Quantum Time Machines Are Expectation Values of the Scalar ﬁeld Squared and Stress- Quantum-Mechanically Stable

Home , Misner space

arXiv:gr-qc/9712033v1 6 Dec 1997 hudpeettm rvlt cu.S a,i pt of spite in far, So physics occur. of protection to laws travel chronology the time which a prevent to of actu- should according formulation has [13], the and This conjecture to construct led spacetime fluctuations. ally the vacuum di- instable stress-energy quantum the makes the of of emergence vergences the is machines time which strings conditions boundary [12]. cosmic unphysical to parallel-moving moreover, tachyonic subject, in a are having mass of of need the center with is related severe difficulty More the in real [10,11]. commonplace gravity a semiclassical become and be has quantum energy has longer tunnels negative matter no as the where could problem regions This on of density. required emergence negative-energy the pressure to inward leads and large the by of onset develop). the CTC’s is where (that region [8] nonchronal horizon the fluctu- chronology vacuum the of on tensor ations stress-energy the the from of originating divergences instabilities im- more quantum and, the [9] portantly, matter (CTC’s) the curves induced for timelike at an conditions closed show allowing energy shortcomings: classical models following the these two of violation of the all of one [8], trav- least while time arise in would backward that elling causality classical the of olation on based machine time its [6]. a [2], ringholes ultimately, Roman so-called motion [4], and, relative machine [5] Gott’s Politzer’s at rings the the set [3], [1], extension strings Grant’s al. cosmic et proposed, of Thorne been [2-7] couple of since use- wormhole have papers potentially the machines Several of including time subject. influx for scientific the models recent of on ful a those [8,9] to to books and books rise from jumped fiction giving has science machine journals, of time a pages of the notion the [1], sever yfrteauetpolmwt oto h proposed the of most with problem acutest the far By produced generally are conditions energy of Violation vi- potential with related difficulties those all of Apart Yurt- and Thorne Morris, by papers seminal the After ASnme() 42.z 04.62.+v 04.20.Gz, number(s): the PACS obstruction. which an for t longer space, no just Misner is modified with conjecture the machine protection of time out ren stable constructed the ev quantum-mechanically be which regular A in becomes space fields be zon. scalar Misner direction massless Lorentzian spatial complex modified closed quantized a the of in period results the that striction h otnaino inrsaeit h ulda eini region Euclidean the into space Misner of continuation The .INTRODUCTION I. osj ueird netgcoe Cient´ıficas, Serrano Investigaciones de Superior Consejo etod ´sc Mge aań,Isiuod Matemát de Catalán”, F´ısica Instituto “Miguel de Centro unu iemachine time Quantum er .González-D´ıaz F. Pedro Dcme ,1997) 6, (December 1 usd hogotteppruisaeue uhthat such used are units IV, paper dis- c the also Sec. is Throughout machine quan- in time quantized these cussed. conclude a We of that concept checked the everywhere. where is regular It are the tities in space. propagating the field Misner in scalar III modified massless quan- complex Sec. a for in of the evaluated case tensor are hence stress-energy fluctuations and vaccum the tum function of for Hadamard value structure The expectation modified the a space. to in Misner back resulting continued then region, and Lorentzian space- Eu- fixed distinct are the the directions of into time periodicity space the where Misner sector of clidean continuation the consider it. of out size constructed every- sub-microscopic be with regular machines can is time We only spacetime but quantum this holes. where, for in black tensor currently fluctuations of stress-energy being vacuum the horizon quantization that event checked the the have time for for is which the [19] that pattern assumed if as a on same to acceded closed according the quantized the be is of only the coordinate period can and the region time-dependent, now that nonchronal becomes is direction space distin- spatial Misner The respect with usual spacetime to account. considered the into of of feature properties taken guishing thermal are the gravity out where Euclidean [18], constructed space spacetime Misner nonchronal from a in divergences devices. spacetime classical with fields combination matter inappropriate quantum fluctua- from of vacuum artifact quantum are merely to tions due tensor divergences the stress-energy that of and frame- proper, the gravity in quantum of inapplicable work be to conjecture expect would Hawking’s one the everywhere, machines. strong locality entail causal time must of proposed violations [17] hitherto foam all spacetime because for Actually, is semiclassical it is as con- the holds physics, However, protection this chronology [16]. [14,15], where forcefully realm it rather survived violate has to jecture intended attempts some ¯ = h ae sognzda olw.I e.I we II Sec. In follows. as organized is paper The these of rid get to attempt an consider we paper this In h = oe iedpnet hsrestriction This time-dependent. comes rwee vno h hoooyhori- chronology the on even erywhere, G 1. = ent ml h oooia re- topological the imply to seen s eilsia akn’ chronology Hawking’s semiclassical 2,206Mdi (SPAIN) Madrid 28006 121, esbmcocpcsz a then may size sub-microscopic he raie teseeg esrfor tensor stress-energy ormalized csyF´ısica Fundamental, y icas II. THE MODIFIED MISNER SPACE ds2 = (dy˜0)2 + (dy˜1)2 + (dy˜2)2 + (dy˜3)2. (2.8) − It has been stressed [20] that Misner space encom- However, the topology that corresponds to Euclideanized passes many remarkable pathologies in the structure of Misner space is different from that is associated with (2.3). For (2.8), the topology is S1 S1 R2. This spacetime, including those of wormholes and ringholes × × and other spacetimes that contain potentially interesting can be seen by noting that in the coordinates of metric time machines. Therefore, the spacetime structure and (2.1), this metric has an apparent singularity at t = 0. topology of the Misner space itself remain being subjects One can still extend metric (2.1) beyond t = 0 by using which focuse current attention. It is known that the met- new coordinates defined by [21] ric of Misner space can be written as [21] T = t2, V = ln t + x1, (2.9) ds2 = dt2 + t2(dx1)2 + (dx2)2 + (dx3)2, (2.1) − with which metric (2.1) transforms into where 0

0 1 2 3 0 Metric (2.12) will become positive definite provided we (y ,y ,y ,y ) (y cosh(na) 2 2 ↔ allow the continuation u = iζ; hence we have ζ + w = T V and, from (2.9), it can be seen that the section on − + y1 sinh(na),y0 sinh(na)+ y1 cosh(na),y2,y3), (2.4) which ζ and w are both real will correspond to 0 t 1, 2π x1 0. That section defines a Misner instanton≤ ≤ − ≤ ≤ which corresponds to the identifications (t, x1, x2, x3) describing transitions on the chronal region. Using then (t, x1 + na,x2, x3) in Misner coordinates. ↔ (2.5) and the second of (2.13) we finally obtain: Let us now consider the Euclidean continuation of met- 2iζw ric (2.1). Doing that will allow us to investigate how pe- exp = riodical properties of the distinct space directions in the −ζ2 + w2 Euclidean sector transform when they are rotated back to the Lorentzian region. Metric (2.1) becomes definite 2 2 −2 w ζ π/2+ χ positive if we introduce the rotation τ τ exp − exp i . (2.14) w2 + ζ2 τ 2 t = iτ, x1 = iχ. (2.5) It follows that on the Euclidean sector, both τ and χ are Then, periodic, with respective periods 2 2 2 2 2 2 3 2 1 ds = dτ + τ (dχ) + (dx ) + (dx ) . (2.6) Π = , Π =2πτ 2 = g(τ). τ 2 χ An interesting property of the Euclideanized Misner space is that its covering space preserves the Lorentzian Therefore, the topology of the Euclidean Misner space is S1 S1 R2, and one should expect that an observer in signature of the Minkowskian covering for the original × × Misner space. From (2.2) and (2.5) we have now the Misner space would detect a thermal bath at temperature coordinate transformations: TM = 2. Rotating back to the Lorentzian sector (i.e. taking y˜0 = iτ cos χ, y˜1 = τ sin χ, y˜2 = x2, y˜3 = x3, (2.7) τ = it and χ = ix1) we see that t becomes no longer − periodic,− though x−1 still keeps a periodic character, with so that a period which should be given by

2 a = g(t) =2πt2. (2.15) with α the automorphic parameter and γ representing the | | symmetry (periodicity identifications) transformation of Thus, the continuation to the Euclidean section of Misner the modified Misner space; i.e.: the transformation will space shows that the Lorentzian Misner space itself must be taken to satisfy the periodicity condition for a period 1 be modified in that the period of coordinate x should no 2πt2. We shall follow the analysis carried out by Sushkov longer be an arbitrary constant, but it depends on time [14] and look at complex scalar fields that obey the field t according to (2.15). This change manifests in that the equation extension of the region covered by metric (2.1) accross the chronology boundary t = 0 into a new region that ✷φ = ✷φ¯ =0. contains CTC’s becomes modified in that the twisted null geodesics [21] spiral round and round now each time On the covering space coordinates, this equation admits with a larger frequency as they approach t = 0, to fi- the general positive-frequency solution: nally diverge at this point. On the Minkowskian covering 0 1 2 3 space this modification translates as a concentration onto φ(y ,y ,y ,y )= a single point of all the identified points on the surface 0 2 1 2 σ = (y ) (y ) = 0 of the left and right chronology A(k0, k1, k2, k3) 0 1 2 3 − i( k0y +k1y +k2y +k3y ) horizons. Actually, the need of making period a propor- dk1dk2dk3 3 e − , 4π 2 k tional to t2 arises from the kind of instantonic quanti- Z 0 zation implied by continuing into the Euclidean sector, (3.2) so that the modified Misner space can be regarded as 2 2 2 essentially semiclassical. where k0 = k1 + k2 + k3 and A(k0, k1, k2, k3) is a given spectral function. Removal of the resulting singularity at t = 0 can only p be achieved if we let time t to be quantized so that The demand that φ satisfies condition (3.1) amounts either (i) the point at t = 0 is replaced by a mini- to the functional relation mum nonzero throat (i.e. by transforming t according 0 2 1 2 0 2 2 2 2 2 φ(y cosh(2πt )+ y sinh(2πt ),y sinh(2πt ) to t t + R0, with R0 a given small constant which defines→ the throat, so restoring the usual Misner space sit- uation where twisted null geodesics terminated at t = 0, + y1 cosh(2πt2),y2,y3)= e2πiαφ(y0,y1,y2,y3), (3.3) leading to the existence of two inequivalent geodesically incomplete analytic extensions of (2.1), V = ln t + x1, with which automorphicity for the solutions will hold pro- which are locally inextendible [21], or (ii),± if we± want to vided establish full equivalence between the two possible an- A(k cosh(2πt2) k sinh(2πt2), k sinh(2πt2) alytic extensions V (so that all null geodesics do not 0 − 1 − 0 terminate at t = 0± but are extendible beyond it), then time t must be quantized in such a way that though the + k cosh(2πt2), k , k )= e2πiαA(k , k , k , k ). (3.4) two extensions cover the whole space only some particu- 1 2 3 0 1 2 3 lar values of time are allowed. This admits a general solution which is formally the same A simplest ansatz that implements this requirement is as that obtained by Sushkov [14], i.e. t2 = (n + β)C, ∞ iν A(k , k , k , k )= Cn(k , k )(k k ) , (3.5) 0 1 2 3 2 3 0 − 1 where n =0, 1, 2, 3, ..., β is an arbitrary parameter of or- n=0 der unity, and C defines again a throat at n = 0. In the X next section we shall see that it is such an ansatz what (where again the Cn(k2, k3) are arbitrary functions of k2 is actually required if we want a scalar field propagat- and k3), but in which the frequency ν would in principle ing in the modified Misner space to be expandible in an have explicit dependence on time: orthonormal basis with time-independent frequency. n + α ν = . (3.6) − t2 III. THE HADAMARD FUNCTION Demand of time-independence for frequency ν implies quantization of time t according to the law

In what follows we shall consider the propagation of 2 2 a complex scalar field φ in the modified Misner space t = (n + α)t0, (3.7) discussed in Sec. II. For the sake of generality we will where t2 is an arbitrary constant. This quantization con- require φ to satisfy the automorphic condition [22] (X 0 dition for time t is exactly of the same form as that of t, x1, x2, x3) ≡ the ansatz used in Sec. II in order to avoid the singu- 1 larity at t = 0, and leads finally to a frequancy given by 2πiα 2 φ(γX)= e φ(X), 0 α , (3.1) ν = t− . ≤ ≤ 2 − 0

3 2 Inserting (3.5) into (3.2) and carrying first out the in- e2πnt cos(2πα) 1 Ψn(α, t)= 2 2 − , (3.12) tegration over k1 and then transforming to Misner coor- e4πnt 2e2πnt cos(2πα)+1 dinates [14], we obtain finally for the automorphic field − and φ(t, x1, x2, x3)= 1 σ(Xµ, X˜ ν )= (t t˜)2 2{− − ∞ 1 2 3 dk2 dk3Cñ(k2, k3)ϕ(t, x , x , x ), (3.8) 1 1 2 2 2 3 3 2 n=0 +2tt˜[cosh(x x˜ ) 1]+(x x˜ ) + (x x˜ ) , X Z Z − − − − } where (3.13)

(2) −2 1 2 3 1 2 3 i( t0 x +k2x +k3x ) ϕ(t, x , x , x )= D(n)H −2 (kt)e − , it0 µ ν 1 2 − σ0(X , X˜ )= (t t˜) (3.9) 2{− −

(2) 2 1 1 2 2 2 3 3 2 with Hiν (kt) the Bessel function of the third kind, D(n) + t (x x˜ ) + (x x˜ ) + (x x˜ ) . (3.14) 2 2 − − − } a normalizing coefficient, and k = k2 + k3. Adopting the definition of scalar product of Ref. 14, one can now The final result of this calculation is: show that the solutions (3.9) form anp orthonormal basis (1) β ˜ γ 1 1 1 in a Hilbert space provided we choose Gren(X , X )= 4π2 β ˜ γ − β ˜ γ σ(X , X ) σ0(X , X ) √2 √2 π D(n)= π(n+α) = . (3.10) 2 t2 2t ∞ ∞ 1 1 8πte 2 8π√n + αt0e 0 1 cosh[n(x x˜ )] sin(n arccos χ) + − Ψn(α), π2 2 2 n=1 n √n + α√n˜ + αt0 1 χ Noting that for two generally different sets of Misner X X˜=1 − coordinates (X xi and X˜ x˜i , i = 0, 1, 2, 3) the p (3.15) corresponding arbitrary≡ { } frequencies≡ { ν }andν ˜ are the same, we can finally calculate the renormalized Hadamard func- where tion. This will be taken to be given by (µ,ν = 0, 1, 2, 3, 2πn(n+α)t2 e 0 cos(2πα) 1 β,γ =1, 2, 3) Ψ (α)= , n 4πn(n+α)t2 2πn(n+α)t2 − e 0 2e 0 cos(2πα)+1 − ∞ (1) β ˜ γ (1) µ ˜ ν (3.16) Gren(X , X )= Gren(X , X ) n˜=0 ˜ t=√n+αt0,t=√n˜+αt0 X β ˜ γ 1 2 1 1 σ(X , X )= 2αt0[cosh(x x˜ ) 1] ∞ ∞ 2{ − − µ ν µ ν = dk2dk3 ϕ(X )¯ϕ(X˜ )+ ϕ(X˜ )¯ϕ(X ) n=0 n 2 2 2 3 3 2 X X˜=0 Z Z h i + (x x˜ ) + (x x˜ ) , (3.17) − − }

G(1)(Xµ, X˜ ν) 1 0 ˜ β ˜ γ 2 1 1 2 2 2 − t=√n+αt0,t=√n˜+αt0 σ0(X , X )= αt0(x x˜ ) + (x x˜ ) 2{ − −

1 1 1 = + (x3 x˜3)2 . (3.18) 2 µ ν µ ν 4π σ(X , X˜ ) − σ0(X , X˜ ) ˜ − } t=t=√αt0 (1) β ˜ γ It follows that the value of Gren(X , X ) remains finite on the whole modified Misner space for any value 1 1 1 ∞ ∞ cosh[n(x x˜ )] sin(n arccos χ) of α, even α = 0, provided the arbitrary constant t is + − Ψn(α, t) , 0 π2 tt˜ 1 χ2 nonzero. This result inmediately reflects in the remark- n=1 n˜=1 − A X X able implication that the renormalized vacuum expecta- p (3.11) 2 2 tion values of the scalar field squared, φ = 0 φ 0 , and stress-energy tensor, T = 0 T h0 ,i remainh | both| i where the subscript A means evaluation at µν µν finite as well on the nonsingularh i modifiedh | | i Misner space. A t = √n + αt , t˜ = √n˜ + αt , These quantities are evaluated as follows: { 0 0} (1) µ ν 1 ∞ nΨ(α) Gren(X , X˜ ) is the Hadamard function obtained in Ref. 2 (1) β ˜ γ φ = lim Gren(X , X )= 2 2 , (3.19) h i X˜ X π t n + α 14, with Ψn(α, t) given now by 0 n=1 → X

4 1 α A chronology horizon with nonzero width can only be Tβγ = lim [(1 ξ) µ ˜ ν + (2ξ )gµν α ˜ h i X˜ X{ − ∇ ∇ − 2 ∇ ∇ restored if time is quantized according to a rule which → parallels that assumed for the quantization of the black hole event horizon [19]. 1 ✷ (1) µ ˜ ν 2ξ µ ν + ξgµν ] Gren(X , X ) We can show that the region t < 0 of the modified − ∇ ∇ 2 t=√n+αt0,t˜=√n˜+αt0 } Misner space keeps its nonchronal character everywhere 2 only if αt0 < 1, that is in the quantum-gravity regime. Consider the line L defined by t = ωx1, x2 = x3 = 0, 1 ∞ diag(L, 3L,M,M) 1 1 − = 2 4 2 + 2 F (α, t0), (3.20) which will be timelike everywhere provided 2πω < 1, π t (n + α) π 2 3 0 n=1 and the circle L defined by t = √αt > 0, x = x = 0, X 0 0 which is also timelike everywhere if √αt0 < 1. It is then where F (α, t0) is a complicate function of α and ξ arising 1 √α easy to see that on L1, the point Q(x = ω ) precedes from the time derivatives of the function Ψn(α, t), which 1 √α is regular for all values of α and ξ; ξ is the coupling the point P (x = 2π + ω ), but these two points are parameter for the scalar field (ξ = 0 for minimal coupling also on circle L0 where P precedes Q, provided ω > 0. 1 Therefore, there will be CTC’s always that the condition and ξ = 6 for conformal coupling) and L and M are as given in Ref. 14, but with Ψ (α, a) replaced for Ψ (α), n n 1 > 2πω √αt , as given in (3.16), that is ≥ 0 holds, and hence the above conclusion follows. 3 1 ∞ n n 3ξn The modified Misner space could be therefore trans- L = − Ψn(α) π2 3 − 2 formed into a time machine by simply replacing the iden- n=1 X tified flat planes for identified spheres or tori, and allowing one of the resulting spherical or toroidal mouths to 1 ∞ 2n + n3 9ξn move relative to the respective other mouth. This would M = Ψn(α). π2 3 − 2 be equivalent to extract two spheres or tori from three- n=1 X dimensional Euclidean space and identify the sphere or For the particular case of non-twisted fields, α = 0, ex- torus surfaces, while they are set in relative motion, so pression (3.12) reduces to when you enter the surface of one you find yourself emerg- ing from the surface of the other. In Minkowski space- 1 πn2t2 time, the time machine is obtained identifying the two Ψ(t )= e2 0 1 − , 0 − world tubes swept out by the approaching (or receding) spheres or tori, with events at the same Lorentz time 2 so that the quantities φ and Tβγ are finite even when identified. The result would be what one may call quan- h i h i α = 0. tum time machine. The chronology horizon of this con- Thus, all the divergences coming from quantum vac- struct is not on just the single surface at t = 0, but uum fluctuations that take place on the chronology hori- is spread throughout a time strip with nonzero width, zon of the classical Misner space whose periodic direction around the surface at t = 0. has constant period are smoothed out on the modified, no We have also checked that the quantum vacuum po- longer classical Misner space where the period along the larization has no catastrophic effects anywhere in the periodic direction is time-dependent, for which case the modified Misner space, so quantum time machines are expectation values of the scalar field squared and stress- quantum-mechanically stable. One would moreover ex- energy tensor are regular everywhere for any value of α pect that when their size approach the Planck length and ξ, provided t0 = 0. scale, these machines are spontaneously created in the 6 quantum-gravity framework of the spacetime foam [17]. We do not know the probability for the existence of such IV. CONCLUSIONS: THE QUANTUM TIME machines, but the very definition of the foam requires vi- MACHINE olation of causal locality everywhere, and hence one can assume the existence of submicroscopic quantum time In this paper we have considered the change of topol- machines as a pre-requisite for the existence of the space- ogy implied by the continuation from Lorentzian Misner time foam itself, and may therefore adscribe a probability space into its Euclidean counterpart; i.e.: from S1 R3 to of order unity for the existence of time machines in the S1 S1 R2. It was seen that the spatial direction× which foam. is also× periodic× in the Euclidean continuation has a pe- The possibility that a future technology [1], or present riod that depends on the Euclidean time and rotates back natural process taking place somewhere in the universe to an also time-dependent period on the Lorentzian sec- [23], be able to pull a quantum time machine with ex- tor. This induces a change in the structure of the Misner tremely large spacetime curvature out from the quantum space, giving rise to a true singularity at t = 0, instead spacetime foam, and then grow it up to macroscopic size of the apparent singularity on the chronology horizon. [1], will largely reside on the probability of existence that

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ACKNOWLEDGMENTS

For useful comments, the author thanks L.J. Garay and G.A. Mena Marugn of IMAFF. This research was sup- ported by DGICYT under Research Projects No. PB94- 0107 and, partially, No. PB93-0139.

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