On a new approach for constructing wormholes in Einstein Born Infeld gravity

Transcription

1 Eu. Phys. J. C 6 76:6 DOI.4/epjc/s Regula Aticle - Theoetical Physics On a new appoach fo constucting womholes in Einstein Bon Infeld gavity Jin Young Kim,a, Mu-In Pak,b Depatment of Physics, Kunsan National Univesity, Kunsan 545, Koea Reseach Institute fo Basic Science, Sogang Univesity, Seoul -74, Koea Received: 7 Septembe 6 / Accepted: 6 Novembe 6 / Published online: 5 Novembe 6 The Authos 6. This aticle is published with open access at Spingelink.com Abstact We study a new appoach fo the womhole constuction in Einstein Bon Infeld gavity, which does not equie exotic mattes in the Einstein equation. The Bon Infeld field equation is not modified by coodinate independent conditions of continuous metic tenso and its deivatives, even though the Bon Infeld fields have discontinuities in thei deivatives at the thoat in geneal. We study the elation of the newly intoduced conditions with the usual continuity equation fo the enegy-momentum tenso and the gavitational Bianchi identity. We find that thee is no violation of enegy conditions fo the Bon Infeld fields contay to the usual appoaches. The exoticity of the enegy-momentum tenso is not essential fo sustaining womholes. Some open poblems ae discussed. Intoduction Womholes ae non-singula space-time stuctues connecting two o moe diffeent univeses o pats of the same univese ]. Howeve, in the conventional appoaches fo tavesable womholes, thee ae poblems in the natualness. Fist, the hypothetical exotic mattes, which violate the enegy conditions in the standad geneal elativity GR but ae essential fo sustaining the thoats of womholes, have neve been discoveed 7]. Second, the usual cuts and pastes of the thoats and subsequent putting of the hypothetical mattes to the thoats by hand to satisfy the Einstein equation 8,9] ae too atificial to be consideed as a natual pocess. Moeove, we do not know much about the fomation mechanism of womholes due to the gavitationally epulsive natue of the exotic mattes. To cicumvent the obstacle by exotic mattes, one may conside othe extended theoies of gavity so that the exotic a jykim@kunsan.ac.k b muinpak@gmail.com mattes ae natually geneated. Fo example, one may conside non-minimally coupled scala tenso gavity theoies 8], highe cuvatue gavities 9 5], highe dimensional gavities 6 ], and bane-wolds theoies 4 6]. Fo a moe complete and moden eview, see Ref. 7] and the efeences theein. On the othe hand, to cue the poblem of atificiality of the conventional cuts and pastes constuction of thoats of womholes, a new appoach fo the womhole constuction has been poposed ecently 8,9]. In the new appoach, the thoat is defined as the place whee the solutions ae smoothly joined. Thee, the metic and its deivatives ae continuous so that the exotic mattes ae not intoduced at the thoats. Fom the new definition, thoats cannot be constucted abitaily contay to the conventional cuts and pastes appoach. Fo example, we conside a fou dimensional spheically symmetic womhole connecting two emote pats of the same univese o mathematically equivalently, the eflection symmetic two univeses. The metic is descibed by ds = N c dt d f dθ sin θdφ with two coodinate patches, each one coveing the ange,. Then the adius of thoat is defined as dn d = d f d = with the usual matching condition N = N, f = f by demanding that the metic and the Schwazschild coodinates be continuous, though not geneally smooth, acoss the thoat. If thee exists a coodinate patch M in which the singulaities ae absent fo all values of, one can constuct a smooth egula womhole-like geomety, which may

2 6 Page of Eu. Phys. J. C 6 76 :6 o may not be tavesable depending on the location of,by joining M and its mio patch M at the thoat. Hee, it is impotant to note that, in the new appoach, f does not need to vanish, in contast to the Mois Thone appoach 7], while the quantities dn /d, d f /d in do not need to vanish in both the Mois Thone appoach 7] and the cuts and pastes appoach of Visse 8,9]. So, in the new appoach, we have two options fo exact solutions of the same Einstein equation when the thoat point exists. We may conside eithe the usual solutions fo compact objects, like the black holes which may have singulaities but usually shielded by the hoizons geneally, o womhole-like objects which do not have singulaities in the whole space-time domain. We do not have a pioi eason to choose only one of the two options so that we can conside both options equally. This is in contast to the Mois Thone-type tavesable womholes, which cannot co-exist with the solutions of black holes with the same conseved chages. In this context, we call the new type of womholes as natual womholes and the thei thoats as the natual thoats to distinguish these fom the peviously known womholes. Oiginally the new womhole solutions 8] have been studied in Hořava o Hořava Lifshitz gavity, which has been poposed as a enomalizable quantum gavity based on diffeent scaling dimensions between the space and time coodinates 4]. But the new appoach can be applied to othe gavity theoies as well if the natual thoat exists. Howeve, the existence of the thoat fo black holes with otation would be vey difficult unless some coincidences occu. Fo the Ke black hole in GR, thee ae moe metic functions and thee is no solution fo the thoat whee all the metic functions join smoothly simultaneously. This implies that the womhole thoat in the spheically symmetic configuations could easily be destoyed by adding othe conseved chages. Convesely, the womhole thoat could be fomed only afte losing all chages except the mass 9]. It has been also agued that the situation with electomagnetic chages would be simila because thee ae additional gauge fields to be joined smoothly at the same thoat. One can easily show that thee is no solution fo the thoat whee the electomagnetic fields F μν join smoothly, df μν /d =, F μν = F μν, in addition to the conditions fo the metic. The pupose of this pape is to see whethe the concept of natual womholes could also be genealized to chaged cases o not. In paticula, in ode to see the geneic ole of chages fo non-linea electomagnetic fields at shot distances, we conside the Bon Infeld BI action, instead of the usual Maxwell action, without modifying the Einstein Hilbet action at shot distances. The oganization of this pape is as follows. In Sect.,we set up the basic equations of Einstein Bon Infeld gavity with a cosmological constant and study the exact solutions fo spheically symmetic black holes and thei physical popeties, including the black hole themodynamics and phase stuctues. In Sect., we study the natual womhole geomety as the solution of the Einstein equation without intoducing additional mattes at the thoat based on the constuction sketched in Sect.. We show that the BI field equation without additional tems at the thoat is still valid even with the thoat fom some easonable conditions of coodinate independent continuity of metic tensos, the pesevation of the usual gavitational Bianchi identity, and the continuity equation of the enegy-momentum tensos fo the BI fields. In Sect. 4, we conclude with seveal discussions. Thoughout this pape, we use the conventional units fo the speed of light c and the Boltzman constant k B, c = k B =, but keep the Newton constant G and the Planck constant h unless stated othewise. Black hole solutions in Einstein Bon Infeld gavity Bon Infeld action of non-linea electodynamics was oiginally intoduced to solve the infinite self-enegy of a point chage in Maxwell s linea electodynamics 4,4]. When this action is combined with Einstein action to study the gavity of chaged objects, the shot-distance behavio of the metic is modified 4,44]. With the ecent development of D-banes, the Bon Infeld-type action has attacted enewed inteests as an effective action fo low-enegy supesting theoy 45 47]. Thee have been extensive studies about black hole solutions in the Einstein gavity coupled to Bon Infeld electodynamics Einstein Bon Infeld EBI gavity] 48 58]. In this section, we biefly descibe EBI gavity and its black hole solutions. Since ou esults do not depend much on the space-time dimensions D 4, we conside the EBI gavity action with a cosmological constant in D = dimensions fo simplicity, S = d 4 x ] R g 6πG LF, 4 whee LF is the BI Lagangian density given by LF = 4β F μν F μν β. 5 Hee, the paamete β is a coupling constant with dimensions length] which needs to flow to infinity to ecove the usual Maxwell electodynamics with LF = F μν F μν F at low enegies. The coection tems fom a finite β epesent the effect of the non-linea BI fields. This situation is simila to the Lagangian fo a elativistic fee paticle mc v /c, which educes to the Newtonian Lagangian mv fo c limit. As the speed of light c

3 Eu. Phys. J. C 6 76 :6 Page of 6 gives the uppe limit fo the speed v of a paticle, the paamete β gives the uppe limit fo the stength of fields F if F <. Taking 6πG = fo simplicity, the equations of motion ae obtained: μ Fμν F β =, 6 R μν Rg μν g μν = T μν, 7 whee the enegy-momentum tenso fo BI fields is given by T μν = g μν LF 4F ρμfν ρ. 8 F β Let us now conside a static and spheically symmetic solution with the metic ansatz ds = N dt f d dθ sin θdφ. 9 Fo the static electically chaged case whee the only nonvanishing component of the field stength tenso is F t E, one can find N = f. Then the geneal solution fo the BI electic field is obtained fom Eq. 6 as Q E =, 4 Q β whee Q is the integation constant that epesents the electic chage localized at the oigin. Note that the electic field is finite in the limit, E β fo β < due to the non-linea effect of BI fields while it educes to the usual Coulomb field E Q/ in the limit β Fig.. Now, fom and, the Einstein equation 7 educes to d d f = β 4 Q β, and a simple integation ove gives f = C β Q β 4 d, whee C is an integation constant. Hee, the static popety is the esult of the BI field equation 6. If we elax this assumption, we need to conside the magnetic components of the field stength as well Fig. The plots of E fo vaying β with a fixed Q. In paticula, we conside β = 5,, /, /, top to bottom with Q = With a staightfowad integation, one can expess the solution in a compact fom in tems of the incomplete elliptic integal of the fist kind, f = C β Q β 4 4 iβ Q iβ EllipticF Q, i. 4 Fo small, this may be expanded as f = β Q C β β 5Q 4 O 6, 5 which shows a mass-like tem C/. The paamete C is called an intinsic mass 49] and epesents the mass-like behavio nea the oigin. Howeve, it is impotant to note that this is not the usual ADM mass defined at lage. In ode to see this, it is convenient to eoganize the integal in as = so that the integal is convegent in the limit. Then we have anothe expession fo the solution f = M β Q β 4 4 Q F, 4 ; 5 4 ; Q β 4, 6 in tems of the hypegeometic function 5 5,56]. Hee, M is anothe mass paamete, defined by M C M, 7 M = β Q 5 π Ɣ Ɣ, whee M comes fom the pat of the integal. One can find that the total mass M, which is the sum of the intinsic

4 6 Page 4 of Eu. Phys. J. C 6 76 :6 mass C and the finite self-enegy of a point chage M,is the usual ADM mass defined by the asymptotic behavio of the metic at lage f = M Q Q4 β 6 O. 9 This shows that the physical paametes, like the mass and the cosmological constant, ae shifted at the shot distances due to non-linea effects of BI fields: The fouth tem on the ight-hand side of 5 epesents the space with an effective cosmological constant eff = β, which can behave like an anti de Sitte space eff < nea the oigin even though it behaves like a flat = o a de Sitte > space at the asymptotic infinity. The solution 4 o6 has a cuvatue singulaity at the oigin, =, R = 4β Q 4 β 6β Q O 4, R μν R μν = 8β Q 4 8β Q β 4 4 β 6β 4 O, R μναβ R μναβ = 48C 6 β QC 5 6β Q 4 O. Note that the leading singulaity nea = is of the ode O 6 in the Riemann tenso squae, which is the same as that of Schwazschild Sch black hole. But this singulaity is absent fo the maginal case of C = M = M due to the egula behavio of the solution nea the oigin 5, which is the same as that of Reissne Nodstom RN black hole. On the othe hand, it can be shown that thee is no cuvatue singulaity at the hoizon, which is defined as the solution fo f = in ou case, as expected 5]. The solution can have two hoizons geneally and the Hawking tempeatue fo the oute hoizon is given by T H h 4π = h 4π d f d = β Q β 4 ]. Thee exists an extemal black hole limit of vanishing tempeatue, whee the oute hoizon meets the inne hoizon at One cannot diectly obtain the singula behavio of RN black hole nea = fom as one cannot obtain RN black hole solution fom 5 bytaking β limit because β < has been implicitly assumed in the expansions of 5and. One can obtain RN esults only fom 9 which is valid always unless β =. = β β / 4β β Q /4 4β fo a non-vanishing cosmological constant and = Q 4β fo vanishing. At the extemal point, the ADM mass M M = β Q β 4 ] 4 Q F, 4 ; 5 4 ; Q β 4 4 gets the minimum M = ] Q F, 4 ; 5 4 ; Q β 4. 5 The fist law of black hole themodynamics is found as dm = T ds BH A dq, 6 with the usual Bekenstein Hawking entopy fomula by S BH = π hg, 7 and the scala potential 5] A = F, 4 ; 5 4 ; Q β 4. 8 Now, fom the esult, one can classify the black holes in tems of the values of β Q and the cosmological constant. i β Q > /: In this case, the black hole may have one nondegeneate hoizon type I, o two non-degeneate nonextemal hoizons o one degeneate extemal hoizons type II, depending on the mass fo the asymptotically AdS < o flat =, i.e., the type I fo M M and the type II fo M < M Fig. a, b. Thee can be a phase tansition fom a Schwazschild Sch-like type I black hole to Reissne Nodstöm RN-like type II black hole when the mass becomes smalle than the maginal mass M, which is the mass value at = in Fig.. When the mass is smalle than the extemal mass M, which is the mass value at the extemal point in Fig. top cuve, the singulaity at = becomes naked as usual. In the limit β, only the RN-like type II black holes ae possible as in GR. On the othe hand, fo asymptotically ds > case, the situation is moe complicated due to the cosmological hoizon. In this case, thee can exist maximally thee hoizons two smalle ones fo black holes and the lagest one fo the

5 Eu. Phys. J. C 6 76 :6 Page 5 of Fig. The plots of f fo vaying M with a fixed β Q > / and cosmological constant <left, = cente, >ight. We conside M =,.95,., M,.5 top to bottom withq =,β =, M =.6...,and =/5 fo the AdS cases Fig. The plots of the ADM mass M vs. the black hole hoizon adius fo vaying β Q with a fixed cosmological constant. Fo small, thee is no significant diffeence fo diffeent values of cosmological constant left. The maginal mass M is given by the mass value at =. The top two cuves epesent M > M, M = M fo β Q > /, βq = /, espectively, with the extemal mass M, wheeas the bottom two cuves epesent the cases whee M is absent fo β Q < /. We conside β Q = /, /, /4.5, / top to bottom with β =. The effect of cosmological constant is impotant fo lage ight: <thin cuve, = medium cuve, > thick cuve. We conside =/5 fo the AdS cases cosmological hoizon, depending on the mass and the cosmological constant Fig. c. ii β Q = /: In this case, only the Sch-like type I black holes ae possible Fig. 4. It is peculia that the hoizon shinks to zeo size fo the maginal case M = M,even though its mass has still a non-vanishing value Fig.. This does not mean that the metic is a flat Minkowskian. This athe means that all the gavitational enegy is stoed in the BI electic field as a self-enegy. When the mass is smalle than the maginal mass M, the singulaity at = becomes naked. Note that this is the case whee the GR limit β does not exist. iii β Q < /: This case is simila to the case ii, except that the maginal case has no even a point hoizon so that its singulaity is naked always Fig. 5. Constuction of natual womhole solutions The stategy to constuct a natual womhole is to find the thoat adius defined by with the matching condition 8,9]. Based on this definition, cases i and ii in Figs. and 4, espectively, and the case iii in Fig. 5a AdS case show the thoats depending on the mass M fo given values of β Q and. Moe details ae as follows. Fo the case i of β Q > /, thee is no thoat fo the Schlike type I black holes with M > M down to the maginal case of M = M, whee a point-like thoat is geneated at the oigin. The zeo-size = thoat can gow as the black hole loses its mass. But, down to the extemal case of M = M < M, whee the inne and oute hoizons meet at and Hawking tempeatue vanishes Fig. 6, the thoat is inside the oute hoizon and non-tavesable. As the InthedScaseFig.c, egadless of the existence of the inside thoat, thee is anothe outside adius which satisfies the thoat condition. But this cannot be consideed as the womhole thoat because it does not meet the equiement that the thoat occupies the minimum adial coodinate, which has been assumed implicitly in the definition of Sect.. In pinciple, it could also be possible to have a thoat at the maximum adius so that one univese is smoothly connected to anothe by just taveling towad the cosmological hoizon. But, though inteesting, this is not ou main concen and we will not conside this possibility in this pape.

6 6 Page 6 of Eu. Phys. J. C 6 76 : Fig. 4 The plots of f fo vaying M with a fixed β Q = / and cosmological constant <left, = cente, >ight. We conside M =,.85, M,.9,. top to bottom with Q =,β = /, M = ,and =/5 fo the AdS cases Fig. 5 The plots of f fo vaying M with a fixed β Q < / and cosmological constant <left, = cente, >ight. We conside M =,.7, M,.75,. top to bottom with Q =,β = /, M =.74...,and =/5 fo the AdS cases Fig. 6 The plots of the Hawking tempeatue T H vs. the black hole hoizon adius fo vaying β Q with a fixed cosmological constant. We conside β Q = /, /4.5, /, / top to bottom cuves with β =, = /5 left: AdS case, = cente: flat case, = /5 ight: ds case black hole loses its mass futhe, thee emeges and gows a tavesable thoat fom the extemal black hole hoizon. 4 Fo the AdS case Fig. a, the thoat gows and pesists until its mass M vanishes. Howeve, fo the flat Fig. b and ds Fig. c cases, the thoat does not gow to an indefinitely lage size. Thee is a maximum size at a cetain mass M befoe M vanishes, i.e., M > M >. If we futhe educe the mass below M, only the space-time with a naked singulaity is a possible solution. So, the usual naked singulaity solution fo M < M may be eplaced by the non-singula 4 Fo a discussion as egads a dynamical geneation of this unusual pocess, which is beyond the usual linea petubation analysis, see Ref. 9] and the efeences theein. This pocess is diffeent fom the dynamical womhole pictue suggested by Haywad whee womholes emege fom bifucating, i.e. non-extemal black hole hoizons 9,59]. womhole geomety until a cetain non-negative mass M is eached in flat and ds cases. This may suggest the genealized cosmic censoship 8,9] by the existence of a womholelike stuctue aound the oigin, which has been thought to be a singula point of black hole solutions whee all the usual physical laws could beak down and might be egaded as an indication of the incompleteness of geneal elativity. Fo the case ii of β Q = /, a point-like thoat can be geneated at the oigin, contay to the geneation of a finitesize thoat fo the case i. But its gowing pocess is the same as the case i, showing the existence of a maximum size of the thoat with the mass M > fo flat and ds cases, and M = fo AdS case Fig. 4. Fo the case iii of β Q < /, the natual womhole geomety exists only fo AdS case Fig. 5a and its gowing

7 Eu. Phys. J. C 6 76 :6 Page 7 of 6 behavio is simila to the case ii. On the othe hand, fo flat and ds cases Fig. 5b, c, only the Sch-like type I black hole solution fo M > M o the naked singulaity solution fo < M < M can exist. So fa, we have consideed geometic aspects in constucting the natual womhole. Hee, it is impotant to note that the ight-hand side of the Einstein equation depends only on the values of the BI field stength themselves, not on thei deivatives. This means that Eq. 7 is still valid even in the pesence of the thoat, whee the BI fields do not join smoothly, i.e., thei spatial deivatives ae discontinuous Fig.. This would be valid fo any kind of matte field including gauge field if the coupling with the matte and gavity does not contain deivatives. In othe wods, the same solution of the Einstein equation 7 without the thoat can also be the exact solution fo each patch of the womhole geomety even when consideing matte coupled system. Howeve, the above agument would not hold fo the BI field equation 6, which depends on the deivatives of BI fields. Equation 6 fo the womhole geomety with a thoat can be modified due to the discontinuities of deivatives of BI fields. To find the possible modification tem, let us intoduce anothe coodinate η which spans the whole womhole spacetimes with a geodesically complete anges, by joining two coodinate patches, each one coveing, with the thoat at η = fo =. Fo example, we may conside the Ellis-type metic ansatz fo a tavesable womhole, i.e., a thoat-type geomety that physical objects can popagate though, ds = N ldt dl l dθ sin θdφ, 9 whee l is the pope length fom the thoat given by l = d/ f. Then, in the egion close to the thoat η =, one may expand the oiginal coodinate fo each patch as d η = η d dη dη η. Hee, we note that d/dη, d/dη, and d /dη > in ode that the minimum adius exists. In geneal, thee ae two options fo the choice of η coodinate. a Fist, if we conside the egula coodinate tansfomation η of without singulaity, we need to conside d/dη = d/dη = but its invese tansfomation η becomes singula at the thoat, dη/d top cuve in Fig. 7 9]. b Second, if we conside the case of d/dη = bottom cuve in Fig. 7, we have a singulaity a delta func- Fig. 7 The plots of possible η functions tion singulaity due to the discontinuity of d/dη in the thid tem of so that the coodinate tansfomation η becomes singula at the thoat. The second option was not consideed befoe 9]butit seems that both options ae possible on the geneal gound. Fo example, if we conside the deivative of the metic tenso g μν with espect to the pope length l in the Schwazschild coodinate dg μν d dg μν = dl dl d = f dg μν d, one can join the metic smoothly, dg μν /dl = dg μν / dl =, by consideing eithe f = Mois Thone appoach 7] o dg μν /d = dg μν /d = new appoach in this pape 8,9]. But, independently of the choice of the two options, we may geneally obtain the adial component of the BI electic field and its deivative, E = E ɛη E ɛ η, de = de ɛη de ɛ η d d d dη E ] dη δη, d d whee we have used the continuity of the field E = E and dɛη/dη =δη with the step function ɛη Fig. 8. In geneal, singulaities may ente into the BI field equation when it involves the adial deivative d d fo the womhole geomety with a thoat, egadless of the choice of η coodinate unless we fine-tune the coodinate η 9] so that dη = d dη. 4 d

8 6 Page 8 of Eu. Phys. J. C 6 76 :6 4 eta 4 Fig. 8 The plots of Eη and E η = Eη ˆη functions Note that the condition 4 has been implicitly assumed in the Mois Thone constuction fo the fist option, though dη/d becomes infinity 7]. Now, in ode to undestand the condition 4 in ou constuction fo the second option, let us conside the second deivatives of the metic tenso g μν with espect to η d g μν dη = d d g μν d dη dη d dg μν d. 5 Then the continuity of the second deivatives with espect to η and at the thoat note that the second tem in 5 vanishes in ou case], d g μν d gμν dη = d g μν = d dη d g μν d,, 6 equies the condition 4 natually. Note that, in the usual case of the fist option, the fist tem in 5 vanishes and one can obtain anothe condition d d dη = dη fom the continuity of the fist deivative with espect to, dg μν dg d μν = d, which need not to be vanished 9]. On the othe hand, in ou case of the second option, the condition 4 is also necessay in ode to have the continuity of the metic tenso and its deivatives in a coodinate independent way. Hee, it is impotant to note that thee is no constaint on d which is cucial fo the discussion dη of the enegy condition in Sect. 4. Anothe impotant consequence of the condition 4 is the consistency of the field equation 6 with the Einstein equation 7. In the usual equations of motion, we need the Bianchi identity fo the Riemann tensos and the continuity equation fo the matte s enegy-momentum tensos μ T μν =. 7 But, instead of consideing the continuity equation 7, thee is an easie way to check it. It is to conside the geneal elation befoe implementing 7, p θ = p φ = ρ d d ρ μt μ, 8 fo the enegy-momentum tenso Tν μ = diag ρ, p, p θ, p φ with ρ = p = 4β, E /β p θ = p φ = 4β E /β. 9 Note that, when the usual continuity equation 7 isused, we have the conventional elation see Ref. 6], fo example, p θ = p φ = ρ d ρ. 4 d In othe wods, if the usual elation 4 is violated, the continuity equation 7 is also violated exactly at the same place whee 4 fails. Actually, in ou case, 4 fails at the thoat if the condition 4 is not consideed! Fom Eq. 9, p θ, p φ of 4 depend only on the field E, not on its deivatives de/d so that they ae not affected by the pesence of the thoat. Howeve, the ight hand side of 4 depends on de/d, aswellase, which intoduces the additional delta-function tem at the thoat. This means that, in ode to have a consistent equation 4, we need a counte tem on the ight-hand side to cancel the singula tem at the thoat implying the violation of the continuity equation, μ T μν δη. 4 Because the Einstein equation 7 is not affected by the existence of the natual womhole, this violation of the continuity equation means that the Bianchi identity fo the cuvatue tenso fails also at the thoat, μ G μν δη, 4

9 Eu. Phys. J. C 6 76 :6 Page 9 of 6 with the Einstein tenso G μν = R μν Rg μν. In electodynamics, the Bianchi identity μ F μν =, with the dual field stength F μν = ɛμνσρ F σρ, is violated at the location of the magnetic monopole. In ou case, this electomagnetic Bianchi identity woks as one can easily check fom the solution since F μν =. But we may have a violation of the gavitational Bianchi identity at the thoat if we do not equie the condition 4. 5 This can be consideed as futhe suppot of the condition 4 in ou new appoach. 4 Enegy conditions In the usual set-up of womholes, exotic mattes violating the enegy conditions ae essential to sustain the thoats. 6 Even when we look fo womhole solutions without the explicit intoduction of exotic mattes, we need some effective matte tems in the Einstein equation which play the ole of exotic mattes. Womhole solutions in highe-deivative gavities is one example. In that case, highe-deivative tems act as the effective enegy-momentum tenso which can violate the enegy conditions geneally. Howeve, this is not the case in ou constuction of natual womholes and all the enegy conditions can be nomal. To see this, let us conside ρ p θ = ρ p φ = 4β E /β E /β, 4 ρ p θ = 4β E /β E /β, 44 ρ i p i = p θ = 4β E /β, 45 wheewehaveusedρ p = fom 9. Then it is easy to see that the quantities in 44, 45, as well as ρ in 9, ae all non-negative fo the whole ange of E β. InFig.9, we have plotted the case fo two hoizons in the RN-like type II black hole but the esult is the same as the Sch-like type I case. This shows that the stong enegy condition SEC: ρ p i, ρ i p i, which includes the null enegy condition NEC: ρ p i, is satisfied as well as the weak enegy condition WEC: ρ, ρ p i and the dominant enegy condition DEC: ρ p i o equivalently ρ p i. In ode to undestand the physical oigin of this esult, which is unusual in the conventional womhole physics, let 5 This will apply also to stings and othe distibutional souces in GR 6]. We thank Jennie Taschen fo pointing this out. 6 This esult assumes topologically tivial and static mattes. So, thee could still exist some oom fo avoiding the no-go esult by elaxing those assumptions 6]. Fig. 9 The plots of ρ p θ, ρ, ρ p θ,thee thick cuves, left to ight, ρ i p i thin dotted cuve, and f thin solid cuve fo β =, Q = us conside the usual Mois Thone ansatz instead of 9], ds = e φ dt d b / dθ sin θdφ. 46 Then the enegy density and pessues ae given by ] db ρ = d, 47 p = b b ] dφ d, 48 { p θ = p φ = b d φ d dφ dφ d d ] db dφ d b d }, 49 fom the Einstein equation, G μν g μν = T μν /. An impotant combination of the quantities in 47 49, which is cucial fo the violation of enegy condition in the usual appoach 9], is ρ p = eφ d e φ b ]. 5 d In the usual appoach of the fist option, the quantity in the backet vanishes at the thoat = since, when we conside the pope length l, fo example, d/dl = b / =, i.e., b =. But, in ode that b < fo > so that is the space-like coodinate fo an obseve outside the thoat, the quantity in the backet needs to be positive, i.e., it has positive deivative, 7 7 At the thoat =, this is equivalent to d /dl = d d b/ >, which is known as the flaing-out condition 7].

10 6 Page of Eu. Phys. J. C 6 76 :6 d e φ b ] >, 5 d which educes to ρ p <, 5 so that all the enegy conditions ae violated. Howeve, this is not the case in ou appoach of the second option, whee the quantity in the backet does not vanish at the thoat fo the tavesable womhole. The adial deivative of the quantity in the backet, which is a positive quantity fo >, can have any values. Actually, in ou example, we have ρ p =. One can also obtain simila esults fo othe combinations of the enegy and pessues as in Fig. 9. This shows that the exoticity of the enegy-momentum tenso fo mattes o effective mattes fom the modified gavities is not essential in the new appoach. 5 Discussion We have studied a new appoach to constuct a natual womhole geomety in EBI gavity without intoducing additional exotic mattes at the thoat. BI field equations ae not modified at the thoat fo coodinate independent conditions of continuous metic tenso and its deivatives, even though BI fields have discontinuities in thei deivatives geneally. If we do not equie the newly intoduced conditions, the modification tem exists and it poduces the violation of the gavitational Bianchi identity. This suppots the necessity of the new conditions in ou constuction. Remakably, thee is no violation of enegy conditions and the enegy-momentum tensos ae nomal. It is shown that this is the cucial effect of the new appoach and the exoticity of the enegy-momentum tensos fo the mattes is not essential fo natual womholes. It would be a challenging poblem to see whethe the nonexoticity in the enegy-momentum tensos is a sign fo the stability of womholes o not. Seveal futhe emaks ae in ode. Fist, the new appoach fo womholes has been fist studied in Hořava gavity 8], which has been poposed as a enomalizable quantum gavity model, whee the effective enegy-momentum tensos fom the highe spatial deivative contibutions violate the usual enegy conditions in geneal. Thee, it has been agued that the Hořava gavity womhole is the esult of micoscopic womholes ceated by negative enegy quanta which have enteed the black hole hoizon in Hawking adiation pocess. 8 In othe wods, the existence of 8 Hawking adiation pocess involves vitual pais of paticles nea the event hoizon, one of the pai entes into the black hole while the othe escapes. The outgoing paticle can be obseved as a eal paticle with a positive enegy with espect to an obseve at infinity. Then the ingoing paticle must have a negative enegy in ode that the enegy is womholes may eflect the quantum gavity effect of black holes. Howeve, we have found that the natual womhole constuction is still valid in the classically chaged black holes in GR and the usual enegy conditions can be satisfied too. This indicates that the fomation mechanism fo the natual chaged womholes has a classical oigin in contast to vacuum womholes in Hořava gavity and othe highe cuvatue gavities. So, thee ae two possible oigins fo womholes, one fo the classical and the othe fo the quantum mechanical ones, and both effects need to be consideed geneally. Actually, in ou womholes fo EBI gavity, thee is the uppe bound fo the electic field which may eflect the quantum mechanical pai ceation of photons 9 at shot distances so that both the classical and the quantum effects ae involved in the natual womholes. Second, we note that the new appoach may have some difficulties when extending to highe-deivative gavity theoies due to possible singulaities at the thoat. Fo example, when thid-ode spatial deivatives appea in the Einstein equation, we need to conside d g μν dl = d d g μν dl d d d g μν dl dl d d dl d dg μν d. 5 In the usual Mois Thone appoach, only the thid tem suvives at the hoizon due to d /dl = and it emains finite by the continuity condition d /dl = d /dl. Similaly, one may genealize this agument to abitaily highe-ode deivatives in the Einstein equation. On the othe hand, in the new appoach the thid tem vanishes but the fist two tems suvive at the thoat due to dg μν /d =. Howeve, the second tem can cause a singulaity due to the discontinuity of d /dl unless we conside the case of d g μν /d =, which does not seem to be quite geneic. This might imply that the new appoach is not sufficiently complete as to descibe all the geneic highe-deivative gavity theoies. O this might imply that the natual womholes ae unstable fo the case whee the highe deivatives ae involved like otating black holes in Hořava gavity o othe highe cuvatue gavities, as has been pointed out by one of us ecently 9]. Footnote 8 continued conseved 6,64]. This implies that the negative enegy paticle that falls into black hole can play the ole of exotic matte fo womhole fomation 9]. 9 In the sting theoy context, whee BI type action is consideed as its low enegy effective action, this coesponds to the pai ceation of open stings.

11 Eu. Phys. J. C 6 76 :6 Page of 6 Note added: Afte finishing this pape, a elated pape 65] appeaed whose analysis of black hole in fou dimensions is patly ovelapping with ous. Acknowledgements JYK appeciates National Cente fo Theoetical Sciences Hsinchu and C. Q. Geng fo hospitality duing the pepaation of this wok. MIP appeciates The Intedisciplinay Cente fo Theoetical Study Hefei whee this wok was completed and JianXin Lu, Li-Ming Cao, and othe faculty membes fo hospitality duing his visit. MIP was suppoted by Basic Science Reseach Pogam though the National Reseach Foundation of Koea NRF funded by the Ministy of Education, Science and Technology 6RAB44. Open Access This aticle is distibuted unde the tems of the Ceative Commons Attibution 4. Intenational License ons.og/licenses/by/4./, which pemits unesticted use, distibution, and epoduction in any medium, povided you give appopiate cedit to the oiginal authos and the souce, povide a link to the Ceative Commons license, and indicate if changes wee made. Funded by SCOAP. Refeences. J.A. Wheele, Ann. Phys. N.Y., H.G. Ellis, J. Math. Phys. 4, H.G. Ellis, Gen. Relativ. Gavity, K.A. Bonnikov, Acta Phys. Opl. B 4, T. Kodama, Phys. Rev. D 8, G. Clement, Gen. Relat. Gavity, M.S. Mois, K.S. Thone, Am. J. Phys. 56, M. Visse, Nucl. Phys. B 8, M. Visse, Loenzian womholes AIP Pess, 995. A.G. Agnese, M. La Camea, Phys. Rev. D 5, 995. L.A. Anchodoqui, S.E. Peez Begliaffa, D.F. Toes. Phys. Rev. D 55, K.K. Nandi, B. Bhattachajee, S.M.K. Alam, J. Evans, Phys. Rev. D 57, P.E. Bloomfield, Phys. Rev. D 59, K.K. Nandi, Phys. Rev. D 59, K.K. Nandi, A. Islam, J. Evans, Phys. Rev. D 55, K.K. Nandi, Y.-Z. Zhang, Phys. Rev. D 7, P. Kanti, B. Kleihaus, J. Kunz, Phys. Rev. Lett. 7, 7 8. P. Kanti, B. Kleihaus, J. Kunz, Phys. Rev. D 85, D. Hochbeg, Phys. Lett. B 5, H. Fukutaka, K. Tanaka, K. Ghooku, Phys. Lett. B, D.H. Coule, K.-I. Maeda, Class. Quantum Gavity 7, K. Ghooku, T. Soma, Phys. Rev. D 46, N. Fuey, A. DeBenedictis, Class. Quantum Gavity, 5 4. F.S.N. Lobo, Class. Quantum Gavity 5, F. Duplessis, D.A. Easson, Phys. Rev. D 9, A. Chodos, S. Detweile, Gen. Relat. Gavity 4, G. Clement, Gen. Relat. Gavity 6, B. Bhawal, S. Ka, Phys. Rev. D 46, S. Ka, D. Sahdev, Phys. Rev. D 5, V.D. Dzhunushaliev, D. Singleton, Phys. Rev. D 59, A. DeBenedictis, A. Das, Nucl. Phys. B 65, 79. H. Maeda, M. Nozawa, Phys. Rev. D 78, M.R. Mehdizadeh, M.K. Zangeneh, F.S.N. Lobo, Phys. Rev. D 9, K.A. Bonnikov, S.-W. Kim, Phys. Rev. D 67, M. La Camea, Phys. Lett. B 57, 7 6. F.S.N. Lobo, Phys. Rev. D 75, F.S.N. Lobo, axiv:64.8 g-qc] 8. M.B. Cantcheff, N.E. Gandi, M. Stula, Phys. Rev. D 8, S.W. Kim, M.-I. Pak, Phys. Lett. B 75, 5 4. P. Hořava, Phys. Rev. D 79, M. Bon, L. Infeld, Poc. R. Soc. Lond. A 4, M. Bon, L. Infeld, Poc. R. Soc. Lond. A 44, B. Hoffmann, Phys. Rev. 47, B. Hoffmann, L. Infeld, Phys. Rev. 5, R.G. Leigh, Mod. Phys. Lett. A 4, E.S. Fadkin, A.A. Tseylin, Phys. Lett. B 6, A.A. Tseylin, Nucl. Phys. B 76, A. Gacia, H. Salaza, J.F. Plebanski, Nuovo Cimento 84, M. Demianski, Found. Phys. 6, H.P. de Oliveia, Class. Quantum Gavity, D.A. Rasheed, axiv:hep-th/ S. Fedinando, D. Kug, Gen. Relat. Gavity 5, 9 5. T.K. Dey, Phys. Lett. B 595, R.G. Cai, D.W. Pang, A. Wang, Phys. Rev. D 7, Y.S. Myung, Y.-W. Kim, Y.-J. Pak, Phys. Rev. D 78, S. Gunasekaan, R.B. Mann, D. Kubiznak, JHEP, 57. D.C. Zou, S.J. Zhang, B. Wang, Phys. Rev. D 89, S. Fenando, Int. J. Mod. Phys. D, S.A. Haywad, Int. J. Mod. Phys. D 8, P. Nicolini, A. Smailagic, E. Spallucci, Phys. Lett. B 6, R.P. Geoch, J.H. Taschen, Phys. Rev. D 6, E. Ayon-Beato, F. Canfoa, J. Zanelli, Phys. Lett. B 75, 6 6. S.W. Hawking, Natue 48, S.W. Hawking, Commun. Math. Phys. 4, S. Li, H. Lu, H. Wei, axiv:66.7 hep-th]