 2V ) V ) b +)+b + ) 2 1 b +) 2V ) + 1 2V ) V ) b )+b ) 2 1 b ) 2V ) b +) 1 b ) 2V ) Φ + ) 2V ) Φ ). 2.90)

20 30 F.S.N. Lobo nd the flux tem is given by Ξ) = 1 4π [Φ + ) 1 b +) 2V ) + Φ ) 1 b ) 2V ). 2.91) The thee quntities {σ), P), Ξ)} e elted by the diffeentil consevtion lw, so t most two of them e independent Lineized Eqution of Motion Conside lineiztion ound n ssumed sttic solution 0 to the eqution of motion 1 2ȧ2 + V ) = 0, nd so lso solution of ä = V ). A Tylo expnsion of V ) ound 0 to second ode yields V ) = V 0 ) + V 0 ) 0 ) V 0 ) 0 ) 2 + O[ 0 ) 3, 2.92) nd since we e expnding ound sttic solution, ȧ 0 =ä 0 = 0, we utomticlly hve V 0 ) = V 0 ) = 0, which educes Eq. 2.92) to V ) = 1 2 V 0 ) 0 ) 2 + O[ 0 ) ) The ssumed sttic solution t 0 is stble if nd only if V ) hs locl minimum t 0, which equies V 0 )>0, which is the pimy citeion fo womhole stbility. Fo instnce, it is extemely useful to expess m s ) nd m s ) by the following expessions: m s ) =+m s) nd m + 2 { b+ )/) + 2V ) 1 b+ )/ 2V ) + b )/) + 2V } ), 1 b )/ 2V ) 2.94) { s ) = b+ )/) + 2V ) 1 b+ )/ 2V ) + b )/) + 2V } ) 1 b )/ 2V ) + { [b+ )/) + 2V ) 2 4 [1 b + )/ 2V ) + [b )/) + 2V ) 2 } 3/2 [1 b )/ 2V ) 3/2 + { b+ )/) + 2V ) 2 1 b+ )/ 2V ) + b )/) + 2V } ). 2.95) 1 b )/ 2V ) Doing so llows us to esily study lineized stbility, nd to develop simple inequlity on m s 0) using the constint V 0 )>0. Simil fomule hold fo

21 2 Womhole Bsics 31 σ ), σ ),fop ), P ), nd fo Ξ ), Ξ ). In view of the edundncies coming fom the eltions m s ) = 4πσ) 2 nd the diffeentil consevtion lw, the only inteesting quntities e Ξ ), Ξ ). Fo pcticl clcultions, it is extemely useful to conside the dimensionless quntity m s )/ nd then to expess [m s )/ nd [m s )/. It is similly useful to conside 4πΞ), nd then evlute [4π Ξ) nd [4π Ξ) see Ref. [21 fo moe detils). We shll evlute these quntities t the ssumed stble solution The Mste Equtions Fo pcticl clcultions it is moe useful to wok with m s )/, so tht in view of the bove, in ode to hve stble sttic solution t 0 we must hve while m s 0 )/ 0 = { 1 b + 0 ) b 0 ) 0 }, 2.96) [m s )/ 0 =+ 1 2 { b+ )/) 1 b+ )/ + b )/) 1 b )/ } 0, 2.97) nd the stbility condition V 0 ) 0 is tnslted into [m s )/ { [b+ )/) 2 [1 b + )/ + [b )/) 2 } 3/2 [1 b )/ 0 3/2 { b+ )/) 1 b+ )/ + b )/). 2.98) 1 b )/)} 0 In the bsence of extenl foces this inequlity o the equivlent one fo m s 0) bove) is the only stbility constint one equies. Howeve, once one hs extenl foces Φ ± = 0), thee is dditionl infomtion: [4π Ξ) { } 0 =+ Φ + ) 1 b + )/ + Φ ) 0 1 b )/ { 1 Φ + 2 ) b + )/) 1 b+ )/ + Φ ) b )/) }, 2.99) 1 b )/ 0 nd povided Φ ± 0) 0)

22 32 F.S.N. Lobo [4π Ξ) 0 { } Φ + ) 1 b + )/ + Φ ) 0 1 b )/ { } Φ + ) b + )/) 1 b+ )/ + Φ ) b )/) 1 b )/ 1 4 { 1 2 {Φ + ) [b + )/) 2 [1 b + )/ 3/2 + Φ ) [b )/) 2 } [1 b )/ 0 3/2 Φ + ) b + )/) 1 b+ )/ + Φ ) b )/) 1 b )/ } ) If Φ ± 0) 0, we simply hve [4π Ξ) 0 { } Φ + ) 1 b + )/ + Φ ) 0 1 b )/ { } Φ + ) b + )/) 1 b+ )/ + Φ ) b )/) 1 b )/ 1 4 { 1 2 {Φ + ) [b + )/) 2 [1 b + )/ 3/2 + Φ ) [b )/) 2 } [1 b )/ 0 3/2 Φ + ) b + )/) 1 b+ )/ + Φ ) b )/) 1 b )/ } ) Note tht these lst thee equtions e entiely vcuous in the bsence of extenl foces, which is why they hve not ppeed in the litetue until now. In discussing specific exmples one now meely needs to pply the genel fomlism descibed bove. Sevel exmples e pticully impotnt, some to emphsize the fetues specific to possible symmety between the two univeses used in tvesble womhole constuction, some to emphsize the impotnce of NEC nonvioltion in the bulk, nd some to ssess the simplifictions due to symmety between the two symptotic egions [21, Discussion These lineized stbility conditions eflect n extemely genel, flexible nd obust fmewok, which is well-dpted to genel spheiclly symmetic thin-shell tvesble womholes nd, in this context, the constuction confines the exotic mteil to the thin shell. The ltte, while constined by spheicl symmety, is llowed to move feely within the fou-dimensionl spcetimes, which pemits fully dynmic nlysis. Note tht to this effect, the pesence of flux tem hs been, lthough widely ignoed in the litetue, consideed in get detil. This flux tem coesponds to the net discontinuity in the consevtion lw of the sufce stesses of the bulk momentum flux, nd is physiclly intepeted s the wok done by extenl foces on the thin shell.

23 2 Womhole Bsics 33 Reltive to the lineized stbility nlysis, we hve evesed the logic flow typiclly consideed in the litetue, nd intoduced novel ppoch to the nlysis. We ecll tht the stndd pocedue extensively used in the litetue is to define pmetiztion of the stbility of equilibium, so s not to specify n eqution of stte on the boundy sufce [ Moe specificlly, the pmete ησ) = dp/dσ is usully defined, nd the stndd physicl intepettion of η is tht of the speed of sound. In this section, the thn dopting the ltte ppoch, we consideed tht the stbility of the womhole is fundmentlly linked to the behviou of the sufce mss m s ) of the thin shell of exotic mtte, esiding on the womhole thot, vi pi of stbility inequlities. Moe specificlly, we hve consideed the sufce mss s function of the potentil. This novel pocedue implicitly mkes demnds on the eqution of stte of the mtte esiding on the tnsition lye, nd demonsttes in full genelity tht the stbility of thin-shell womholes is equivlent to choosing suitble popeties fo the mteil esiding on the thin shell. Futhemoe, specific pplictions wee exploed nd we efe the ede to Ref. [21 fo moe detils. Acknowledgements FSNL cknowledges finncil suppot of the Fundção p Ciênci e Tecnologi though n Investigdo FCT Resech contct, with efeence IF/00859/2012, funded by FCT/MCTES Potugl). Refeences 1. Mois MS, Thone KS. Womholes in spce-time nd thei use fo intestell tvel: tool fo teching genel eltivity. Am J Phys. 1988;56: Visse M. Loentzin womholes: fom Einstein to Hwking. New Yok: Ameicn Institute of Physics; Lobo FSN. Exotic solutions in Genel Reltivity: Tvesble womholes nd wp dive spcetimes, Clssicl nd Quntum Gvity Resech, 1 78, 2008), Nov Sci. Pub. ISBN Hochbeg D, Visse M. Null enegy condition in dynmic womholes. Phys Rev Lett. 1998;81: Hochbeg D, Visse M. Geometic stuctue of the geneic sttic tvesble womhole thot. Phys Rev D. 1997;56: Sushkov SV. Womholes suppoted by phntom enegy. Phys Rev D. 2005;71: Xiv:g-qc/ Lobo FSN. Phntom enegy tvesble womholes. Phys Rev D. 2005;71: Xiv:g-qc/ Lobo FSN. Stbility of phntom womholes. Phys Rev D. 2005;71: Xiv:g-qc/ Lobo FSN. Chplygin tvesble womholes. Phys Rev D. 2006;73: Xiv:g-qc/ Lobo FSN. Vn de Wls quintessence sts. Phys Rev D. 2007;75: Xiv:g-qc/ Tiple FJ. Enegy conditions nd spcetime singulities. Phys Rev D. 1978;17: Visse M, K S, Ddhich N. Tvesble womholes with bitily smll enegy condition violtions. Phys Rev Lett. 2003;90: Xiv:g-qc/

24 34 F.S.N. Lobo 13. Hko T, Lobo FSN, Mk MK, Sushkov SV. Modified-gvity womholes without exotic mtte. Phys Rev D. 2013;87: Isel W. Singul hypesufces nd thin shells in genel eltivity, Nuovo Cimento 44B, ); nd coections in ibid.48b, ). 15. Visse M. Tvesble womholes: some simple exmples. Phys Rev D. 1989;39: Visse M. Tvesble womholes fom sugiclly modified Schwzschild spcetimes. Nucl Phys B. 1989;328: Visse M. Quntum mechnicl stbiliztion of Minkowski signtue womholes. Phys Lett B. 1990;242: Poisson E, Visse M. Thin shell womholes: lineiztion stbility. Phys Rev D. 1995;52:7318 [Xiv:g-qc/ Eio EF, Romeo GE. Lineized stbility of chged thin-shell womoles. Gen Rel Gv. 2004;36: Lobo FSN, Cwfod P. Lineized stbility nlysis of thin shell womholes with cosmologicl constnt. Clss Qunt Gv. 2004;21: Gci NM, Lobo FSN, Visse M. Geneic spheiclly symmetic dynmic thin-shell tvesble womholes in stndd genel eltivity. Phys Rev D. 2012;86: Ishk M, Lke K. Stbility of tnspent spheiclly symmetic thin shells nd womholes. Phys Rev D. 2002;65: Visse M, Wiltshie DL. Stble gvsts: n ltentive to blck holes? Clss Qunt Gv. 2004;21: Bouhmdi-Lpez M, Lobo FSN, Mtn-Mouno P. Womholes minimlly violting the null enegy condition. JCAP. 2014;141111):007.

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