Quantum time machine Pedro F. Gonz´alez-D´ıaz Centro de F´ısica “Miguel Catal´an”, Instituto de Matem´aticas y F´ısica Fundamental, Consejo Superior de Investigaciones Cient´ıficas, Serrano 121, 28006 Madrid (SPAIN) (December 6, 1997) The continuation of Misner space into the Euclidean region is seen to imply the topological re- striction that the period of the closed spatial direction becomes time-dependent. This restriction results in a modified Lorentzian Misner space in which the renormalized stress-energy tensor for quantized complex massless scalar fields becomes regular everywhere, even on the chronology hori- zon. A quantum-mechanically stable time machine with just the sub-microscopic size may then be constructed out of the modified Misner space, for which the semiclassical Hawking’s chronology protection conjecture is no longer an obstruction. PACS number(s): 04.20.Gz, 04.62.+v I. INTRODUCTION some attempts intended to violate it [14,15], this con- jecture has survived rather forcefully [16]. However, the After the seminal papers by Morris, Thorne and Yurt- realm where chronology protection holds is semiclassical sever [1], the notion of a time machine has jumped from physics, as it is for all hitherto proposed time machines. the pages of science fiction books to those of scientific Actually, because spacetime foam [17] must entail strong journals, giving rise to a recent influx of papers [2-7] violations of causal locality everywhere, one would expect and books [8,9] on the subject. Several potentially use- the Hawking’s conjecture to be inapplicable in the frame- ful models for time machines have since been proposed, work of quantum gravity proper, and that the divergences including the wormhole of Thorne et al. [1], the Gott’s of stress-energy tensor due to quantum vacuum fluctua- couple of cosmic strings set at relative motion [2], its tions merely are artifact from inappropriate combination Grant’s extension [3], the Politzer’s machine [4], Roman of quantum matter fields with classical spacetime devices. rings [5] and, ultimately, a time machine based on the In this paper we consider an attempt to get rid of these so-called ringholes [6]. divergences in a nonchronal spacetime constructed out Apart of all those difficulties related with potential vi- from Misner space [18], where the thermal properties of olation of classical causality that would arise while trav- Euclidean gravity are taken into account. The distin- elling backward in time [8], all of these models show at guishing feature of the considered spacetime with respect least one of the two following shortcomings: an induced to usual Misner space is that the period of the closed violation of the classical energy conditions for the matter spatial direction becomes now time-dependent, and the allowing closed timelike curves (CTC’s) [9] and, more im- nonchronal region can only be acceded on if the time portantly, the quantum instabilities originating from the coordinate is quantized according to a pattern which is arXiv:gr-qc/9712033v1 6 Dec 1997 divergences of the stress-energy tensor of vacuum fluctu- the same as that for the quantization being currently ations on the chronology horizon [8] (that is the onset of assumed [19] for the event horizon of black holes. We the nonchronal region where CTC’s develop). have checked that the stress-energy tensor for quantum Violation of energy conditions are generally produced vacuum fluctuations in this spacetime is regular every- by the large inward pressure required on the tunnels where, but only time machines with sub-microscopic size and leads to the emergence of regions where matter has can be constructed out of it. negative-energy density. This could no longer be a real The paper is organized as follows. In Sec. II we problem as negative energy has become commonplace in consider the continuation of Misner space into the Eu- quantum and semiclassical gravity [10,11]. More severe is clidean sector where the periodicity of the distinct space- the difficulty related with the need of having a tachyonic time directions are fixed and then continued back to the center of mass in parallel-moving cosmic strings which Lorentzian region, resulting in a modified structure for are subject, moreover, to unphysical boundary conditions Misner space. The Hadamard function and hence the [12]. expectation value of the stress-energy tensor for quan- By far the acutest problem with most of the proposed tum vaccum fluctuations are evaluated in Sec. III in the time machines is the emergence of the stress-energy di- case of a complex massless scalar field propagating in the vergences of the quantum vacuum fluctuations. This modified Misner space. It is checked that these quan- makes instable the spacetime construct and has actu- tities are regular everywhere. We conclude in Sec. IV, ally led to the formulation of a chronology protection where the concept of a quantized time machine is also dis- conjecture [13], according to which the laws of physics cussed. Throughout the paper units are used such that should prevent time travel to occur. So far, in spite of c =¯h = G = 1. 1 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 <t< , 0 x2, x3 , and 0 x1 2π. ∞ ≤ ≤ ∞ ≤ ≤ ds2 = dV dT + T dV 2. (2.10) With respect to Minkowski space, the fact that x1 is pe- − riodic is the unique distinguishing property originating In order to investigate the periodic properties of the from its topology S1 R3. Otherwise, they are identi- × directions in the Euclideanized Misner space, one can cal. This can be seen by introducing the covering space introduce now the extended coordinate definitions of the through the coordinate transformations Lorentzian Misner extended metric 0 1 0 1 2 2 3 3 y = t cosh(x ), y = t sinh(x ), y = x , y = x , u + w = T, u w = V, (2.11) − (2.2) with which (2.10) becomes where the metric becomes ds2 = du2 + dw2 + (u + w)(du dw)2 (2.12) − − ds2 = (dy0)2 + (dy1)2 + (dy2)2 + (dy3)2. (2.3) − and 1 If the period associated with direction x is a, then the u w V topology of the Misner space implies the identification of u2 w2 = T V, − = . (2.13) points on the covering space given by − u + w T 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.