Particle creation near the chronology horizon† Sergey V. Sushkov Department of Geometry, Kazan State Pedagogical University, Mezhlauk 1 st., Kazan 420021, Russia e-mail: [email protected] Submitted to Physical Review D, January, 1998 We investigate the phenomenon of particle creation of the massless scalar field in the model of spacetime in which, depending on the model’s parameter ao, the chronology horizon could be formed. The model represents a two-dimensional curved spacetime with the topology R1 ×S1 whichisasymptotically flatinthepastandinthefuture. Thespacetimeisglobally hyperbolicand has no causal pathologies if ao < 1, and closed timelike curves appear in the spacetime if ao ≥ 1. We obtain the spectrum of created particles in the case ao <1. In the limit ao →1 this spectrum 8 gives the number of particles created into mode n near the chronology horizon. The main result 9 we have obtained is that the number of scalar particles created into each mode as well as the full 9 numberof particles remain finiteat themoment of forming of the chronology horizon. 1 n PACS numbers: 0420G, 0462 a J 9 I. INTRODUCTION spacetime – the “Roman ring” of traversable wormholes – for which the vacuum polarization can be made arbi- 1 v In1988Morris,Thorne,andYurtsever[1]haddemon- trarilysmallallthewaytothechronologyhorizon;inad- 5 strated how one could manufacture closed timelike dition,Li[11]showedthattherenormalizedstress-energy 2 curves in a spacetime containing two relatively moving tensor may be smoothed out by introducing absorption 0 traversable wormholes. That work kindled considerable material, such that the spacetime with a time machine 1 interest in the question whether it is possible, in princi- may be stable against vacuum fluctuations. 0 8 ple, to construct a “time machine” — i.e. whether, by These examples indicate that it is impossible to prove 9 performingoperationsinaboundedregionofaninitially thechronologyprotectionconjecturetakingintoaccount / “ordinary”spacetime,itispossible tobringabouta “fu- only effects of the vacuum polarization. Recently Visser c q ture” in which there will be closed timelike curves. So [12,10]has givenargumentsthat the solvingofthe prob- - far, in spite of many-year attempts to answer this ques- lemofchronologyprotectionisimpossiblewithinthecon- gr tion, there does not exist full and clear understanding textofsemi-classicaltheoryofgravityandrequiresafully : of the problem. On the first stage of investigations, it developed theory of quantum gravity. v seemedthatthequantumfieldtheorycouldgiveamech- Nevertheless in this work we remain in the framework i X anismwhichwouldbeabletoprotectchronology. Hawk- of semi-classical quantum gravity. We shall discuss such r ing has suggested the chronology protection conjecture theaspectofthequantumfieldtheoryinspacetimeswith a which states that the laws of physics will alwaysprevent the time machine which was so far not investigated. Up the formation of closed timelike curves [2,3]. His argu- tothis time, variousaspectsofvacuumpolarizationnear ments were based on the supposition that the renormal- the chronology horizon have mainly been investigating. ized vacuum expectation values of a stress-energytensor Here we consider dynamics of the chronology horizon of a quantum field must diverge at the chronology hori- forming. As is known, time-dependent processes in the zon which separates a region with closed timelike curves presence of quantum fields are accompanied by a pair fromaregionwithoutthem. Variousattemptsatproving creation. So one may expect that particles of quantized Hawking’sconjecturehavebeenmade[4],culminatingin fields will be created in the process of the chronology singularity theorems of Kay, Radzikowski, and Wald [6]. horizon forming. The aim of this work is to determine However,inrecenttime, anumber ofexamplesofcon- thespectrumofparticlesandtrytoanswerthequestion: figurations with the bounded renormalized stress-energy Can the particle creation stop the chronology horizon tensornearthechronologyhorizonhasbeengiven[7–11]. forming? For instance,in our works[8,9]it wasshownthat, in the Weusetheunitsc=G=h¯ =1throughoutthepaper. caseofautomorphicfields,thereexistsaspecificchoiceof quantum state for which the renormalized stress-energy II. A MODEL OF SPACETIME tensor vanishes at the chronology horizon (in fact it is zerointhewholespacetime);Krasnikov[7]exhibitedsev- eral 2D models, and Visser [10] presented a 4D model of Here we shall discuss a model of spacetime in which the chronology horizon is being formed. Let us consider 1 a strip {η ∈(−∞,+∞), ξ ∈[0,L]} on the η-ξ plane and As follows from the asymptotical properties (2) of a(η) assume that the points on the bounds γ−: ξ = 0 and in the in-region the equations (8) take the form: γ+: ξ =L are to be identified: (η,0)≡(η,L). After this procedure we obtain a manifold M with a topology of η+ξ =const, η−ξ =const, (9) cylinder: R1×S1. Introduceonthismanifoldthemetric and in the out-region they are ds2 =dη2+2a(η)dηdξ−(1−a2(η))dξ2, (1) η+(1+a )ξ =const, η−(1−a )ξ =const. (10) o o where a(η) is a monotonically increasing function of η which has the following asymptotic behavior: Analysingoftheequations(8)togetherwiththeirasymp- totical forms (9) and (10) reveals the following quali- a(η)→0 if η →−∞, tative picture: The light future cone in the past (see (2) a(η)→a if η →+∞, Eqs.(9)) has the angle α = 90◦ and has no inclination o 1 (i.e., the angle between the cone’s axis and the η-axis is whereaoissomeconstant. Furtherweshallcalltheman- zero); then, the cone is enlarging and inclining so that ifold M as the spacetime M with the metric (1). Note in the future (see Eqs.(10)) its angle becomes equal to that the metric’s coefficients in Eq.(1) do not depend on α =π−arctan(1+a )−arctan(1−a ) andthe inclina- 1 o o ξ, so they themself and their derivatives are taking the tion’s angle becomes equal to α = 1[arctan(1+a )− 2 2 o same values at the points (η,0) and (η,L). Hence, the arctan(1−a )]. There are two qualitatively different o internalmetricsandexternalcurvaturesatbothlinesγ− cases: (i) a < 1 and (ii) a ≥ 1. In the first case, o o and γ+ are identical. This guarantees the regularity of a < 1, this rotation of the cone on the η-ξ plane does o the spacetime M. not lead to the appearance of causal pathologies, i.e. all The metric (1) describes a curved spacetime which future-directed world lines remain unclosed. In the sec- is asymptotically flat in the past, η → −∞ (the “in- ondcase,a ≥1,thesituationiscardinallychanged(this o region”), and in the future, η → ∞ (the “out-region”). caseis illustratedby the spacetime diagraminthe figure Really, as follows from (2), if η → −∞, the metric (1) 1). takes exactly the Minkowski form ds2 =dη2−dξ2. (3) In the future, η →∞, the metric (1) reads ds2 =dη2+2a dηdξ−(1−a2)dξ2, (4) o o and takes the Minkowski form ds2 =dt2−dx2 (5) in new ‘Minkowski’ coordinates t=η+a ξ, x=ξ. (6) o Note thatthe spacetime M couldbe consideredasthe factor space: M = M/R. Here M is an universal cov- ering spacetime for M, in our case it is the whole plane (η,ξ), and R is the equivalence relation: (η,ξ+L)≡(η,ξ) (7) Fig.1. ConsidernowthecausalstructureofthespacetimeM. With this aim we have to define null (lightlike) curves Namely, now there is such a moment of time (the value which form a light cone at a point. The equations for ofη =η∗)whenoneofthecone’ssidetakesa‘horizontal’ null curves could be found from the condition ds2 = 0. position, i.e. one of the null curve’s equations becomes There exist exactly two null curves which go via each η =const. Buttheselinesareclosed(seetheEq.(7)),and point of the spacetime and their equations read hence,atthemomentη∗ the closednullcurvesappearin our model. We shall speak that a time machine is being η dη˜ formed at this moment of time. If η = ∞ (a = 1) ξ+ =const, ∗ o 1+a(η˜) the time machine is formed in the infinitely far future. Z (8) η dη˜ Otherwise, it is formed at the time η∗ <∞. In this case ξ− =const. at later times η >η the closedline η =const lies inside ∗ 1−a(η˜) Z 2 of the light cone, i.e. the region η > η∗ contains closed ω =|kn|andφ(n+,in) andφn(−,in) arepositiveandnegative timelike curves. frequencysolutions(the“in-modes”)inthein-region,re- Thus we may conclude we haveconstructedthe model spectively. The in-modes form the basis in Hilbert space of spacetime in which, depending on the parameter a , H with the scalar product o the chronology horizon is being formed at some moment ofNtiomwe.letusinvestigateabehaviorofquantizedfieldsin (φn,φn′)=−i (φn∂↔µφ∗n′)dΣµ, (18) Z this model. wheredΣµ =dΣnµ,withdΣbeingthevolumeelementin agivenspacelikehypersurface,andnµ beingthetimelike unit vector normal to this hypersurface. Choosing the III. A PARTICLE CREATION hypersurface η = const we may write down the scalar product (18) in the in-region as follows: A. Scalar field: solutions of the wave equation L ∂φ∗ ∂φ Consider a conformal massless scalar field φ for which (φn,φn′)=−i dξ φn ∂ηn′ − ∂ηnφ∗n′ (19) the Lagrangianis Z0 (cid:18) (cid:19)η=const 1 The in-modes are orthonormal provided L= ∇ φ∇µφ. (11) µ 2 1 D(in) = , (20) The scalar field φ obeys the wave equation n 4π|n+α| 2φ=0. (12) where α=0 for an ordinpary scalar field and α= 1 for a 2 twisted one. Inaddition,itfollowsfromtheidentificationrule(7)and Analogously we may find a solution of the wave equa- from the quadric form of the lagrangian (11) that the tion (15) in the out-region. There a(η) → a and scalar field has to obey the periodic condition (the ordi- o a′(η)→0, and the Eq.(15) reduces to nary field) φ(η,ξ+L)=φ(η,ξ) (13) (1−a2o)∂η2+2ao∂η∂ξ−∂ξ2 φ(η,ξ)=0, (21) h i or the antiperiodic one (the twisted field) The complete set of solutions of this equation is φ(η,ξ+L)=−φ(η,ξ). (14) φ(±,out) =D(out)eiknξe−iknaoβ−1ηe∓iωβ−1η, (22) n n In the metric (1) the wave equation (12) reads where we denote (1−a2)∂η2+2a∂η∂ξ−∂ξ2−2aa′∂η+a′∂ξ φ(η,ξ)=0, β =1−a2o, (23) h i (15) (+,out) (−,out) and φ and φ are positive and negative fre- n n where ∂ = ∂ , ∂ = ∂ and a prime denotes the deriva- quencysolutions(the“out-modes”)intheout-region,re- η ∂η ξ ∂ξ spectively. The out-modes (22) formthe basis in Hilbert tive on η, a′ =da/dη. space H with the scalar product (18) which in the out- First of all, let us solve this equation in the asymp- region reads totical regions. In the in-region, where a(η) → 0 and a′(η)→0, the Eq.(15) reduces to L ∂φ∗ ∂φ [∂η2−∂ξ2]φ(η,ξ)=0, (16) (φn,φn′)=−iZ0 dξ(cid:26)(cid:18)φn ∂ηn′ − ∂ηnφ∗n′(cid:19) The complete set of solutions of this equation is + 1−aoa2 φn∂∂φξ∗n′ − ∂∂φξnφ∗n′ (24) φ(±,in) =D(in)eiknξe∓iωη, (17) o (cid:18) (cid:19)(cid:27)η=const n n The out-modes are orthonormal provided where 2πn β k = , n=±1,±2,... (ordinary field) D(out) = , (25) n L n s4π|n+α| or Now let us solve the Eq.(15) in a general case. Noting k = 2π(n+ 12), n=0,±1,±2,... (twisted field), that the metric (1) is invariant under translations in the n L 3 ξ-direction and taking into account the periodic or an- typical time of variation of the function a(η) from one tiperiodic conditions (13), (14) we may find solutions in asymptotical value to another one. The conditions (32) the following form: reduce now to φ (η,ξ)=u (η)eiknξ. (26) 4 a2 n n k2 >> o. (35) n 27T2 Substituting the expression (26) into the wave equation (15) we obtain the equation for u (η): Substituting the expression(34)into Eq.(33)wecould n rewrite the equation (33) as (1−a2)u′′+2a(ik −a′)u′−ik (ik −a′)u=0. (27) n n n 4k2v Introduce a new function v(η) by the relation v′′+ n =0. (36) (2−a2(1+tanhγη))2 o η a(ik −a′) u(η)=v(η)exp − n dη˜ . (28) A solution of this equation could be found in terms of 1−a2 (cid:18) Z (cid:19) hypergeometric series. In paticular, to find a solution whichwill be positive frequency in the in-regionwe have Note that in the in-region the relation (28) has the to choose the solution of the equation (36) as follows: asymptotical form u(η)=v(η), (29) vn(in)(η)=Dn(in)(−z)−iµ(1−βz)σF(q,r;s;βz), (37) whereas in the out-region it reads where D(in) is defined by (20) and n u(η)=v(η)e−iknaoβ−1η (30) z =−e2γη, µ= ω , β =1−a2 2γ o After substituting Eq.(28) into Eq.(27) we obtain the 1 1 (1−β)2 second-order differential equation for v(η) in a so-called s=1−2iµ, σ = + −µ2 (38) normal form v′′+Ω2(η)v =0: 2 s4 β2 1−β 1+β q =σ+iµ , r =σ−iµ , k2 +a′2+a′′a(1−a2) β β v′′+ n v =0. (31) (1−a2)2 (cid:18) (cid:19) and F(q,r;s;z) is a Gaussian hypergeometric function. Further we shall solve this equation only for modes for Taking into account the relations (28) and (26) we can which the conditions now write down the solution of the wave equation (15) as k2 >>a′2, k2 >>a′′a(1−a2) (32) n n η a(ik −a′) f (η,ξ)=D(in)eiknξe−iωηexp − n dη˜ are fulfilled. That is we shall only consider the modes n n 1−a2 (cid:18) Z (cid:19) whose wave length is much less than a typical scale of ×(1−βz)σF(q,r;s;βz) (39) variation of the function a(η). Taking into account the conditions (32) we could now neglect in (31) the terms NotethatF(q,r;s;0)=1foranyq,rands. Nowitisnot a′2 and a′′a(1−a2) and rewrite∗ difficult to see that this solution in the in-region, where η → −∞ or z → −0, has the following asymptotical k2v v′′+ n =0. (33) form: f (η,ξ) ≈ D(in)eiknξe−iωη, which coincides with (1−a2)2 n n the positive frequency in-modes φ(+,in). n Now let us restrict our considerationby the special form of the function a(η): B. Bogolubov coefficients 1 a2(η)= a2(1+tanhγη), (34) 2 o Now let us remind some mathematical aspects for de- whereγ isaparameter. Itiseasytoseethatthefunction scribing of the physical phenomenon of partical creation a(η) defined by Eq.(34) possesses the necessary asymp- by a time-depend gravitationalfield. totical behavior (2). The quantity T = (2γ)−1 gives the So, we have obtained the set of solutions {fn} which are positive frequency (the “in-modes”) in the past. Let {F } be positive frequency solutions (the “out-modes”) n in the future. (We do not need to know an explicit form ∗Let me emphasize that the approximation which we use ofthesesolutions. Itwillbeenoughtoknowtheirasymp- is no WKB approximation. Remind that the last means, toticpropertiesintheout-region,i.e. Fn ≈φ(n+,out).) We roughly speaking(!), neglecting terms with Ω′, Ω′′, ... in the maychoosethesetwosetsofsolutionstobeorthonormal, solution of theequation v′′+Ω2(η)v=0. so that 4 (fn,fn′)=(Fn,Fn′)=δnn′ consider the analitical continuation of the hypergeomet- (fn∗,fn∗′)=(Fn∗,Fn∗′)=−δnn′ ric function F(q,r;s;z) into the region of large values of |z| [13] (f ,f∗)=(F ,F∗)=0. (40) n n′ n n′ F(q,r;s;z)= The in-modes may be expanded in terms of the out- Γ(s)Γ(r−q) 1 modes: (−z)−qF(q,1−s+q;1−r+q; ) Γ(r)Γ(s−q) z f = (α F +β F∗). (41) n nm m nm m Γ(s)Γ(q−r) 1 m + (−z)−rF(r,1−s+r;1−q+r; ). (48) X Γ(q)Γ(s−r) z Inserting this expansioninto the orthogonalityrelations, After substituting this expression into the Eq.(39) it is Eq. (40), leads to the conditions not difficult to see that the asymptotic of the in-modes (αnmα∗n′m−βnmβn∗′m)=δnn′, (42) in the out-region is Xm fn(η,ξ)≈Dn(out)eiknξe−iknaoβ−1η and × (−β)−iµβ−1(1−β) Γ(s)Γ(r−q) e−iωβ−1η β1/2Γ(r)Γ(s−q) (αnmαn′m−βnmβn′m)=0. (43) (cid:18) Xm +(−β)iµβ−1(1+β) Γ(s)Γ(q−r) eiωβ−1η . (49) Thefieldoperator,φ,maybeexpandedintermsofeither β1/2Γ(q)Γ(s−r) (cid:19) the {fn} or the {Fn}: Comparing the last expression with the asymptotical form of the out-modes, Eq.(22), we may write φ= (a f +a†f∗)= (b F +b†F∗). (44) n n n n n n n n Xn Xn φ(nin)(η,ξ) =Anφ(n+,out)+Bnφ∗n(+,out) =A F +B F∗ (50) The a and a† are annihilation and creation operators, n n n −n n n respectively, in the in-region, whereas the bn and b†n are where the corresponding operators for the out-region. The in- vacuumstateisdefinedbya |0i =0, ∀n,anddescribes A =(−β)−iµβ−1(1−β) Γ(s)Γ(r−q) , (51) n in n β1/2Γ(r)Γ(s−q) the situationwhen no particlesare presentinitially. The out-vacuum state is defined by b |0i = 0, ∀n, and n out describes the situation when no particles are present at B =(−β)iµβ−1(1+β) Γ(s)Γ(q−r) , (52) late times. Noting that a = (φ,f ) and b = (φ,F ), n β1/2Γ(q)Γ(s−r) n n n n we may expandthe two sets of creationand annihilation As follows from Eq.(41), the coefficients A and B operator in terms of one another as n n are the related to the Bogolubov coefficients by α = nm A δ , β = B δ . The number of particles cre- a = (α∗ b −β∗ b† ), (45) n nm nm n n,−m n nm m nm m ated into mode n is now determined as m X hN i=|B |2. (53) or n n Using the Eq.(52) gives b = (α a +β∗ a†). (46) m nm n nm n 2 n Γ(s)Γ(q−r) X hN i=β−1 (54) n Γ(q)Γ(s−r) This is a Bogolubov transformation, and the αnm and (cid:12) (cid:12) βnm are called the Bogolubov coefficients. Tocarryoutcalculations(cid:12)(cid:12)intheabovee(cid:12)(cid:12)xpressionweshall Letus assumethat noparticle werepresentbefore the use the following formula(cid:12)e [13]: (cid:12) gravitational field is turned on. In the Heisenberg ap- π π proach|0iin is the state of the system for alltime. How- |Γ(iy)|2 = ysinhπy, |Γ(12 +iy)|2 = coshπy, ever, the physical number operator which counts parti- πy cles in the out-region is Nm = b†mbm. Thus the number |Γ(1+iy)|2 = . (55) of particles created into mode m is sinhπy By using this formulae and the relations (38) we may hNmi=inh0|b†mbm|0iin = |βnm|2. (47) finally obtain the spectrum of created particles: n X coshπω(1−β) +coshπ ω(1−β) 2−1 To find the Bogolubovcoefficients in our case we have γβ γβ hN i= . (56) todetermineanasymptoticalformofthein-modesinthe n 2sinhπω sinqh(cid:0)πω (cid:1) γ γβ out-region where η →∞ or z →−∞. With this aim we 5 C. Particle creation near the chronology horizon ACKNOWLEDGMENT Now let us analyse how much particles ofthe massless I would like to thank my colleague Sosov E. N. for scalar field is created at the moment of the chronology helpful discussions. This work was supported in part by horizon forming. As was mentioned above, the value of the Russian Foundation for Basic Research under grant the parameter a determines either the time machine is No. 96-02-17066. o formed or not. If a < 1 then the chronology horizon is o absent in the spacetime, but if a = 1 then it appears o in the infinitely far future. The expression (56) gives us the spectrum of particles created in the out-region. The spectrumdependsontheparametera (β =1−a2),and o o in the limit a →1the expresion(56)will determine the o number ofparticles creatednearthe chronologyhorizon. † The work hasbeen presented in the15th Meeting ofthe Going to the limit a →1 in Eq.(56) gives InternationalSocietyonGeneralRelativityandGravita- o tion, 16-21 December, 1997, IUCAA,Pune 1 [1] M. S. Morris, K. S. Thorne, and U. Yurtsever, hN i= . (57) n sinhπω Phys. Rev.Lett. 61, 1446 (1988). γ [2] S. W. Hawking, “The Chronology Protection Conjec- ture”,inProceedingsofthe6thMarcelGrossmannMeet- Thus we see that the number of particles created into ing, Kyoto, Japan, June 23-29, 1991, edited by H. Sato, mode n near the chronology horizon is finite. We may (World Scientific, Singapore, 1992). also conclude that the full number of particles N = [3] S. W. Hawking, Phys. Rev. D46, 603 (1992). hN i will be finite because the spectrum (57) is ex- n n [4] The reader could find a review of various aspects of the ponentially decreasing. P timemachineproblemaswellasalotofreferencesinan exellent monograph [5] [5] M. Visser, Lorentzian wormholes—from Einstein to IV. CONCLUSION Hawking, (AIPPress, NewYork,1995). [6] B. S. Kay, M. Radzikowski, and R. M. Wald, Quantum Let us summarize. In this work we have constructed FieldTheoryonSpacetimeswithaCompactlyGenerated the model of spacetime in which, depending on the Cauchy Horizon, 1996, gr-qc/9603012. model’s parameter a , the chronology horizon could be [7] S.V.Krasnikov,Phys.Rev.D54,7322-7327, (1996); gr- o formed; there are no causal pathologies if a < 1, and qc/9508038. o the chronology horizon appears at the moment of time [8] S. V. Sushkov, Class. Quant. Grav. 12, 1685-1697, η < ∞ if a > 1. In the case a = 1 closed lightlike (1995). ∗ o o curves are formedin the infinitely far future. The model [9] S. V. Sushkov, Class. Quant. Grav. 14, 523-534, (1997); represents a two-dimensional curved spacetime with the gr-qc/9509056. topologyR1×S1 whichisasymptoticallyflatinthe past [10] M.Visser,Traversablewormholes: theRomanring,1997, gr-qc/9702043 andin the future, butwhich is non-flatinthe intermedi- [11] L.-X.Li,Musttimemachinebeunstableagainstvacuum ate region. As a consequence, in this spacetime the cre- fluctuations? 1997, gr-qc/9703024. ation of particles of quantized fields by the gravitational [12] M. Visser, A reliability horizon for semi-classical quan- field is possible. We have studied the particle creation tum gravity,1997, gr-qc/9702041. of a massless scalar field in the case a < 1, i.e. in the o [13] M. Abramowitz, and I. A. Stegun, Handbook of Math- case when the spacetime is globally hyperbolic and has ematical Functions, (US National Bureau of Standards, no causal pathologies. As a result, the spectrum of cre- Washington, 1964) ated particles has been obtained (see Eq.(56)). In the limit a → 1 this spectrum gives the number of par- o ticles created into mode n near the chronology horizon (see Eq.(57)). The main result we have obtained is that the number of scalar particles created into each mode as well as the full number of particles remain finite at the moment of forming of the chronology horizon. This result might mean that the phenomenon of particle cre- ation could not prevent the formation of time machine. However, to do the final conclusion one has to take into account a backreaction of created particles on a space- time metric. 6