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A Class of Time-Machine Solutions with a Compact Vacuum Core
Abstract
We present a class of curved-spacetime vacuum solutions which develop closed timelike curves at some particular moment. We then use these vacuum solutions to construct a time-machine model. The causality violation occurs inside an empty torus, which constitutes the time-machine core. The matter field surrounding this empty torus satisfies the weak, dominant, and strong energy conditions. The model is regular, asymptotically flat, and topologically trivial. Stability remains the main open question.
... In these cases, the metric is given and the stress-energy tensor is derived a posteriori from the right-hand side of the Einstein equations; the undesired, exotic features emerging from this construction (typically, the violation of the standard energy conditions) are regarded as secondary issues. Probably, the most influential papers in this class are those of Ori and Soen [32,33,34,37]. In particular, [34,37] present a time machine with a toroidal spatial core, surrounded by a region where the spacetime metric is conformally flat. ...
... Probably, the most influential papers in this class are those of Ori and Soen [32,33,34,37]. In particular, [34,37] present a time machine with a toroidal spatial core, surrounded by a region where the spacetime metric is conformally flat. CTCs are developed inside the toroidal region when the external time coordinate reaches a specific value, and violations of the energy conditions appear simultaneously. ...
... Dealing with the above mentioned paradoxes and problems is not among our purposes; here, we just propose to enrich the second class of spacetimes with time travels, introducing a new model (which violates the energy conditions). In setting up this model, we were mainly stimulated by the paper of Tippett-Tsang [42] (the work of Mallary, Khanna and Price [24] also gave us some general motivation to consider this subject); later on we realized that our construction has a closer contact with the model of Ori-Soen [33,34,37]. Our model is topologically trivial, possesses no curvature singularity, and is both space and time orientable; it consists of a toroidal "time machine", which contains CTCs and is surrounded by flat Minkowski space. ...
Inspired by some recent works of Tippett-Tsang and Mallary-Khanna-Price, we present a new spacetime model containing closed timelike curves (CTCs). This model is obtained postulating an ad hoc Lorentzian metric on , which differs from the Minkowski metric only inside a spacetime region bounded by two concentric tori. The resulting spacetime is topologically trivial, free of curvature singularities and is both time and space orientable; besides, the inner region enclosed by the smaller torus is flat and displays geodesic CTCs. Our model shares some similarities with the time machine of Ori and Soen but it has the advantage of a higher symmetry in the metric, allowing for the explicit computation of a class of geodesics. The most remarkable feature emerging from this computation is the presence of future-oriented timelike geodesics starting from a point in the outer Minkowskian region, moving to the inner spacetime region with CTCs, and then returning to the initial spatial position at an earlier time; this means that time travel to the past can be performed by free fall across our time machine. The amount of time travelled into the past is determined quantitatively; this amount can be made arbitrarily large keeping non-large the proper duration of the travel. An important drawback of the model is the violation of the classical energy conditions, a common feature of many time machines. Other problems emerge from our computations of the required (negative) energy densities and of the tidal accelerations; these are small only if the time machine is gigantic.
... The violation of causality from exact solutions of general relativity is not limited to Gödel or Gödel-type solutions. Other models also demonstrate this phenomenon, such as Kerr and Kerr-Newman black holes [4,5], the Van-Stockum model [6], cosmic strings [7], the Tipler rotating cylinder [8], the Bonnor and Steadman space-time [9], the Ori time-machine metric [10], and several others detailed in [11,12,13,14]. Another important solution to highlight here is the one proposed by Gott and Li [15]. ...
... We see that for t = const = T < 0, the closed curve is spacelike and becomes light-like or null for T = 0. Moreover, for t = const = T > 0, the closed curve is time-like, and hence, form closed time-like curves at an instant of time analogue to Ori time-machine model [10]. These time-like closed curves evolves from an initial space-like hypersurface in a well-behaved manner. ...
... These time-like closed curves evolves from an initial space-like hypersurface in a well-behaved manner. We know that a hypersurface t = const = T is spacelike when g tt < 0 and time-like whenever g tt > 0 [10]. We finds ...
In this paper, our objective is to explore a time-machine space-time formulated in general relativity, as introduced by Li (Phys. Rev. D {\bf 59}, 084016 (1999)), within the context of modified gravity theories. We consider Ricci-inverse gravity of all Classes of models, {\it i.e.}, (i) Class-{\bf I}: , (ii) Class-{\bf II}: model, and (iii) Class-{\bf III}: model, where is the anti-curvature tensor, the reciprocal of the Ricci tensor, , is its scalar, and are the coupling constants. In fact, we show that Li time-machine space-time serve as valid solution within this Ricci-inverse gravity theory. Thus, this new theory allows the formation of closed time-like curves analogue to general relativity, representing a possible time machine model in Ricci-inverse modified gravity theoretically.
... 'True' time machine spacetimes are those where CTCs form at some particular instant of time. The Ori time machine spacetime [9], which is locally isometric to plane wave spacetimes, is a prime example in this category. Some other examples would be the spacetimes discussed in [10][11][12][13][14][15][16][17]. ...
... It should be mentioned here that there are quite a number of causality violating solutions where there are no violations of the standard energy conditions of GR. A handful of such solutions would be the spacetimes discussed in [9,13,17,26,27]. The point of discussion here is that the violation of energy conditions is not a prerequisite for obtaining CTCs [9,28]. ...
... A handful of such solutions would be the spacetimes discussed in [9,13,17,26,27]. The point of discussion here is that the violation of energy conditions is not a prerequisite for obtaining CTCs [9,28]. The spacetime discussed in this work satisfies the energy conditions. ...
We present an axially symmetric spacetime which contains closed timelike curves, and hence violates the causality condition. The metric belongs to type III in the Petrov classification scheme with vanishing expansion, shear and twist. The matter-energy represents a pure radiation field with a negative cosmological constant. The spacetime is asymptotically anti-de Sitter space in the radial direction.
... Among the time-machine spacetimes, we mention two: the first being Ori's compact core [17] which is represented by a vacuum metric locally isometric to pp waves and second, which is more relevant to the present work, the Misner space [22] in 2D. This is essentially a two dimensional metric (hence flat) with peculiar identifications. ...
... These curves are null for t = 0, spacelike throughout for t = t 0 < 0, but become timelike for t = t 0 > 0, which indicates the presence of closed timelike curves (CTC). Hence CTC form at a definite instant of time satisfy t = t 0 > 0. It is crucial to have analysis that the above CTC evolve from a spacelike t = constant hypersurface (and thus t is a [17]. This can be ascertained by calculating the norm of the vector ∇ µ t (or by determining the sign of the component g tt in the inverse metric tensor g µν [17]). ...
... Hence CTC form at a definite instant of time satisfy t = t 0 > 0. It is crucial to have analysis that the above CTC evolve from a spacelike t = constant hypersurface (and thus t is a [17]. This can be ascertained by calculating the norm of the vector ∇ µ t (or by determining the sign of the component g tt in the inverse metric tensor g µν [17]). We find from (2) that g tt = t e −2 α r 2 + β 2 z 2 e −6 α r 2 . ...
We present a symmetric spacetime, admitting closed timelike curves (CTCs) which appear after a certain instant of time, i.e., a time-machine spacetime. These closed timelike curves evolve from an initial spacelike hypersurface on the planes z = constant in a causally well-behaved manner. The spacetime discussed here is free from curvature singularities and a 4D generalization of the Misner space in curved spacetime. The matter field is of pure radiation with cosmological constant.
... The one discussed in Refs. [19][20][21][22] is in the latter category. ...
... All the components of the Riemann tensor R μν ρσ are constant and the determinant of the corresponding metric tensor g μν is det g = −1, including at r = 0. Hence the space-time is regular everywhere and the maximum analytical extension of the space-time is possible to the region with negative r, i.e., the domain of r is r ∈] − ∞, +∞[. Thus the presented model is regular, assuming asymptotically flat, and topologically trivial (see also Ref. [19]). The space-time is free from curvature singularities since the curvature invariants, such as R μνρσ R μνρσ and R μνρσ R ρσ λτ R μν λτ are constant, being equal to zero. ...
... become timelike and CTCs are formed. This is identical to the manner in which CTCs are formed in Ref. [19]. Moreover, these curves evolve from an spacelike t = const. ...
We present a cylindrically symmetric space-time, representing pure radiation as that of matterenergy that satisfies the energy conditions. We show that the space-time admits circular closed timelike geodesics (CTGs) by solving geodesic equations and other techniques. The space-time is regular everywhere, free from curvature singularities and has nondiverging, shear-free null geodesic congruences. The stability of CTGs under linear perturbations is studied and they are found to be linearly stable. Additionally, a modification of the space-time is studied and it is shown that closed timelike curves appear after a certain instant of time, i.e., a time-machine space-time.
... In literature, there are a number of vacuum and nonvacuum 2 Advances in High Energy Physics spacetimes admitting CTCs which have been constructed without naked singularity. Small samples of these are in [21][22][23][24][25][26][27][28][29][30][31][32][33]. Recently, a vacuum spacetime with a naked curvature singularity admitting CTCs appeared in [34] which may represent a Cosmic Time Machine. ...
... Recently, a vacuum spacetime with a naked curvature singularity admitting CTCs appeared in [34] which may represent a Cosmic Time Machine. One way of classifying causality violating spacetimes would be to categorize the metrics either as eternal time machines in which CTCs always exist (in this class would be [21,22]) or as time machine spacetimes in which CTCs appear after a certain instant of time (in this class would be [25,26]). However, many of the spacetimes admitting CTC suffer from one or more severe physical problems. ...
... These curves are null (or null geodesics) at = 0 = 0, spacelike throughout = 0 < 0, but become timelike for = 0 > 0, which indicates the presence of CTCs. Hence, CTCs form at a definite instant of time satisfying = 0 > 0. The above analysis is valid provided that the CTCs evolve from an initially spacelike = const hypersurface [26]. A hypersurface = const is spacelike provided < 0 for < 0, timelike provided > 0 for > 0, and null hypersurface provided = 0 for = 0. From metric (23), we found that ...
We present a cylindrically symmetric, Petrov type D, nonexpanding, shear free and vorticity free solution of Einstein's field equations. The spacetime is asymptotically flat radially and regular everywhere except on the symmetry axis where it possesses a naked curvature singularity. The energy momentum tensor of the spacetime is that for an anisotropic fluid which satisfies the different energy conditions. This spacetime is used to generate a rotating spacetime which admits closed timelike curves and may represent a Cosmic Time Machine.
... A small sample of these would be the ones discussed in Refs. [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. One way of classifying such causality violating spacetimes would be to categorize the metrics as either eternal time machine in which CTC always exist (for examples Refs. ...
... In the latter category would be the one discussed in Refs. [11][12][13][14]. ...
... Among the time machine spacetimes, we mention two: the first being Ori's compact core [13] which is represented by a vacuum metric, locally isometric to plane-waves spacetime and second one is the Misner space [23] in 2D spacetime. The Misner space is essentially a twodimensional metric (hence flat) with peculiar identifications. ...
Here we present a cyclicly symmetric non-vacuum spacetime, admitting closed timelike curves (CTCs) which appear after a certain instant of time, i.e., a time-machine spacetime. The spacetime is asymptotically flat, free-from curvature singularities and a four-dimensional extension of the Misner space in curved spacetime. The spacetime is of type II in the Petrov classification scheme and the matter field pure radiation satisfy the energy condition.
... It has been shown that Hawking's no go theorem on the formation of timelike curves can be circumvented either by relaxing the assumption on the compact generation of the horizon or by admitting violation of the null energy condition, see [6,8,[25][26][27][28] (we shall say more on this in Sect. 3). ...
... In this sense the term 'space region' used in the previous paragraph is appropriate. Nevertheless, Amos Ori in a series of papers [27,28] has criticized the previous argument maintaining that the assumption of local time machine creation would have to be expressed by the following concept, which he terms compact construction. Here S 0 represents the region were the actions of the advanced civilization leading to the formation of closed timelike curves took place. ...
A sufficiently general definition for the future and past boundaries of the chronology violating region is given. In comparison to previous studies, this work does not assume that the complement of the chronology violating set is globally hyperbolic. The boundary of the chronology violating set is studied and several propositions are obtained which confirm the reasonability of the definition. Some singularity theorems related to chronology violation are considered. Among the other results we prove that compactly generated horizons are compactly constructed.
... The violation of causality from exact solutions of general relativity is not limited to Gödel or Gödel-type solutions. Other models also demonstrate this phenomenon, such as Kerr and Kerr-Newman black holes [4,5], Lanczos-van Stockum model [6,7], cosmic strings [8], Tipler rotating cylinder [9], Bonnor and Steadman spacetime [10], the Ori time-machine metric [11], and several others detailed in [12][13][14][15]. Another important solution to highlight here is the one proposed by Gott and Li [16]. ...
... This Misner-like space contains CTCs, with regions of CTCs separated from non-CTC regions by chronology horizons. Following this, Ori developed a four-dimensional generalization of Misner space in curved space-time, which is also a vacuum solution to the field equations of General Relativity [11]. Subsequently, a number of four-dimensional curved space-times, representing generalizations of the twodimensional Misner space, have been presented in the literature (see, for example, [12][13][14][15] and related references therein). ...
In this paper, our objective is to explore a time-machine space-time formulated in general relativity, as introduced by Li (Phys. Rev. D 59, 084016 (1999)), within the context of modified gravity theories. We consider Ricci-inverse gravity of all Classes of models, i.e., (i) Class-I: f(ℛ, 𝒜) = (ℛ + κℛ² + β 𝒜), (ii) Class-II: f(ℛ, Aμν Aμν ) = (ℛ + κℛ² + γ Aμν Aμν ) model, and (iii) Class-III: f(ℛ, 𝒜, Aμν Aμν ) = (ℛ + κℛ² + β𝒜 + δ𝒜² + γ Aμν Aμν ) model, where Aμν is the anti-curvature tensor, the reciprocal of the Ricci tensor, Rμν , 𝒜 = gμν Aμν is its scalar, and β, κ, γ, δ are the coupling constants. Moreover, we consider f(ℛ) modified gravity theory and investigate the same time-machine space-time. In fact, we show that Li time-machine space-time serve as valid solutions both in Ricci-inverse and f(ℛ) modified gravity theories. Thus, both theory allows the formation of closed time-like curves analogue to general relativity, thereby representing a possible time-machine model in these gravity theories theoretically.
... Interests in CTCs were revived in the works of Thorne and Novikov in the context of the Morris-Thorne wormholes which are thought to occur in the presence of exotic matter (or in modified gravity) [3][4][5][6][7][8]. Despite the problematic nature of CTCs, the mathematical validity sparked considerable interest among gravitational theorists [9][10][11][12][13][14][15][16][17][18][19]. The concept of CTCs serves as a "gedanken experiment", compelling us to grapple with and reevaluate the foundational principles of general relativity and its modifications. ...
... A timelike path involving 'time travel' for microscopic particles would pose no such threat and should be allowed to contribute in the Feynman sum of transition amplitudes for these particles, as we will formulate shortly. The implications of foreknowledge on macroscopic animate objects constitute a subject of ongoing debate and deliberation, nevertheless [2,6,7,[9][10][11][12][13][14][15][16][17][18][19]. We venture one view: macroscopic objects might undergo a strong decoherence process upon passing through the wormhole throat. ...
... One can show that the closed curves {t, r, ϕ, z} → {t, r 0 , ϕ + 2 n π, z 0 } is time-like in the time-like region t > 0, spacelike in the region t < 0, and null at t = 0. Thus, the space-time admits CT Cs that are formed at an instant of time in time-like region t > 0 analogue to the Ori's time-machine space-time [32]. Now, the space-time (3.1) is a non-vacuum solution to the field equations, which we will discuss below. ...
... (iii) For the function H(r) = (c 3 r 2 + c 4 r), where c 4 > 0, the energy density of Vaidya radiation becomes ρ = 1 2 c 0 (c 1 + Λ c 2 c 4 ), which is a constant and positive provided c 1 > (−Λ) c 2 c 4 since Λ < 0. (iv) For the function H(r) = (c 3 r 2 − c 4 r), the energy density will be ρ = 1 2 c 0 (c 1 − Λ c 2 c 4 ) which is always positive since Λ < 0. Thus, space-time (3.1) with (3.2) represents a family of type-N space-times with Vaidya null dust or radiation as matter-energy content and a negative cosmological constant in general relativity theory. This family of type-N space-time admits closed time-like curves at certain instant of time t > 0, thus, act as time-machine model analogue to Ori's time-machine [32]. Next, we will study this space-time within the framework of RI gravity and analyze the results. ...
... Interests in CTCs were revived in the works of Thorne and Novikov in the context of the Morris-Thorne wormholes which are thought to occur in the presence of exotic matter (or in modified gravity) [3][4][5][6][7][8]. Despite the problematic nature of CTCs, the mathematical validity sparked considerable interest among gravitational theorists [9][10][11][12][13][14][15][16][17][18][19]. The concept of CTCs serves as a "gedanken experiment", compelling us to grapple with and reevaluate the foundational principles of general relativity and its modifications. ...
... A timelike path involving 'time travel' for microscopic particles would pose no such threat and should be allowed to contribute in the Feynman sum of transition amplitudes for these particles, as we will formulate shortly. The implications of foreknowledge on macroscopic animate objects constitute a subject of ongoing debate and deliberation, nevertheless [2,6,7,[9][10][11][12][13][14][15][16][17][18][19]. We venture one view: macroscopic objects might undergo a strong decoherence pro- cess upon passing through the wormhole throat. ...
This Letter aims to advance novel properties of a class of Closed Timelike Curves recently discovered in scalar-tensor gravity [Universe 9, 467 (2023)]. Therein, it was shown that a wormhole acts a gateway between two worlds, where the two asymptotically flat sheets in the Kruskal-Szekeres diagram are glued antipodally along directions -- time t and the polar and azimuth angles of the 2-sphere -- to form a wormhole throat. This contrasts with the standard embedding diagram which typically glues the sheets only along the and directions. Crucially, due to the `gluing' along the t direction, the wormhole becomes a portal connecting the two spacetime sheets with physical time flows, enabling the emergence of closed timelike loops which straddle the throat. We shall point out that this portal permits the possibility of backward propagation of information time. This feature is ubiquitous for wormholes in scalar-tensor theories. In addition, we formulate the Feynman sum for transition amplitudes of microscopic particles in the proximity of a wormhole throat in which we account for timelike paths that experience time reversal.
... The second class is spacetimes designed specifically for time travel. These include metrics due to Ori and co-workers [12][13][14], the wormhole spacetimes discussed earlier, the spacetime of Mallary et al. [15], and Tippett and Tsang's spacetime [11]. The third class is spacetimes that are designed to produce superluminal travel but allow time travel as a natural consequence. ...
... Hence, it must be −1 to pass the backwards-in-t test. Note that e and ℓ are not independent because of the constraint of circular motion; rather, e = −u t may be read off from Equation (12). 2 The radicand test is ℓ 2 ≥ ϵg ϕϕ , and the future-pointing test is ⟨u, u orient ⟩ < 0. ...
We investigate timelike and null geodesics within the rotating “time machine” spacetime proposed by Ralph, T.C.; et al. Phys. Rev. D 2020, 102, 124013. This is a rotating analogue of Alcubierre’s warp drive spacetime. We obtain geodesics that begin and end in the surrounding flat space region, yet achieve time travel relative to static observers there. This is a global property, as the geodesics remain locally future-pointing, as well as timelike or null.
... where ̺ 0 , z 0 are constants. Therefore, the space-time (4) reduces to 2D Misner-like space [42,43] given by ...
... Hence, the Misner space admits closed time-like at an instant of time, T = const = T 0 > 0. In our case, for the chosen space-time (4), the closed curves defined by {̺, φ, z} → {̺ 0 , φ + 2 n π, z 0 } is spacelike for τ < 0 and time-like for τ > 0. Hence, our fourdimensional space-time (4) admits closed time-like curves at an instant of time analogue to the two-dimensional Misner space. These closed time-like curves evolve from an initial spacelike hypersurface in a causally well-behaved manner [43]. This point is clear from the following discussion. ...
Our focus is on a specific type-N space–time that exhibits closed time-like curves in general relativity theory within the framework of Ricci-inverse gravity model. The matter-energy content is solely composed of a pure radiation field, and it adheres to the energy conditions while featuring a negative cosmological constant. One of the key findings in this investigation is the non-zero determinant of the Ricci tensor (R_{\mu\nu}), which implies the existence of an anti-curvature tensor (A^{\mu\nu}) and, as a consequence, an anti-curvature scalar (A). Furthermore, we establish that this type-N space–time serves as a solution within modified gravity theories via the Ricci-inverse model, which involves adjustments to the cosmological constant (\rho) of the radiation field expressed in terms of a coupling constant. As a result, our findings suggest that causality violations remain possible within the framework of this Ricci-inverse gravity model, alongside the predictions of general relativity.
... A true time machine space-time is the one in which CTCs evolve at a particular instant of time from an initial space-like hypersurface in a causally well-behaved manner satisfying all the energy conditions with known type of matter fields. In this category, the Ori time machine spacetime [37] is considered to be most remarkable. But the matter sources satisfying all the energy conditions are of unknown type in this space-time. ...
... We check whether the CTCs evolve from an initially space-like t = constant hypersurface (and thus t is a time coordinate). This is determined by calculating the norm of the vector ∇ μ t [37] (or alternately from the value of g tt in the inverse metric tensor g μν ). A hypersurface t = constant is space-like when g tt < 0 at t < 0, time-like when g tt > 0 for t > 0, and null g tt = 0 for t = 0. ...
In this paper, we present a type D, nonvanishing cosmological constant, vacuum solution of Einstein’s field equations, extension of an axially symmetric, asymptotically flat vacuum metric with a curvature singularity. The space-time admits closed time-like curves (CTCs) that appear after a certain instant of time from an initial space-like hypersurface, indicating it represents a time-machine space-time. We wish to discuss the physical properties and show that this solution can be interpreted as gravitational waves of Coulomb-type propagate on anti-de Sitter space backgrounds. Our treatment focuses on the analysis of the equation of geodesic deviations.
... In turn such schemes have been criticised on various grounds [3,4]. So far the argument appears undecided [5][6][7][8]. In the end it seems all such discussions require assumptions to be made about the nature of the so far undiscovered complete theory of quantum gravity. ...
... Whilst one could, in principle, transform between reference frames in order to patch together the desired evolutions, this is challenging in practice. Other solutions are similar, with various exotic spacetimes allowing for CTCs in principle [10] but not containing them explicitly, or not allowing an explicit connection between the CTC and chronology preserving objects in flat space, but rather requiring a patching together of different reference frames [7]. Exceptions are recent ad hoc metrics such as in Ref. [9]. ...
We present a global metric describing closed timelike curves embedded in Minkowski spacetime. Physically, the metric represents an Alcubierre warp drive on a rotating platform. The physical realizability of such a metric is uncertain due to the exotic matter required to produce it. Never-the-less we suggest that this metric will have applications in more rigorously studying the behavior of quantum fields interacting with closed timelike curves.
... There are time-like curves provided ds 2 < 0 for t = t 0 > 0, spacelike provided ds 2 > 0 for t = t 0 < 0, and null curve ds 2 = 0 for t = t 0 = 0. Therefore the closed curves defined by (t, r, ψ, z) ∼ (t 0 , r 0 , ψ + ψ 0 , z 0 ) being time-like in the region t = t 0 > 0, formed closed time-like curves. Noted the Gott's time-machine space-time generated closed timelike curves by imposing one of the coordinate ψ is periodic identifying ψ ∼ ψ + ψ 0 with period ψ 0 < 2 π [29] (see also [30,31]). These time-like closed curves evolve from an initial spacelike t = const < 0 hypersurface [30,31]. ...
... Noted the Gott's time-machine space-time generated closed timelike curves by imposing one of the coordinate ψ is periodic identifying ψ ∼ ψ + ψ 0 with period ψ 0 < 2 π [29] (see also [30,31]). These time-like closed curves evolve from an initial spacelike t = const < 0 hypersurface [30,31]. We find from metric (7) that the metric component g 00 is given by ...
An anti-de Sitter background four-dimensional type N solution of the Einstein’s field equations, is presented. The matter-energy content pure radiation field satisfies the null energy condition (NEC), and the metric is free-from curvature divergence. In addition, the metric admits a non-expanding, non-twisting and shear-free geodesic null congruence which is not covariantly constant. The space-time admits closed time-like curves which appear after a certain instant of time in a causally well-behaved manner. Finally, the physical interpretation of the solution, based on the study of the equation of the geodesics deviation, is analyzed.
... In these cases, the metric is given and the stress-energy tensor is derived a posteriori from the righthand side of the Einstein equations; the undesired, exotic features emerging from this construction (typically, the violation of the standard energy conditions) are regarded as secondary issues. Probably, the most influential papers in this class are those of Ori and Soen [30,31,32,34]. In particular, [32, 34] present a time machine with a toric spatial core, surrounded by a region where the spacetime metric is conformally flat. ...
... Dealing with the above mentioned paradoxes and problems is not among our purposes; here, we just propose to enrich the second class of spacetimes with time travels, introducing a new model (which, as expected, violates the energy conditions). In setting up this model, we were mainly stimulated by the paper of Tippet-Tsang [39]; however, later on we realized that our construction has a closer contact with the model of Ori-Soen [32,34]. Our model is topologically trivial, possesses no curvature singularity, and is both space and time orientable; it consists of a toric "time machine", which contains CTCs and is surrounded by flat Minkowski space. ...
Inspired by some recent works of Tippett and Tsang, and of Price et al., we present a new spacetime model containing closed timelike curves (CTCs). This model is obtained postulating an ad hoc Lorentzian metric on , which differs from the Minkowski metric only inside a spacetime region bounded by two concentric tori. The resulting spacetime is topologically trivial, free of curvature singularities and is both time and space orientable; besides, the inner region enclosed by the smaller torus is flat and displays geodesic CTCs. Our model shares some similarities with the time machine of Ori and Soen but it has the advantage of a higher symmetry in the metric, allowing for the explicit computation of a class of geodesics. The most remarkable feature emerging from this computation is the presence of future-oriented timelike geodesics starting from a point in the outer Minkowskian region, moving to the inner spacetime region with CTCs, and then returning to the initial spatial position at an earlier time; this means that time travel to the past can be performed by free fall across our time machine. The amount of time travelled into the past is determined quantitatively; this amount can be made arbitrarily large keeping non-large the proper duration of the travel. An important drawback of the model is the violation of the classical energy conditions, a common feature of most time machines. Other problems emerge from our computations of the required (negative) energy densities and tidal forces; these are found to be small on a human scale only if the time machine has an astronomical size.
... These curves are null at = 0 = 0 and spacelike throughout = 0 < 0 but become timelike for = 0 > 0, which indicates the presence of closed timelike curves (CTCs). Hence CTCs form at a definite instant of time satisfying = 0 > 0. The above analysis is valid provided that the CTCs evolve from an initial spacelike = hypersurface (thus is a time coordinate) [63]. This can be determined by calculating the norm of the vector ∇ (or by determining the sign of in the metric tensor ] [63]). ...
... Hence CTCs form at a definite instant of time satisfying = 0 > 0. The above analysis is valid provided that the CTCs evolve from an initial spacelike = hypersurface (thus is a time coordinate) [63]. This can be determined by calculating the norm of the vector ∇ (or by determining the sign of in the metric tensor ] [63]). A hypersurface = is spacelike provided < 0 for < 0, timelike provided > 0 for > 0, and null hypersurface provided = 0 for = 0. ...
In this article, we present a gravitational collapse null dust solution of the Einstein field equations. The spacetime is regular everywhere except on the symmetry axis where it possesses a naked curvature singularity, and admits one parameter isometry group, a generator of axial symmetry along the cylinder which has closed orbits. The space- time admits closed timelike curves (CTCs) which develop at some particular moment in a causally well-behaved manner and may represent a Cosmic Time Machine. The radial geodesics near to the singularity, and the gravitational Lensing (GL) will be discussed. The physical interpretation of this solution, based on the study of the equation of the geodesic deviation, will be presented. It was demonstrated that, this solution depends on the local gravitational field consisting of two components with amplitude , .
... CTCs have also 2 Advances in High Energy Physics been studied in the context of Brans-Dicke gravity [19], where rigid rotation of matter has been shown to generate these closed curves. Ori and coworkers [20][21][22][23] have obtained solutions with CTCs which appear after a certain instant, thereby raising the possibility of constructing a workable time machine at least in theory. We place space-time with CTCs (or CTGs or CNGs) into two categories. ...
... Hence, the space-time evolves from a partial Cauchy surface (i.e., Cauchy spacelike hypersurface) into a null hypersurface that is causally well-behaved up to a moment, that is, a null hypersurface = 0, and the formation of CTCs takes place from causally well-behaved initial conditions. This type of CTCs is different from those in the Gödel universe [34], where the CTCs preexist, and similar to those in [21], where they form at some moment. The CTCs of the space-time discussed in this paper are analogous to those formed in the Misner space [33]. ...
We present an axially symmetric, asymptotically flat empty space solution of the Einstein field equations containing a naked singularity. The spacetime is regular everywhere except on the symmetry axis where it possess a true curvature singularity. The spacetime is of type D in the Petrov classification scheme and is locally isometric to the metrics of case IV in the Kinnersley classification of type D vacuum metrics. Additionally, the spacetime also shows the evolution of closed timelike curves (CTCs) from an initial hypersurface free from CTCs.
... Strictly speaking, CTCs are worldlines that are closed in both space and time, meaning that a particle that enters one when t = 0 also exits it when t = 0 at the same spatial point. Metrics specifically designed to allow CTCs have been found since Gödel, with the most famous example being the traversable wormhole [4], among many others [5][6][7][8][9]. ...
We present two modifications to the rotating Alcubierre metric [1], which was shown to permit closed timelike curves (CTCs). We find that if the rotation rate of the spacetime is made spatially dependent, in certain cases there exist simple approximate timelike geodesics that are also CTCs, provided that the velocity of the warp bubble varies slowly. The second modification is essentially the original Alcubierre metric [ 2 ] with a periodic boundary, resulting in a cylindrical spacetime which can also be related to the rotating Alcubierre metric. Furthermore, this spacetime contains simple exact timelike geodesics that are also CTCs. In both modifications, the CTC geodesics that we have found allow for a simple model of a particle interacting with a CTC for a finite proper time interval, entering and exiting in chronology-respecting space. Given the simplicity of both of these exotic spacetimes, despite their questionable physical realisability, we suggest that they may be useful in studies of quantum fields near CTCs.
... Another extremely interesting feature of traversable wormholes is that they may be hypothetically manipulated to induce closed timelike curves (CTCs) [6][7][8][9]. In fact, General Relativity is contaminated with non-trivial geometries, which generate CTCs [10][11][12][13][14][15][16][17][18], which allow time travel, in the sense that an observer that travels on a trajectory in spacetime along this curve and may return to an event before his departure [19]. ...
The recently obtained special Buchdahl-inspired metric Phys. Rev. D 107, 104008 (2023) describes asymptotically flat spacetimes in pure Ricci-squared gravity. The metric depends on a new (Buchdahl) parameter k˜ of higher-derivative characteristic, and reduces to the Schwarzschild metric, for k˜=0. For the case k˜∈(−1,0), it was shown that it describes a traversable Morris–Thorne–Buchdahl (MTB) wormhole Eur. Phys. J. C 83, 626 (2023), where the weak energy condition is formally violated. In this paper, we briefly review the special Buchdahl-inspired metric, with focuses on the construction of the Kruskal–Szekeres (KS) diagram and the situation for a wormhole to emerge. Interestingly, the MTB wormhole structure appears to permit the formation of closed timelike curves (CTCs). More specifically, a CTC straddles the throat, comprising of two segments positioned in opposite quadrants of the KS diagram. The closed timelike loop thus passes through the wormhole throat twice, causing two reversals in the time direction experienced by the (timelike) traveller on the CTC. The key to constructing a CTC lies in identifying any given pair of antipodal points (T,X) and (−T,−X) on the wormhole throat in the KS diagram as corresponding to the same spacetime event. It is interesting to note that the Campanelli–Lousto metric in Brans–Dicke gravity is known to support two-way traversable wormholes, and the formation of the CTCs presented herein is equally applicable to the Campanelli–Lousto solution.
... violate the energy conditions for the given range of γ . All physical quantities given in (8) are finite at r = r 0 and vanishes for r → ∞. ...
In this paper, we explore the quantum system of non-relativistic particles in a unique scenario: a circularly symmetric and static three-dimensional wormhole space-time accompanied by cosmic strings. We focus on a specific case where the redshift function [Formula: see text] to be zero and defining the shape function as [Formula: see text]. After establishing this background space-time, we investigate the behavior of a harmonic oscillator within the same wormhole context. By doing so, we observe the effects of the cosmic string and wormhole throat radius on the eigenvalue solution of the oscillator's eigenvalue problem. The primary finding is that these cosmic features lead to modifications in the energy spectrum and wave functions of the system, breaking the degeneracy of energy levels that would typically be present in a more conventional setting. As a particular case, we present the specific energy level [Formula: see text] and the corresponding wave function [Formula: see text], which are associated with the ground state of the quantum system. These results highlight the fascinating and unique properties of the harmonic oscillator in the background of a circularly symmetric, static wormhole space-time with cosmic strings.
... Although it is an open question whether CTCs are possible in our universe [18][19][20][21][22], considering dynamics beyond the ordinary temporal view is relevant to other research areas as well. In a theory that combines quantum physics with general relativity, it is expected that spacetime loses its classical properties [23,24], possibly leading to indefinite causal structures [25][26][27]. ...
The theory of general relativity predicts the existence of closed time-like curves (CTCs), which theoretically would allow an observer to travel back in time and interact with their past self. This raises the question of whether this could create a grandfather paradox, in which the observer interacts in such a way to prevent their own time travel. Previous research has proposed a framework for deterministic, reversible, dynamics compatible with non-trivial time travel, where observers in distinct regions of spacetime can perform arbitrary local operations with no contradiction arising. However, only scenarios with up to three regions have been fully characterised, revealing only one type of process where the observers can verify to both be in the past and future of each other. Here we extend this characterisation to an arbitrary number of regions and find that there exist several inequivalent processes that can only arise due to non-trivial time travel. This supports the view that complex dynamics is possible in the presence of CTCs, compatible with free choice of local operations and free of inconsistencies.
... and the coordinate ψ is chosen to be cyclic (and/or periodic locally), that is, each ψ is identified with ψ + ψ 0 for a certain parameter ψ 0 > 0, or 0 ≤ ψ ≤ ψ 0 [53] (see also [21,22]). For the space-time study here, we have assumed that X → 0 as r → 0, where X = |η μ η ν g μν | = |g ψψ | and ∂ ψ is a spacelike Killing vector [54]. ...
A family of type N exact solution of the Einstein's field equations, regular everywhere except on the symmetry axis where it possesses a naked curvature singularity, is present. The stress-energy tensor is of the anisotropic fluid coupled with pure radiation field satisfy the different energy conditions and the physical parameters diverge r→0. The space-time admitting a non-expanding, non-twisting, and shear-free geodesic null congruence and belongs to a special class of type N Kundt metrics. The space-time is geodesically complete along the radiation direction in the constant z-planes and exhibits geometrically different properties from the known pp-waves. The present family of solution admits closed time-like curves (CTC) which appear after a certain instant of time and the space-time is a four-dimensional generalization of the Misner space metric in curved space-time.
... Although it is an open question whether CTCs are possible in our universe [18][19][20][21][22], considering dynamics beyond the ordinary temporal view is relevant to other research areas as well. In a theory that combines quantum physics with general relativity, it is expected that spacetime loses its classical properties [23,24], possibly leading to indefinite causal structures [25][26][27]. ...
The theory of general relativity predicts the existence of closed time-like curves (CTCs), which theoretically would allow an observer to travel back in time and interact with their past self. This raises the question of whether this could create a grandfather paradox, in which the observer interacts in such a way to prevent their own time travel. Previous research has proposed a framework for deterministic, reversible, dynamics in the presence of CTCs, where observers in distinct regions of spacetime can perform arbitrary local operations with no contradiction arising. However, only scenarios with up to three regions have been fully characterised, revealing only one type of process where the observers can verify to both be in the past and future of each other. Here we extend this characterisation to an arbitrary number of regions and find that there exist several inequivalent processes that can only arise in the presence of CTCs. This supports the view that complex dynamics is possible in the presence of CTCs, compatible with free choice of local operations and free of inconsistencies.
... Specifically designed Lorentzian geometries, describing CTCs at the price of violating the usual energy conditions. Examples of such geometries are those analysed by Ori and Soen [5,32,33,35,37], and by Tippett and Tsang [42]. Third class. ...
This work is essentially a review of a new spacetime model with closed causal curves, recently presented in another paper. The spacetime at issue is topologically trivial, free of curvature singularities, and even time and space orientable. Besides summarizing previous results on causal geodesics, tidal accelerations, and violations of the energy conditions, here redshift/blueshift effects and the Hawking–Ellis classification of the stress–energy tensor are examined.
... Specifically designed Lorentzian geometries, describing CTCs at the price of violating the usual energy conditions. Examples of such geometries are those analysed by Ori and Soen [5,32,33,35,37], and by Tippett and Tsang [42]. Third class. ...
This work is essentially a review of a new spacetime model with closed causal curves, recently presented in another paper (Class. Quantum Grav. \textbf{35}(16) (2018), 165003). The spacetime at issue is topologically trivial, free of curvature singularities, and even time and space orientable. Besides summarizing previous results on causal geodesics, tidal accelerations and violations of the energy conditions, here redshift/blueshift effects and the Hawking-Ellis classification of the stress-energy tensor are examined.
... Second one being a time machine spacetime, where CTC appears after a certain instant of time in a causally well behaved manner (e.g. [7,8,11]). However, some known solutions of the field equations with CTC are considered unphysical because of their unrealistic or exotic matter-energy source which are violate the weak energy condition (WEC). ...
We present a topologically trivial nonvacuum solution of the Einstein’s field equations in curved spacetime with stress-energy tensor Type I fluid, satisfying the energy conditions. The metric admits closed timelike curves which appear after a certain instant of time, and the spacetime is a four-dimensional generalization of flat Misner space in curved spacetime.
... x 0 , z 0 are constants. Here the y coordinate is chosen periodic, that is each y identified y + y 0 for a certain parameter y 0 > 0 (see [21]). From the metric (1), we get ...
We present a topologically trivial, non-vacuum solution of the Einstein's field equations in four-dimensions, which is regular everywhere. The metric admits circular closed timelike curves, which appear beyond the null curve, and these timelike curves are linearly stable under linear perturbations. Additionally, the spacetime admits null geodesics curve which are not closed, and the metric is of type D in the Petrov classification scheme. The stress-energy tensor anisotropic fluid satisfy the different energy conditions and a generalization of Equation-of-State parameter of perfect fluid . The metric admits a twisting, shearfree, non-exapnding timelike geodesic congruence. Finally, the physical interpretation of this solution, based on the study of the equation of the geodesics deviation, will be presented.
... In 2005, Ori presented in [15] a time machine spacetime which develops closed timelike curves inside a vacuum core part surrounded by a matter field. We focus here on the vacuum core. ...
... One of the most baffling aspects of general relativity is that certain solutions to the Einstein equations contain closed time-like curves (CTCs) [1][2][3][4][5][6][7], where an event can be both in its own future and past. Although it is not known whether CTCs are actually possible in our universe [8][9][10][11][12][13], their mere logical possibility poses the challenge to understand what type of dynamics could be expected in their presence. ...
General relativity predicts the existence of closed time-like curves, along which a material object could travel back in time and interact with its past self. The natural question is whether this possibility leads to inconsistencies: Could the object interact in such a way to prevent its own time travel? If this is the case, self-consistency should forbid certain initial conditions from ever happening, a possibility at odds with the local nature of dynamical laws. Here we consider the most general deterministic dynamics connecting classical degrees of freedom defined on a set of bounded space-time regions, requiring that it is compatible with arbitrary operations performed in the local regions. We find that any such dynamics can be realised through reversible interactions. We further find that consistency with local operations is compatible with non-trivial time travel: Three parties can interact in such a way to be all both in the future and in the past of each other, while being free to perform arbitrary local operations. We briefly discuss the quantum extension of the formalism.
... In this paper, we attempt to write down an axially symmetric metric, locally isometric to the anti-de Sitter space, where closed timelike curves (CTCs) appear after a certain instant -a time-machine spacetime. In this context we distinguish between eternal time-machine spacetimes, such as that of Gödel [2], or the van Stockum metric [3], where CTCs always exist, and true time-machine spacetimes, typically the one discussed in [4], where CTCs appear after a certain instant. Another time-machine spacetime of great interest is the Misner space [5]. ...
We construct an axially symmetric spacetime admitting, after a certain instant, closed timelike curves (CTCs) indicating that it is a time-machine spacetime. The spacetime, which is locally anti-de Sitter, is a four-dimensional extension of the Misner space with identical causality-violating properties. In this spacetime, CTCs evolve from a casually well-behaved initial hypersurface.
... There have been attempts to circumvent the classical no-go theorems on the formation of closed timelike curves either by relaxing the assumption on the compact generation of the horizon or by admitting violation of the null energy condition, see [6,8,[25][26][27]. The reader is warned that on this topic some imprecise or misleading statements can be repeatedly found in the literature; the most relevant example is given by the claim [9] that closed causal curves on compact Cauchy horizons (fountains) would be generic. ...
A sufficiently general definition for the future and past boundaries of the chronology violating region is given. In comparison to previous studies, this work does not assume that the complement of the chronology violating set is globally hyperbolic. The boundary of the chronology violating set is studied and several propositions are obtained which confirm the reasonability of the definition. Some singularity theorems related to chronology violation are considered. Among the other results we prove that compactly generated horizons are compactly constructed.
This research studies primarily centers around a specific vacuum solution of the Einstein’s field equations with a negative cosmological constant, known as a type-D vacuum solution in Anti-de Sitter (AdS) background. The key finding of this study is that the determinant of the Ricci tensor (R_{\mu\nu) possesses a nonzero value. This observation ensures that this vacuum spacetime can be investigated within the framework of Ricci-inverse gravity as well. In fact, it has demonstrated that this type-D vacuum solution serves as an exact solution in Ricci inverse gravity in the background of AdS space. Notably, the negative cosmological constant undergoes modification by the coupling constant within this context. Moreover, we show that this gravity theory permits the formation of closed time-like curves at a specific moment in time analogue to the general relativity case. Consequently, this theory introduces an intriguing time-machine model within the domain of Ricci-inverse gravity theory.
This research study is primarily focus on investigating an exact vacuum solution characterized by a negative cosmological constant, , within the framework of a modified gravity theory. Our specific focus is on a novel theory known as Ricci-inverse gravity. This vacuum solution is representative of an Anti-de Sitter (AdS) space, which, in general relativity, serves as a space–time exhibiting properties akin to a time-machine. Significantly, we demonstrate that this AdS space–time is a valid solution within the Ricci-inverse gravity theory as well. This outcome implies that Ricci-inverse gravity also allows the formation of closed time-like curves, analogous to what is observed in the context of general relativity. The investigation into Ricci-inverse gravity involves introducing the function f(R, A) into the Lagrangian of the system. We consider three distinct scenarios to explore the implications of different choices for the function. The first scenario involves selecting , the second one adopts , and the third scenario employs the function , where and serve as the coupling constants. In each of these scenarios, we derive the modified field equations and subsequently solve them under the assumption of vacuum as the matter-energy content. Notably, our results demonstrate that the cosmological constant undergoes modification based on the coupling constants in each case. Thus, in Ricci-inverse gravity, we establish that this modified Anti-de Sitter (AdS) space–time also serves as a model for a time-machine.
Over the past two decades, substantial efforts have been made to understand the way in which physics enforces the ordinary topology and causal structure that we observe on observed scales – from subnuclear to cosmological. We review the status of topological censorship and the topology of event horizons; chronology protection in classical and semiclassical gravity; and related progress in establishing quantum energy inequalities.
We present a general time machine spacetime in which closed timelike curves are formed at some particular instant of time from an initial spacelike hypersurface from well-behaved initial conditions. We show that a known result in the literature can be obtained as a particular case of the general spacetime. Finally, from the generalized spacetime, we obtain a vacuum solution exhibiting causality violation. The physical properties of this solution are explored in some detail. The formation of closed timelike curves is analogous to that of the two-dimensional Misner space metric.
We present a conformally flat, cylindrically symmetric solution of the Einstein field equations which acts as a time machine, in that the solution predicts the formation of closed timelike curves. The formation of these causality-violating curves is analogous to that of the Misner space in two dimensions.
We present a Petrov type II general spacetime which violates causality in the sense that it allows for the formation of closed timelike curves that appear after a definite instant of time. The metric, which is axially symmetric, admits an expansion-free, twist-free and shear-free null geodesic congruence. From the general metric, we obtain two particular type II metrics. One is a vacuum solution while the other represents a Ricci flat solution with a negative cosmological constant.
We present a global metric describing closed timelike curves embedded in Minkowski spacetime. Physically, the metric represents an Alcubierre warp drive on a rotating platform. The physical realizability of such a metric is uncertain due to the exotic matter required to produce it. Nevertheless we suggest that this metric will have applications in more rigorously studying the behavior of quantum fields interacting with closed timelike curves.
An axially symmetric nonvacuum solution of the Einstein field equations, regular everywhere
and free from curvature divergence is presented. The matter-energy content is a the pure radiation field
satisfying the energy conditions, and the metric is of type N in the Petrov classification scheme. The space-time develops circular closed timelike curves everywhere outside a finite region of space i.e., beyond a
null curve. Furthermore, the physical interpretation of the solution based on the study of the equations
of geodesic deviation is presented. Finally, the von Zeipel cylinders with respect to the Zero Angular
Momentum Observers (ZAMOs) is discussed. In addition, circular null and timelike geodesic of space-time are also presented
A family of type N exact solutions of the Einstein’s field equations with non-zero cosmological constant admitting closed time-like curves, which appear after a certain instant of time, is presented. The space-time is free from curvature divergence, admits a non-expanding, non-twisting, and shear-free geodesics null congruence, and belongs to a special class of type N Kundt metrics. The space-time exhibits geometrically different properties from the known plane-fronted gravitational waves with parallel rays (pp-waves) or plane waves which violate the causality condition. This family of solutions is a four-dimensional generalization of the two-dimensional Misner space metric in curved space-time.
In this paper, we construct a new time-periodic solution and a new time-machine solution according to the general form of the new family of solutions to the Einstein’s vacuum field equations with cosmological constant. The time-periodic solutions have some relations with the loop universe. The time-machine solutions are spacetime configurations including closed time-like curves (CTCs), and allowing physical observers to return to their own past. This problem has been of concern a lot recently. We expect the new solutions obtained in this paper can be applied in modern cosmology and general relativity.
The possibility of time travel is explored within the framework of first order gravity theory in vacuum. We present explicit solutions (to the field equations), whose geodesics allow the proper clock to run backwards. These four-geometries contain degenerate as well as nondegenerate tetrad fields that are sewn together continuously over different regions of the spacetime. Being devoid of matter and also of torsion by construction, these solutions have no exotic field content.
Solutions of “informational paradox” for black holes and “grandfather paradox” for time travel “wormholes” are investigated. The chapter is dealing with the analysis of quantum gravitation theory from the point of view of thermodynamic time arrow. Within this framework black stars, Penrose’s project of new quantum gravitation theory, and anthropic principle are considered.
We see that exact equations of quantum and classical mechanics describe ideal dynamics which is reversible and leads to Poincare’s returns. Real equations of physics describing observable dynamics, e.g., hydrodynamic equations of viscous fluid, are irreversible and exclude Poincare’s returns to the initial state. Besides, these equations describe systems in terms of macroparameters or phase distribution functions of microparameters. For many systems introduction of macroparameters that allow exhaustive describing of dynamics of the system is impossible. Their dynamics becomes unpredictable in principle, sometimes even unpredictable by the probabilistic way. We will refer to dynamics describing such system as unpredictable dynamics. Dynamics of unpredictable systems is not described and not predicted by scientific methods. Thus, the science itself puts boundaries for its applicability. But such systems can intuitively “understand itself” and “predict” the behavior “of its own” or even “communicate with each other” at intuitive level. Perspective of the future of artificial intellect (AI) is considered. It is shown that AI development in the future will be closer rather to art than to science. Complex dissipative systems whose behavior cannot be understood completely in principle will be the basis of AI. Nevertheless, it will not be a barrier for their practical use.
In a recent work, authors prove a yet another no-go theorem that forbids the existence of a universal probabilistic quantum protocol producing a superposition of two unknown quantum states. In this short note, we show that in the presence of closed time like curves, one can indeed create superposition of unknown quantum states and evade the no-go result.
It has been shown elsewhere that in a classical spacetime with multiply connected space slices (wormhole spacetime), closed timelike curves can form generically. The boundary between an initial region of spacetime without closed timelike curves and a later region with them is a Cauchy horizon which can be stable against small classical perturbations. This paper investigates stability against vacuum fluctuations of a quantized field, by calculating the field’s renormalized stress-energy tensor near the Cauchy horizon. The calculation is restricted to a massless, conformally coupled scalar field, but it is argued that the results will be the same to within factors of order unity for other noninteracting quantum fields. The calculation is given in order of magnitude for any spacetime with closed timelike curves, and then a detailed calculation is given for a specific example of such a spacetime: one with a traversable wormhole whose mouths create closed timelike curves by their relative motions. The renormalized stress-energy tensor is found to diverge as one approaches the Cauchy horizon.
However, the divergence is extremely weak: so weak, that as seen in the rest frame of one of the wormhole mouths the vacuum polarization’s gravity distorts the spacetime metric near the mouth by only δgμνVP∼(lP/D)(lP/Δt), where Δt is the proper time until one reaches the Cauchy horizon and D is the distance between the two mouths when the Cauchy horizon forms. For a macroscopic wormhole with D∼1 m, δgμνVP has only grown to lP/D∼10-35 when one is within a Planck length of the horizon. Since the very concept of classical spacetime is normally thought to fail, and be replaced by the quantum foam of quantum gravity on scales Δt≲lP, the authors are led to conjecture that the vacuum-polarization divergence gets cut off by quantum gravity upon reaching the tiny size lP/D, and spacetime remains macroscopically smooth and classical and develops closed timelike curves without difficulty. Hawking, in response to this, has conjectured that the spacetime near the Cauchy horizon remains classical until DΔt (which in a certain sense is frame invariant) gets as small as ∼lP2, and correspondingly until δgμνVP∼1, and that, as a result, the vacuum-polarization divergence will prevent the formation of closed timelike curves. These two conjectures are discussed and contrasted. The attempt to test them might produce insight into candidate theories of quantum gravity.
One of the most serious attempts to find a mechanism “protecting causality” was made in a number of papers beginning from [1, 2] where it is argued that vacuum fluctuations prevent the creation of a time machine by making the expectation value of the stress-energy tensor 〈T〉ren diverge at the Cauchy horizon. In the present talk I show that1.
There are spacetimes with causality violated and with 〈T〉ren, nevertheless, bounded in causal regions.
2.
There is actually no reason to expect that 〈T〉ren diverges in the general case.
We prove two theorems which concern difficulties in the formulation of the quantum theory of a linear scalar field on a spacetime, , with a compactly generated Cauchy horizon. These theorems demonstrate the breakdown of the theory at certain base points of the Cauchy horizon, which are defined as ‘past terminal accumulation points’ of the horizon generators. Thus, the theorems may be interpreted as giving support to Hawking's ‘Chronology Protection Conjecture’, according to which the laws of physics prevent one from manufacturing a ’time machine‘. Specifically, we prove:
Theorem 1.
There is no extension to
of the usual field algebra on the initial globally hyperbolic region which satisfies the condition of F-locality at any base point. In other words, any extension of the field algebra must, in any globally hyperbolic neighbourhood of any base point, differ from the algebra one would define on that neighbourhood according to the rules for globally hyperbolic spacetimes.
Theorem 2.
The two-point distribution for any Hadamard state defined on the initial globally hyperbolic region must (when extended to a distributional bisolution of the covariant Klein-Gordon equation on the full spacetime) be singular at every base point
x
in the sense that the difference between this two point distribution and a local Hadamard distribution cannot be given by a bounded function in any neighbourhood (in M × M) of (x,x).
In consequence of Theorem 2, quantities such as the renormalized expectation value of φ2 or of the stress-energy tensor are necessarily ill-defined or singular at any base point.
The proof of these theorems relies on the ‘Propagation of Singularities’ theorems of Duistermaat and Hörmander.
This is a brief survey of the current status of Stephen Hawking's ``chronology protection conjecture''. That is: ``Why does nature abhor a time machine?'' I'll discuss a few examples of spacetimes containing ``time machines'' (closed causal curves), the sorts of peculiarities that arise, and the reactions of the physics community. While pointing out other possibilities, this article concentrates on the possibility of ``chronology protection''. As Stephen puts it: ``It seems that there is a Chronology Protection Agency which prevents the appearance of closed timelike curves and so makes the universe safe for historians.''
It is shown how, within the framework of general relativity and without the introduction of wormholes, it is possible to modify a spacetime in a way that allows a spaceship to travel with an arbitrarily large speed. By a purely local expansion of spacetime behind the spaceship and an opposite contraction in front of it, motion faster than the speed of light as seen by observers outside the disturbed region is possible. The resulting distortion is reminiscent of the ``warp drive'' of science fiction. However, just as it happens with wormholes, exotic matter will be needed in order to generate a distortion of spacetime like the one discussed here. Comment: 10 pages, 1 figure. Not previously available in gr-qc
Preface; List of tables; Notation; 1. Introduction; Part I. General
Methods: 2. Differential geometry without a metric; 3. Some topics in
Riemannian geometry; 4. The Petrov classification; 5. Classification of
the Ricci tensor and the energy-movement tensor; 6. Vector fields; 7.
The Newman-Penrose and related formalisms; 8. Continuous groups of
transformations; isometry and homothety groups; 9. Invariants and the
characterization of geometrics; 10. Generation techniques; Part II.
Solutions with Groups of Motions: 11. Classification of solutions with
isometries or homotheties; 12. Homogeneous space-times; 13.
Hypersurface-homogeneous space-times; 14. Spatially-homogeneous perfect
fluid cosmologies; 15. Groups G3 on non-null orbits V2. Spherical and
plane symmetry; 16. Spherically-symmetric perfect fluid solutions; 17.
Groups G2 and G1 on non-null orbits; 18. Stationary gravitational
fields; 19. Stationary axisymmetric fields: basic concepts and field
equations; 20. Stationary axisymmetiric vacuum solutions; 21. Non-empty
stationary axisymmetric solutions; 22. Groups G2I on spacelike orbits:
cylindrical symmetry; 23. Inhomogeneous perfect fluid solutions with
symmetry; 24. Groups on null orbits. Plane waves; 25. Collision of plane
waves; Part III. Algebraically Special Solutions: 26. The various
classes of algebraically special solutions. Some algebraically general
solutions; 27. The line element for metrics with
κ=σ=0=R11=R14=R44, Θ+iω≠0; 28.
Robinson-Trautman solutions; 29. Twisting vacuum solutions; 30. Twisting
Einstein-Maxwell and pure radiation fields; 31. Non-diverging solutions
(Kundt's class); 32. Kerr-Schild metrics; 33. Algebraically special
perfect fluid solutions; Part IV. Special Methods: 34. Applications of
generation techniques to general relativity; 35. Special vector and
tensor fields; 36. Solutions with special subspaces; 37. Local isometric
embedding of four-dimensional Riemannian manifolds; Part V. Tables: 38.
The interconnections between the main classification schemes;
References; Index.
It is shown that a region containing closed timelike lines cannot evolve from regular initial data in a singularity-free asymptotically flat space-time. Furthermore, the causality assumption made in the black-hole uniqueness proofs is justified: It is demonstrated that no physically realistc nonsingular black hole can have a causality-violating exterior. (AIP)
A number of important theorems in General Relativity have required a causality assumption; for example, the Geroch topology change theorem, and most of the Hawking-Penrose-Geroch singularity theorems. It is shown in this paper that the causality condition can be replaced by weaker causality conditions, and in some cases removed altogether. In particular, (a) it is shown that if the Einstein equations (and the weak energy condition) hold on the "topology-changing" space-time considered by Geroch, then topology change cannot occur. No causality assumption is needed in the proof. Furthermore, it is shown that if topology change occurs within a finite region, then this change of topology must be accompanied by singularities. (b) It is shown that causality violation cannot prevent the Hawking-Penrose-Geroch singularities unless the causality violation begins "at infinity"-a region which is free of matter and gravitational radiation-and this seems very unlikely.
In 1936 van Stockum solved the Einstein equations Gmunu=-8piTmunu for the gravitational field of a rapidly rotating infinite cylinder. It is shown that such a field violates causality, in the sense that it allows a closed timelike line to connect any two events in spacetime. This suggests that a finite rotating cylinder would also act as a time machine.
Recently, Gott has provided a family of solutions of the Einstein equations describing pairs of parallel cosmic strings in motion. He has shown that if the strings' relative velocity is sufficiently high, there exist closed timelike curves (CTC's) in the spacetime. Here we show that if there are CTC's in such a solution, then every t=const hypersurface in the spacetime intersects CTC's. Therefore, these solutions do not contradict the chronology protection conjecture of Hawking.
It has been suggested that an advanced civilization might have the technology to warp spacetime so that closed timelike curves would appear, allowing travel into the past. This paper examines this possibility in the case that the causality violations appear in a finite region of spacetime without curvature singularities. There will be a Cauchy horizon that is compactly generated and that in general contains one or more closed null geodesics which will be incomplete. One can define geometrical quantities that measure the Lorentz boost and area increase on going round these closed null geodesics. If the causality violation developed from a noncompact initial surface, the averaged weak energy condition must be violated on the Cauchy horizon. This shows that one cannot create closed timelike curves with finite lengths of cosmic string. Even if violations of the weak energy condition are allowed by quantum theory, the expectation value of the energy-momentum tensor would get very large if timelike curves become almost closed. It seems the back reaction would prevent closed timelike curves from appearing. These results strongly support the chronology protection conjecture: The laws of physics do not allow the appearance of closed timelike curves.
We recently presented a model of a compactly generated and topologically trivial time-machine spacetime which satisfies the weak energy condition all the way up to the critical time slice (achronal hypersurface) in which a closed causal curve appears. Here we modify the above model so as to satisfy not only the weak energy condition, but also the dominant energy condition, up to that critical time slice.
It is argued that, if the laws of physics permit an advanced civilization to create and maintain a wormhole in space for interstellar travel, then that wormhole can be converted into a time machine with which causality might be violatable. Whether wormholes can be created and maintained entails deep, ill-understood issues about cosmic censorship, quantum gravity, and quantum field theory, including the question of whether field theory enforces an averaged version of the weak energy condition.
Exact solutions of Einstein's field equations are presented for the general case of two moving straight cosmic strings that do not intersect. The solutions for parallel cosmic strings moving in opposite directions show closed timelike curves (CTCs) that circle the two strings as they pass, allowing observers to visit their own past. Similar results occur for nonparallel strings, and for masses in (2+1)-dimensional spacetime. For finite string loops the possibility that black-hole formation may prevent the formation of CTCs is discussed.
We present a time-machine model in which closed timelike curves evolve, within a bounded region of space, from a well-behaved spacelike initial slice S; this slice (and the entire spacetime) is asymptotically flat and topologically trivial. In addition, this model satisfies the weak energy condition everywhere on S and up until and beyond the time slice (an achronal hypersurface) which displays the causality violation. We discuss the relation of this model to theorems by Tipler and Hawking which place constraints on time-machine solutions.
A space consisting of two rapidly moving cosmic strings has recently been constructed by Gott that contains closed timelike curves. The global structure of this space is analysed and is found that, away from the strings, the space is identical to a generalised Misner space. The vacuum expectation value of the energy momentum tensor for a conformally coupled scalar field is calculated on this generalised Misner space. It is found to diverge very weakly on the Chronology horizon, but more strongly on the polarised hypersurfaces. The divergence on the polarised hypersurfaces is strong enough that when the proper geodesic interval around any polarised hypersurface is of order the Planck length squared, the perturbation to the metric caused by the backreaction will be of order one. Thus we expect the structure of the space will be radically altered by the backreaction before quantum gravitational effects become important. This suggests that Hawking's `Chronology Protection Conjecture' holds for spaces with non-compactly generated Chronology horizon. Comment: 15 pages, plain TeX, 2 figures (not included), DAMTP-R92/35
- Y Soen
- A Ori
Y. Soen and A. Ori, Phys. Rev. D 54, 4858 (1996)
[12] A. Ori, in preparation [13] F
- S W Hawking
S. W. Hawking, Phys. Rev. D 46, 603 (1992)
[12] A. Ori, in preparation
[13] F. J. Tipler, Phys. Rev. Lett 37, 879 (1976);
- J D E Grant
J. D. E. Grant, Phys. Rev. D 47, 2388 (1993)
- F J Tipler
F. J. Tipler, Phys. Rev. D 9, 2203 (1974)
- M Alcubierre
M. Alcubierre, Class. Quant. Grav. 11, L73 (1994)
- A Ori
A. Ori, Phys. Rev. Lett. 71, 2517 (1993)
- A Ori
A. Ori, Phys. Rev. D 44, R2214 (1991)
- S W Kim
- K S Thorne
S. W. Kim and K. S. Thorne, Phys. Rev. D43, 3929 (1991)
- B S Kay
- M J Radzikowski
- R M Wald
B. S. Kay, M. J. Radzikowski, and R. M. Wald, Commun. Math. Phys. 183 533 (1997)
- S V Krasnikov
S. V. Krasnikov, Phys. Rev. D 54, 7322 (1996)