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# A schematic of the timelike observers confined to the interior and exterior of the bubble. Observers A (inside the bubble) and B (outside the bubble) experience the events in dramatically different ways. Arrows indicate the local arrow of time. Within the bubble, A will see the B’s events periodically evolve, and then reverse. Outside the bubble, observer B will see two versions of A emerge from the same location: one’s clock hands will turn clockwise, the other counterclockwise. The two versions of A will then accelerate towards one another and annihilate.

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There are many spacetime geometries in general relativity which contain
closed timelike curves. A layperson might say that retrograde time travel is
possible in such spacetimes. To date no one has discovered a spacetime geometry
which emulates what a layperson would describe as a time machine.
The purpose of this paper is to propose such a space-ti...

## Citations

... General relativity challenges this view. The Einstein equations, describing the relationship between spacetime geometry and mass-energy [1], have counterintuitive solutions containing closed time-like curves (CTCs) [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. An event on such a curve would be both in the future and in the past of itself, preventing an ordinary formulation of dynamics according to an "initial condition" problem. ...

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.

... General relativity challenges this view. The Einstein equations, describing the relationship between spacetime geometry and mass-energy [1], have counterintuitive solutions containing closed time like curves (CTCs) [2][3][4][5][6][7][8][9]. An event on such a curve would be both in the future and in the past of itself, preventing an ordinary formulation of dynamics according to an "initial condition" problem. ...

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.

... The premise that the working mechanisms of mindfulness (as one example of meditation practice) are multi-dimensional [4,35,36,37,38] is accruing, in addition to multi-levelled neural pathway approaches for complex mind states, such as unified non-dual compassion [22], and very subtle mind states during "emptiness insight practices" [33,39]. Non-linear statistics, such as Lyapunov stability theory and stochastic analysis [40], and nonlinear interdependence to _____________________________________________________________________________________ 1 TARDIS: Traversable Acausal Retrograde Domain in Space-Time [47] describing a "vehicle" or "bubble" that traverses the space-time manifold, based on the theory that rather than viewing the Universe as 3D + a 4 th dimension of Time, these dimensions must be imagined as instantaneous and concurrent (popularly depicted as a 'time machine' in science-fiction classics such as "Doctor Who"). reflect dynamical systems complexity [41], are being embraced to investigate the poly-dimensionality and nonlinearity of meditation EEG-substrates. ...

... 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. ...

... 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.

... The spacetime Q 4 p is acausal in the broad sense of lacking a causal structure, but also in the particular, technical, sense that for any pair of points on it, there exists no causal curve connecting them (which, in particular, also implies that it is achronal). The question of the intrinsic (a)causality of spacetime has been studied sometime ago [31], and is a topic of obligated discussion when dealing with the possibility of 'travels in time' [34,54]. Acausal (portions of) spacetimes appears often in relation with wormholes in General Relativity [38]. ...

We construct a family of quantum scalar fields over a $p-$adic
spacetime which satisfy $p-$adic analogues of the G\aa rding--Wightman
axioms. Most of the axioms can be formulated the same way in both, the
Archimedean and non-Archimedean frameworks; however, the axioms depending on
the ordering of the background field must be reformulated, reflecting the
acausality of $p-$adic spacetime. The $p-$adic scalar
fields satisfy certain $p-$adic Klein-Gordon pseudo-differential equations. The
second quantization of the solutions of these Klein-Gordon equations
corresponds exactly to the scalar fields introduced here.

... A second reason is that several mathematical models which allow travel to an earlier time or the transmission of information to the past have been detailed in the technical literature (e.g. Morris et al. 1988;Friedman et al. 1990;Gott 1991;Visser 1996;Mallett 2003;Ralph & Downes 2012;Yuan et al. 2015;Tippett & Tsang 2017) along with popular physics books (e.g. Thorne 1994;Gott 2001;Mallett & Henderson 2008;Kaku 2009;Clegg 2011;Al-Khalili 2016). ...

Abstract It is not uncommon in time travel stories to find that the mechanism by which the time travel is achieved is not invented. A time traveller could journey to his/her own past and give the designs of the time travel machine to his/her earlier self as s/he was given the designs as a younger person. These designs never get thought up by anyone. Such a situation would conflict with the usual conception of the acquisition of knowledge. This situation is called the Temporal Epistemic Anomaly and would arise if knowledge is gained at a time prior to the information in question being transmitted but is not discovered or invented at any time. This article examines the implications of information propagating around a causal chain that is closed in time (which is required to create the Anomaly) and whether this information need have a specific origin point.

... Therein, two flat spacetime geometries are connected via a curved transition region, where the energy conditions are violated; the inner flat region contains CTCs. The model of [39] is simpler than the Ori-Soen spacetime in many aspects, but a price must be paid for this: this spacetime is not time-orientable, and naked curvature singularities appear in the transition region. Third class: ad hoc geometries originally designed to allow hyperfast space travel, which have natural variants possessing CTCs. ...

... 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 $\mathbb{R}^4$, 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.

... The spacetime Q 4 p is acausal in the broad sense of lacking a causal structure, but also in the particular, technical, sense that for any pair of points on it, there exists no causal curve connecting them (which, in particular, also implies that it is achronal). The question of the intrinsic (a)causality of spacetime has been studied sometime ago [31], and is a topic of obligated discussion when dealing with the possibility of 'travels in time' [34,54]. Acausal (portions of) spacetimes appears often in relation with wormholes in General Relativity [38]. ...

We construct a family of quantum scalar fields over a p−adic space-time which satisfy p−adic analogues of the Gårding-Wightman axioms. Most of the axioms can be formulated the same way in both, the Archimedean and non-Archimedean frameworks; however, the axioms depending on the ordering of the background field must be reformulated, reflecting the acausality of p−adic spacetime. The p−adic scalar fields satisfy certain p−adic Klein-Gordon pseudo-differential equations. The second quantization of the solutions of these Klein-Gordon equations corresponds exactly to the scalar fields introduced here.