Chronology protection implementation in analogue gravity

Eur. Phys. J. C (2022) 82:299 https://doi.org/10.1140/epjc/s10052-022-10275-3 Regular Article - Theoretical Physics Chronology protection implementation in analogue gravity Carlos Barceló1,a, Jokin Eguia Sánchez2, Gerardo García-Moreno1,b, Gil Jannes3,c 1Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta de la Astronomía, 18008 Granada, Spain 2Department of Cell Biology and Histology, Faculty of Medicine and Nursing, University of the Basque Country (UPV/EHU), Barrio Sarriena S/N, 48940 Leioa, Spain 3Department of Financial and Actuarial Economics and Statistics, Universidad Complutense de Madrid, Campus Somosaguas s/n, 28223 Pozuelo de Alarcón, Madrid, Spain Received: 31 January 2022 / Accepted: 27 March 2022 © The Author(s) 2022 Abstract Analogue gravity systems offer many insights into gravitational phenomena, both at the classical and at the semiclassical level. The existence of an underlying Minkowskian structure (or Galilean in the non-relativistic limit) in the laboratory has been argued to directly forbid the simulation of geometries with Closed Timelike Curves (CTCs) within analogue systems. We will show that this is not strictly the case. In principle, it is possible to simulate spacetimes with CTCs whenever this does not entail the presence of a chronological horizon separating regions with CTCs from regions that do not have CTCs. We find an Analogue-gravity Chronology protection mechanism very similar in spirit to Hawking’s Chronology Protection hypothesis. We identify the universal behaviour of analogue systems near the formation of such horizons and discuss the further implications that this analysis has from an emergent gravity perspective. Furthermore, we build explicit geometries containing CTCs, for instance spacetimes constructed from two warp-drive configurations, that might be useful for future analysis, both from a theoretical and an experimental point of view. Contents 1 Introduction ...................... 2 Attempts to simulate Gödel spacetime ......... 3 A survey of some spacetimes displaying CTCs amenable to analogue gravity simulation ....... 3.1 CTCs engineering through warp-drive bubbles .. 3.2 Generalized warp-drive regions .......... 3.3 Misner and Misner-like spacetimes ........ 4 Analogue gravity simulations: attempts to simulate CTCs ae-mail: [email protected] be-mail: [email protected] (corresponding author) ce-mail: [email protected] 5 Chronology protection: lessons from analogue gravity 6 Summary and conclusions ............... References ......................... 1 Introduction It is by now well known that many systems akin to condensed matter systems, in the sense of being composed by a large amount of elementary building blocks (atoms or abstract particles), exhibit a behaviour in certain regimes which can be characterized by the presence of some effective fields, classical or quantum, moving in an effectively curved Lorentzian geometry. These behaviours are collectively called “analogue gravity” [1,2]. In its broadest description, the analogue gravity program intends to obtain new insights into gravitational behaviours by analyzing their equivalent counterparts within these analogue frameworks. The reverse direction: acquiring new ideas about laboratory systems by importing gravitational notions and techniques, is also part of the analogue gravity realm. The most paradigmatic analysis within this program has been the theoretical and experimental verification of Hawking radiation within black hole configurations even when these take place in the context of an effective and collective phenomenon. The appearance of spontaneous Hawking radiation in Bose-Einstein condensates has been observed in [3] as originally suggested in [4]. Apart from black holes, since the late 90s, many other types of geometries have also been proposed in different laboratory settings [1], such as rotating geometries [5–7], cosmological solutions [8] with a very recent experimental realization with fluids of light [9], anti-de Sitter spacetime [10], and even warp-drive geometries [11,12]. For a more exhaustive list, see [1] and references therein. 0123456789().: V,-vol 123 299 Page 2 of 17 Eur. Phys. J. C (2022) 82:299 In this paper we are interested precisely in some puzzles that appear when playing with these warp drive geometries. It is well known that one can in principle use warp drives to build time machines [13]. This raises the question whether one could simulate a geometry with Closed Timelike Curves (CTCs) within an analogue system inspired by warp drives. The laboratory systems used to build analogue gravity configurations are always embedded in our locally Minkowskian world. However, the Minkowski structure of our current fundamental description of Nature typically does not even play a role in the analogue geometries: these geometries can be obtained directly within a Galilean description of the laboratory. In both cases, it seems that the causal structure of the background (Minkowski or Galilean) prohibits the simulation of causally pathological spacetimes in the embedded analogue gravity system [14]. However, by analyzing different situations, we will show that this is in fact not strictly the case. The simulation of spacetimes with CTCs per se does not present insurmountable obstacles. The real problem appears when the relevant spacetimes posses a chronological horizon [15], that is, a surface separating a region with CTCs from another with a standard causality. For external or laboratory observers, the inability to generate a region with CTCs manifests itself in the form of divergences in certain properties of the local physics that they experience. On the other hand, for an internal observer without direct access to the underlying causal structure, the inability to produce CTCs manifests itself through some effective protection mechanism. As we will discuss, these mechanisms are reminiscent of Hawking’s Chronology Protection Conjecture. Throughout our discussion, we will revise several simple configurations of spacetimes with CTCs, such as warp drive spacetimes, Gödel spacetime and Misner spacetime. We present versions of these geometries that are amenable to further analysis both in the context of analogue gravity but also from a purely geometrical perspective. In order to maintain an explicit connection with potential laboratory realizations of such CTC spacetimes, we will focus on a concrete substratum, namely a generalized Bose-Einstein condensate with anisotropic masses [1,16]. The inverse acoustic metric for linear perturbations on this quantum fluid can be written gμν =μ ρ0c⎛ ⎝ −1−vi −vjc2hij −vivj⎞ ⎠.(1) Here ρ0is the background fluid density, cthe local speed of sound and vi(x)the velocity of the fluid, which is simply the gradient of the phase of the macroscopic BEC wave function. The matrix hij (or its inverse hij) takes into account any anisotropy acquired by the effective masses of the bosons in the condensate: mij =μhij, where μis an arbitrary conformal constant. In simple (isotropic) BECs, hij is just a multiple of the identity matrix. For a weakly interacting BEC there is also the following relation between c,μ and the effective coupling constant of the condensate λ:c2=λρ/μ. Themetric(1) automatically inherits the stable causality property [15,17] from the background structure [14]. In particular, it contains a globally defined time function tsince gμν∂μt∂νt=−μ ρ0c≤0.(2) This appears to automatically rule out the possibility of simulating metrics containing CTCs. As stated above, we will see that without further qualification this is in fact not strictly true. Moreover, in the cases in which we really find an obstruction, it is interesting to analyze when and how these effective-metric descriptions break down, and how these breakdowns are related to mechanisms of Chronology protection. A brief outline of the remainder of this work is the following. We begin in Sect. 2with the warm-up exercise of trying to simulate a Gödel spacetime and mild deformations thereof in the system described above. We will find that, although it is possible to simulate CTCs, they are trivial in a sense that we will specify concretely. Furthermore, we will find that it is impossible to simulate a modification of Gödel’s spacetime such that a chronologically well-behaved region with no CTCs evolves into a region with CTCs, due to the divergence of the speeds of the fluid required. Motivated by this exercise, we try to analyze whether this is a generic feature of spacetimes containing CTCs. For that purpose, we introduce in Sect. 3a catalogue of geometries amenable to simulation in analogue gravity. Some of them do not have a General Relativistic counterpart. In Sect. 3.1 we describe spacetimes containing CTCs engineered from two warp-drive bubbles. We discuss the impossibility of doing it in 1 +1 spacetime dimensions, with special emphasis on the point that CTCs in such dimensionality require non-trivial topologies. Based on these warp drive tube geometries, we introduce a family of simpler geometries which are qualitatively similar to them but much easier to handle in Sect. 3.2. We conclude Sect. 3with a discussion of 1 +1-dimensional spacetimes in Sect. 3.3. We introduce the archetypal example of a spacetime containing a chronological horizon, Misner’s spacetime, and then discuss how an eternal cylinder with a flat metric can be understood as having “trivial” CTCs by a simple interchange of the time and space coordinates. A reader interested just in the Chronology Protection mechanism in Analogue gravity can safely skip these first sections and jump directly to Sect. 4, which contains a detailed description of the possibility of simulating trivial CTCs in our analogue model, and the impossibility of simulating chronological horizons. Furthermore, we identify the insurmountable difficulty that every 123 Eur. Phys. J. C (2022) 82:299 Page 3 of 17 299 standard analogue gravity model would face when trying to simulate a chronological horizons. In Sect. 5we discuss the interplay of our analysis and Hawking’s Chronology Protection conjecture, its implications for the emergent gravity program and we also connect with recent related discussions in the literature. Finally, we finish in Sect. 6by summarizing the content of the article and describing potential directions for future work. Notation and conventions. We will use the signature (−,+,...,+)for the spacetime metric and follow the Misner–Thorne–Wheeler conventions for the curvature tensors [18]. Greek indices (μ,ν,...)will run from 0 to D, representing spacetime indices, whereas Latin indices (i,j...) will run from 1 to Dand represent spatial indices. Einstein’s summation convention is used throughout the work unless otherwise stated. 2 Attempts to simulate Gödel spacetime As a warm-up exercise we will describe Gödel’s metric as the archetypal example of a geometry which contains CTCs [15,19]. The purpose of this section is twofold. First, we will describe the geometric properties of Gödel’s spacetime. Many of these properties will be shared by any spacetime containing CTCs, thus allowing us to focus the discussion on the essential features for any successful simulation of CTCs in an analogue model. Second, we will dig into the problems that appear when one attempts to simulate such chronologically pathological spacetimes. A more general discussion concerning generic spacetimes displaying CTCs will be provided later. The starting point of this section has a substantial overlap with the unpublished work [20]. An analysis similar to the one presented here for the simulation of Gödel geometry in an optic system was presented in [21]. The identification of the metric components with the physical parameters of the analogue system do not seem to be done in the correct way, and hence the divergences that we observe in the horizons here are absent. Gödel’s spacetime is a solution of the Einstein equations with suitable sources, namely a negative cosmological constant and the energy momentum tensor of a pressureless perfect fluid with a density: ρ∝−. In appropriate coordinates (t,r,φ,z), it can be written as follows [22]: ds2=−dt2+dr2 1+r2 4ω2+r21−r2 4ω2dφ2 +dz2−√2 ωr2dtdφ(3) where ωis a parameter characterizing the solution and related to the density and hence also trivially related to the cosmological constant as ω2=−. That this geometry contains CTCs can be seen as follows: for r≥rC=2ω, the (Killing) vector field ∂φbecomes timelike. Since such a vector needs to be periodically identified to avoid a conical singularity at r=0, we have that φ∼φ+2π. Hence, this vector field has closed orbits. Since it becomes timelike at r≥rC,itis trivial to conclude that the orbits of φfor r>rCare CTCs. It seems that no CTCs pass through the region r<rC. However, we must take into account that this geometry is completely homogeneous, in fact it contains a group of five Killing vector fields acting transitively on the manifold [15]. This means that every point of the manifold can be mapped by a symmetry transformation to any other point on the manifold. Hence, CTCs pass through every single point in this spacetime. However, there are no CTCs confined to the region r<rC, in fact every CTC passing through the region r<rC crosses the cylinder r=rCan even number of times. From the point of view of an acoustic metric, we can realize that this precise system of coordinates allows for a direct realization of Gödel’s metric with suitable fluid parameters. The non-vanishing components of the inverse Gödel metric in these coordinates are gtt =−F(r), gtφ=− 1 √2ω 1 1−r2 4ω2F(r), gφφ =1 r2 1 1−r2 4ω2F(r), grr =1+r2 4ω22 1−r2 4ω2F(r), gzz =1+r2 4ω2 1−r2 4ω2F(r). (4) with F(r)defined as F=1−r2 4ω2 1+r2 4ω2.(5) Comparing Eq. (4) with Eq. (1) we realize that we simply need a motion of the fluid in the φ-direction vi=vφδi φ. Taking into account the change of coordinates to a cylindricallike coordinate system we have that the azimuthal velocity must be vφ=1 √2ω r 1−r2 4ω2 .(6) 123 299 Page 4 of 17 Eur. Phys. J. C (2022) 82:299 On the other hand the three principal directions of the anisotropy matrix hij need to obey c2hrr =1+r2 4ω22 1−r2 4ω2, c2hφφ =1+r2 4ω2 1−r2 4ω22, c2hzz =1+r2 4ω2 1−r2 4ω2.(7) These quantities can be interpreted as three anisotropic sound speeds c2 r,c2 φand c2 z, respectively. In addition, we notice that for the weakly interacting BEC we have λ/c3=F, or equivalently c2=λ2/31+r2 4ω22/3 1−r2 4ω22/3,(8) we finally obtain hrr =λ−2/31+r2 4ω24/3 1−r2 4ω21/3, hφφ =λ−2/31+r2 4ω21/3 1−r2 4ω24/3, hzz =λ−2/31+r2 4ω21/3 1−r2 4ω21/3.(9) Let us analyze the properties of this fluid required to simulate the geometry. The first thing we notice is that the velocity of the fluid becomes infinite at r=rC, where it also changes its sign. Hence the fluid system is singular; the r<rCand r>rCparts of the system are disconnected. However, the CTCs are living entirely in the exterior region of the metric so it may still appear that the analogue system can locally simulate CTCs. However, looking at the speeds of sound we identify an additional issue. The speeds of sound also diverge at r=rC, but moreover c2 r=c2hrr and c2 z=c2hzz become negative for r>rC,socrand czbecome purely imaginary. This means that we no longer have wave-like behaviours, or in other words causal signalling, in those directions within the analogue systems. The whole acoustic picture appears to break down. In that sense, the r=rCcylinder can be understood as a sort of “domain wall”: it separates the interior region in which we have causal signalling from the region in which we have abnormal (exponentially amplified or attenuated) behaviour of the putative sound-like excitations. To design a realistic situation, imagine that we start with a fluid at rest and incite a rotation around the z-axis with the intention of evolving towards the Gödel geometry. In order to do so, we need to obtain a configuration in which there is a separation between a clockwise and an anticlockwise rotating part of the fluid, separated by a surface at r=rC where the velocity needs to approach infinity and moreover the speed of sound also blows up. These requirements essentially imply that the hydrodynamic description of the BEC breaks down. Furthermore, because of the infinite fluid velocity at the r=rCsurface, no signal could cross this surface. However, the physical parameters of the BEC are well defined for r>rC, where one finds CTCs for the BEC excitations. On the one hand, one would need an imaginary sound speed in that region. This can be attained in BECs with attractive interactions [23,24]. On the other hand, two of the effective anisotropic masses of the BEC must be negative. The peculiarity of a particle with a negative mass is that it accelerates backwards when pushed forward.1However, it has been shown experimentally that it is indeed possible to achieve such strange behaviour and create particles with negative effective masses [25]. Therefore, we have a surprising situation. The excitations of a quite strange BEC, with attractive interactions (which in principle would appear to forbid wave phenomena) combined with some negative anisotropic masses, end up behaving as if these excitations live in a perfectly Lorentzian world displaying CTCs. The situation would be equivalent in any other anisotropic fluid, not necessarily quantum.2One would just need that c2 rand c2 zbecome negative in some region while c2 φstays positive. Roughly speaking, this ensures that the r and zcoordinates acquire the same signature as the tcoordinate, leaving the angular coordinate φas the coordinate of different signature, i.e. the time coordinate, and CTCs will develop. From the perspective of the internal observers inside the fluid, “time” would be what for an external (laboratory) observer is simply the angular coordinate. Going back to Gödel’s metric, one could be tempted to modify the parameters of the analogue model and regularize the divergences. A simple example would be the following profiles where a suitable regulating parameter 1isintroduced (for simplicity we restrict our fluid to be effectively two-dimensional): 1Notice that such negative masses are not fundamental, and thus need not result in tachyonic instabilities. 2It is true, however, that engineering a classical fluid to display “negative mass” excitations might be much more complicated, perhaps even impossible in practice. 123 Eur. Phys. J. C (2022) 82:299 Page 5 of 17 299 Fig. 1 The left panel represents the anisotropic speeds of sound and the azimuthal velocity of the fluid required to perform an analogue simulation of Gödel’s geometry for ω=1. All of them display a vertical asymptote at r=rC. The right panel displays the corresponding anisotropic speeds of sound and the azimuthal velocity of the fluid for the regularized Gödel geometry introduced in the text, also for ω=1 and for the regularization parameter =0.01. Whereas the graphics on the left panels all blow up at r=rC, the ones in the right panels are smooth everywhere. All functions are normalized to a maximum value of 1 in the plot vφ=r √2ω 1−r2 4ω2 1−r2 4ω22+2 ,(10) c2 r=1+r2 4ω221−r2 4ω2 1−r2 4ω22+2 ,(11) c2 φ=1+r2 4ω2 1−r2 4ω22+2 .(12) However, it is straightforward to see that the associated acoustic metric (a cousin of Gödel’s metric) is not a regular Lorentzian metric now, it is degenerate at r=rC. Still, strange as it may seem, this acoustic system does approach Gödel metric for rrC. These speeds of sound and velocity of the fluid as well as the corresponding ones for pure Gödel are plotted in Fig. 1. A note of caution might be in order at this point. Although this analysis suggests that it is possible to simulate CTCs in an analogue system (a fluid in this case), these CTCs are, in a sense, trivial. From the point of view of the external observer, the creation of these CTCs corresponds simply to declaring that the internal observer is using in the internal system an angular coordinate as time coordinate. From now on, we will refer to this kind of CTCs as trivial, to distinguish them from CTCs that appear in the causal future of a causally well-behaved region, which are the most interesting ones from a physical point of view. Hawking characterized this type of spacetimes geometrically as those with a compactly generated Cauchy horizon [26].3 Spacetimes with non-trivial CTCs can thus be understood as those with a smooth transition from a region without CTCs to one with CTCs. From an analogue point of view, it seems that the simulation of such non-trivial cases is not possible. Indeed, they would require either a non-regular Lorentzian metric, which could be reproduced within an analogue model, or a well-defined Lorentzian metric but requiring some divergences in the analogue model. The former case would not really constitute a clear proof of principle of the possibility of simulating spacetimes with CTCs, since the CTCs could be understood to be an artifact of the non-smoothness of the 3A compactly generated Cauchy horizon is a Cauchy horizon such that the past extension of its generators enters and remains within a compact subset of the manifold. 123 299 Page 6 of 17 Eur. Phys. J. C (2022) 82:299 metric and thus, in a sense, spurious or at least not directly related to a (non-analogue) relativistic equivalent. On the other hand, the latter case has already been discussed and, based on the example of Gödel’s metric, seems to correspond to trivial CTCs at best, since the two regions need to be causally disconnected. To finish this section let us consider the possibility of simulating a geometry which approaches Gödel’s metric only in a finite range of the laboratory time t. To our knowledge, this geometry does not have a General Relativistic counterpart, in the sense that it is not a solution to the Einstein equations with the energy-momentum tensor of a known matter content. Such behaviour could be achieved with the help of a modulating function f(t), such that we can write a metric ds2=−dt2+dr2 1+f(t)r2 4ω2+r21−f(t)r2 4ω2dφ2 +dz2−f(t)√2 ωr2dtdφ, (13) where we can choose f(t)to be a function with compact support, for instance f(t)=exp −σ (tB−t)(t−tA),(14) which is non-vanishing for t∈(tA,tB). Hence, the metric represents a flat spacetime outside this interval, and develops CTCs within the region (tA,tB). In order to confine the CTCs to a compact region of space also, one could force the metric component gφφ to take negative values only within a finite interval of the r-component, for example through the following replacement gφφ =r21−f(t)r2 4ω2−→ r21−f(t)e−r2 σ2r2 4ω2, (15) where σmust to be sufficiently large in order for the function 1−e−r2 σ2r2 4ω2to display two zeros. This new geometry exhibits CTCs that are confined within a finite region of spacetime. However, for the same arguments explained above, it is not possible to simulate them as acoustic metrics since the fluid would be required to develop a singular velocity. This again illustrates our more general point that it seems impossible to generate metrics with non-trivial CTCs through an analogue metric. 3 A survey of some spacetimes displaying CTCs amenable to analogue gravity simulation In this section we are going to present some geometries containing CTCs which we think are conceptually simpler than Gödel spacetime. Most of these geometries can be found somehow in the literature. However, we think that it is worthy to revise them and present them in a unified way, so that they are amenable to be analyzed from the analogue gravity perspective. First, we will start describing the geometry that results from combining two warp drives and displays CTCs. Motivated by the properties of this geometry, we will introduce a family of spacetimes which are simpler but encapsulate their main geometric features. Finally, we will discuss probably the most paradigmatic example of spacetime containing CTCs: the so-called Misner spacetime. This spacetime is used as a proxy to more convoluted analysis, since its chronological horizon is usually considered to have the general properties a chronological horizon has. Although for most practical purposes this is true, we will put special emphasis here on the fact that Misner spacetime is topologically non-trivial as a manifold (otherwise it could not contain CTCs as we will explain). In higher dimensions, (like 3 +1 spacetime dimensions) it is possible to have spacetimes with CTCs exhibiting a trivial topology. From an analogue gravity perspective, this makes life much easier for their simulation. 3.1 CTCs engineering through warp-drive bubbles Warp drives were originally introduced by Alcubierre [27]. They are based on disturbing a given spacetime within a compact region in such a way that for observers outside that region, observers inside of it move with superluminal speeds. They can be thought as “tachyonic” bubbles that propagate faster than light for external observers. The simplest metrics representing warp drives can be written using the Nátario’s line element: ds2=−dt2+δijc−2dxi−vidtdxj−vjdt.(16) There are some warp drives with non-zero lapse function or with a non-Euclidean metric, although we will not focus on them. In this way, the metric of a warp drive is nicely adapted to be simulated with the acoustic metric of a BEC, as described above, or acoustic metrics showing up in other fluids. This idea has already been suggested in the literature, see for instance [11,12]. Essentially, we need to identify the velocity of the fluid with the shift functions entering the warp drive element. For concreteness, we can think of a warp-drive bub123 Eur. Phys. J. C (2022) 82:299 Page 7 of 17 299 ble whose profile acquires the following form vi(t,xi)=δi xu(t)f(x−x(t))2+y2+z2(17) where f(x)is a compact support function, describing the profile of the bubble which is peaked around the points of the trajectory x(t)=x(0)+t 0 dtu(t), y=z=0.(18) As we just say, the simulation of a single warp drive in a acoustic analogue system is direct [1]. One just need to generate a spacetime region at which the velocity of the flow exceeds the speed of sound. From the internal perspective this allows to travel from one point to another at velocities higher than that of sound (which remember takes the role of the speed of light). It is convenient to rewrite the warp drive metric in Eq. (16) as a perturbation of the flat spacetime metric ημν as gμν =ημν +bμν,(19) with bμν having the following non-vanishing components: b00 =u2(t)f2(x−x(t))2+y2+z2, b01 =b10 =−u(t)f(x−x(t))2+y2+z2.(20) Now, although a single bubble warp drive by itself does not result in any chronology or causality violations, as already noticed in [13] it is relatively easy to engineer a spacetime that contains CTCs by taking advantage of having two warpdrive bubbles in dimensions higher than 1 +1. The idea is similar to the way in which one can send information to the past with a pair of tachyon particles in flat spacetime [28]. What is it then the clash, if any, between warp drive metrics and their analogue simulations? We want to construct regular spacetime geometries based on a combination of two warp drives, in such a way that they contain CTCs. After finding such geometries we will analyze whether it is possible to reproduce them within an analogue model in Sect. 4. The simplest such construction that one can think of a priori involves two warp-drive bubbles in a 1 +1 dimensional setting with trivial R2topology. In the remain of this subsection, we are going to discuss for a moment this 1 +1 potential construction. First, we will naively present it. Then, we will illustrate the obstruction that one finds when one tries to formalize this construction. After that, we will show that this problem cannot be circumvented by presenting some theorems showing that this construction Fig. 2 The figure represents a warp drive tube that starts in a location A and ends in location B. The underlying spacetime can be considered D+1 dimensional with the warp-drive bubble moving in a straight line (along the x-coordinate, in the picture). In 1 +1 dimensions the causal cone would become just two crossing lines but we keep the cone symbol for clarity. This is the simplest construction of a warp drive and as described in the text, it can be simulated in an analogue gravity model without further problems is actually not possible. Finally, we will conclude this subsection by explaining how this construction can be extended to D+1 spacetime dimensions with trivial topology without problems. Let us begin with the most naive way in which one might try to make this configuration. Let us consider a 1+1 dimensional Minkowski background. Let us choose an inertial reference frame Swith Cartesian coordinates (t,x). Furthermore, let us choose two events Aand Bsuch that they are spacelike separated, with coordinates (tA,xA)and (tB,xB), respectively. Without loss of generality, let us assume that tB>tA. This setup is represented pictorially in Fig. 2. We can engineer a warp drive tube connecting the two events. Notice that the lightcones inside the tube are modified with respect to the Minkowskian reference. Furthermore, we emphasize that the tube has some thick walls at which the light cones experience a tilting effect. It is precisely on those walls where the stress energy tensor supporting these configurations necessarily develops some energy conditions violations [27]. We emphasize that the trajectory as seen from outside the tube appears to be spacelike. Of course, observers going from A to Bwithin the bubble would be following strictly timelike trajectories. If one can construct this warp drive, from a purely general relativistic perspective it is also possible to construct an equivalent warp drive configuration in which the coordinate time goes to the past instead of the future [13]. Let us write down such a metric explicitly. Let us start constructing a warp drive metric as the one just described but using 123 299 Page 8 of 17 Eur. Phys. J. C (2022) 82:299 another inertial reference frame Smoving with velocity v in the xdirection. Let us denote with a prime the cartesian coordinates of the reference frame S, i.e. (t,x). In these coordinates, the metric of the bubble takes the simple form g μν of Eq. (16). To find the metric in the coordinates (t,x) adapted to the inertial frame S, we simply need to perform a boost of velocity −vin the x-axis, with vthe relative velocity between Sand S. In this way, the resulting metric reads gμν =ημν +cμν,(21) where cμν is transformed from the prime coordinates to the unprimed ones by an ordinary Lorentz transformation. We highlight that its functional form differs from the bμν tensor introduced in Eq. (19). Actually, we can find its functional form by performing a Lorentz boost in the x-direction of velocity −v, where vis the relative velocity between both frames Sand S. Explicitly, it is described by the following linear transformation in: ()μ ν=cosh φsinh φ sinh φcosh φ,with tanh φ=v, (22) Writing down the transformation we obtain the following cμν tensor c00 =u2(t(t,x), x(t,x), y−y0,z)cosh2φ −2u(t(t,x), x(t,x), y−y0,z)cosh φsinh φ, (23) c01 =c10 =u2(t(t,x), x(t,x), y−y0,z)cosh φsinh φ −u(t(t,x), x(t,x), y−y0,z)cosh2φ −u(t(t,x), x(t,x), y−y0,z)sinh2φ, (24) c11 =u2(t(t,x), x(t,x), y−y0,z)sinh2φ −2u2(t(t,x), x(t,x), y−y0,z)sinh φcosh φ. (25) Notice that the functions tand xdepend on the coordinates t,xin a non trivial manner, and we need to rewrite them in terms of such coordinates. In fact, the coordinates {xμ}are related to the coordinates {xμ}through a Lorentz transformation from Sto Sin which the Lorentz matrix is precisely μ ν. Explicitly, the change of coordinates reads t=tcosh φ−xsinh φ, (26) x=tsinh φ+xcosh φ. (27) Writing everything explicitly without fixing a particular trajectory and shape for the bubbles would not be very illustrative, hence we simply keep everything indicated as done above. We emphasize that one just needs to choose a profile for the bubble and the velocities in order to be able to write down explicitly the metric in global coordinates by substituting in the expressions above. In generic terms we have Fig. 3 The figure represents a warp drive tube that starts in a location Aand ends in location B. The underlying spacetime can be considered D+1 dimensional with the warp-drive bubble moving in a straight line (along the x-coordinate, in the picture). In 1 +1 dimensions the causal cone would become just two crossing lines but we keep the cone symbol for clarity build a warp drive travelling backwards in coordinate time t, pictorially represented in Fig. 3. Note however that one can easily check that the Lorentz transformation we have applied make the new warp drive metric to take a different form from Natario’s line element. For the arguments that follow the precise form of the tubes will not be relevant. Now setting up a combination of a “forward” and a “backward” warp-drive bubbles one can attempt to build a time machine. Let us explicitly illustrate this. We can first set up a “forward” warp drive allowing a faster-than-light travel from A to B. Once the traveller has exit the bubble at B he could immediately enter in a new warp drive, now of a “backward” type, and travel from Ato B. Using another spacelike trajectory, as seen from the external Minkowski spacetime, this second warp drive can take the time-traveller to an event B in pass of the initial event A. In this way a CTC is completed. This setup is pictorially represented in Fig. 4. However, there is a problem concerning this construction. There is a region, the crossing region C, at which one would need to have two different metrics. Actually, this translates into a singular point where the metric is not defined. It is natural then to pose the following question: is it possible to disentangle the crossing point moving around the starting and ending points of the bubbles or/and playing with their shapes and the specific forms of their velocities v1(t), v2(t), in such a way that we find a completely regular Lorentzian metric containing CTCs? The answer to this question is negative. To understand why, it is useful to consider a toy geometry which nicely illustrates the obstruction. Imagine that we write down a geometry which is that of flat spacetime at every 123 Eur. Phys. J. C (2022) 82:299 Page 9 of 17 299 Fig. 4 We represent here two warp drives in 1 +1 dimensions and in such a configuration that they appear to allow for the formation of CTCs. The purple curve represents a generic CTC on this background. The problem with this 1+1 configuration is that the metric in the region where the two warp-drive bubbles cross is ill-defined. As described in the text the simplest geometry with CTCs is either one with a S1×R topology or with topology RD+1in D+1 dimensions, with D>1. In this latter case, we just need to engineer the two warp drives bubbles in different parallel planes to avoid the crossing point except for a circle of radius one around the origin in a given set of Cartesian coordinates (t,x). At such circle, we make the lightcones to make an angle of 45 degrees with the circle at every point. Clearly such geometry is singular since there is a jump in the metric. Even if we try to smooth such geometry by giving the circle a finite size and converting it into a disk, we can find a curve along which the lightcones make a rotation of 360 degrees and, hence, it is impossible to have a smooth metric in the region enclosed by such curve or regularize it in some way. This is depicted schematically in Fig. 5. Now we are in position of stating the following theorem: Theorem Let (M,g)be a two dimensional simply connected spacetime (being Mthe smooth manifold and g its metric). Then, the causality condition automatically holds. Recall [15] that a spacetime is said to satisfy the chronological condition if it does not contain any closed timelike curves, and it is said to obey the causality condition if there are no closed non-spacelike curves. The idea of the proof is already contained in the observation that we have made above: having the structure of lightcones enclosing a compact region, it is impossible to push them inwards or outwards that region without making them zero or singular. The formalization of this statement can be found in [29]. It is possible to even prove a stronger result. In Lorentzian geometry there exist a hierarchy of causality conditions Fig. 5 We represent here the setup described in the text that already shows the difficulty present when trying to build CTCs in a spacetime with a trivial topology. The shaded region represents the region of abnormal behaviour of the lightcones. It seems impossible to regularize the lightcones without removing points from the spacetime, otherwise the metric would need to vanish at some point and hence it would not be a regular spacetime where each of them is stronger than the previous ones. Although the chronology condition is the weakest of such conditions, followed by the causality condition, and they are enough to rule out closed non-spacelike curves, one can still think of spacetimes that are “arbitrarily” close to containing closed causal curves. Hence, these set of stronger conditions attempts to formalize these notion of “almost having closed curves” [15,17]. The strong causality condition [17], which is obeyed by a spacetime if for every point pand every neighbourhood Nof p, there exists a neighbourhood Ocontained in Nsuch that no causal curve intersects Omore than once. By essentially the same arguments exhibited in our proof, one can strengthen the result and prove that every two dimensional time-orientable simply connected spacetime is strongly causal (see Lemma 14.34 of [29]). Actually, it is even possible to strengthen this result under the same hypothesis. A spacetime is said to be stably causal if there exists a timelike vector field tμsuch that the Lorentz metric defined as ˜gμν =gμν −tμtν(which has larger lightcones than gμν at every point) contains no closed causal curves. It can be proved that a spacetime is stably causal if and only if it admits a globally defined time-function, i.e. a function that is strictly increasing along each future directed causal curve [15]. It is also possible to prove that stable causality implies strong causality, i.e. it is a stronger condition. Hence, stable causality is a stronger condition and one might wonder whether it is possible to prove that every two dimensional simply connected spacetime obeys it without further assumptions. In [30] the affirmative answer is provided in the form of Theorem 3.43, where it is shown that every simply connected two dimensional spacetime (M,g) 123 299 Page 16 of 17 Eur. Phys. J. C (2022) 82:299 to the interpretation of the physical system as a gravitational analogue in the first place. We have focused on an acoustic system in which it is essentially the velocity of the fluid which must diverge in order to create the required tilting of the sound cones. We have shown that it is perfectly possible to simulate geometries allowing superluminal behaviours such as warp drives. However, this does not imply directly that one can build an analogue time machine, as we have discussed in detail. It is in fact the formation of a chronological horizon which is forbidden in the analogue implementation, since it is not possible to create a warp drive travelling backward in laboratory time. The obstructions found by exploring the analogue gravity implementation of CTCs resonate with Hawking’s idea of a Chronology Protection mechanism in semiclassical General Relativity. From the point of view presented here, such protection mechanisms arise naturally in frameworks for Emergent Gravity. In these emergent frameworks, there exists a background structure with a more fundamental underlying causality, which naturally prevents the type of manipulations required to create chronological pathologies. Finally, we sum up three important lessons from the present work. (i) Superluminality itself does not imply the possibility of abnormal causal behaviour such as time travel; (ii) Problems in the analogue implementation of chronological horizons appear due to the relative tilting of the causal cones; (iii) The current observational absence of chronological pathologies in our universe is naturally explained in frameworks in which there is a fixed underlying causality beyond the local General Relativistic modifications explored so far. Acknowledgements The authors thank Grisha Volovik for useful conversations and for sharing unpublished notes on the simulation of Gödel spacetime. The authors thank Carlos Sabín and Franco Fiorini for helpful correspondence. GGM and CB thank Luis Garay, Miguel Sánchez and Valentín Boyanov for very useful conversations. GGM thanks Roberto Emparán for an enlightening conversation. Financial support was provided by the Spanish Government through the projects PID2020-118159GB-C43 and PID2020-118159GB-C44, and by the Junta de Andalucía through the project FQM219. C.B. and G.G.M. acknowledge financial support from the State Agency for Research of the Spanish MCIU through the “Center of Excellence Severo Ochoa” award to the Instituto de Astrofísica de Andalucía (SEV-2017-0709). GGM is funded by the Spanish Government fellowship FPU20/01684. Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors’ comment: There is no data in our work to be deposited.] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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