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If time travel is possible, it seems to inevitably lead to paradoxes. These include consistency paradoxes, such as the famous grandfather paradox, and bootstrap paradoxes, where something is created out of nothing. One proposed class of resolutions to these paradoxes allows for multiple histories (or timelines), such that any changes to the past occur in a new history, independent of the one where the time traveler originated. We introduce a simple mathematical model for a spacetime with a time machine, and suggest two possible multiple-histories models, making use of branching spacetimes and covering spaces respectively. We use these models to construct novel and concrete examples of multiple-histories resolutions to time travel paradoxes, and we explore questions such as whether one can ever come back to a previously visited history and whether a finite or infinite number of histories is required. Interestingly, we find that the histories may be finite and cyclic under certain assumptions, in a way which extends the Novikov self-consistency conjecture to multiple histories and exhibits hybrid behavior combining the two. Investigating these cyclic histories, we rigorously determine how many histories are needed to fully resolve time travel paradoxes for particular laws of physics. Finally, we discuss how observers may experimentally distinguish between multiple histories and the Hawking and Novikov conjectures.
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These lecture notes were prepared for a 25-hour course for advanced undergraduate students participating in Perimeter Institutes Undergraduate Summer Program. The lectures cover some of what is currently known about the possibility of superluminal travel and time travel within the context of established science, that is, general relativity and quantum field theory. Previous knowledge of general relativity at the level of a standard undergraduate-level introductory course is recommended, but all the relevant material is included for completion and reference. No previous knowledge of quantum field theory, or anything else beyond the standard undergraduate curriculum, is required. Advanced topics in relativity, such as causal structures, the Raychaudhuri equation, and the energy conditions are presented in detail. Once the required background is covered, concepts related to faster-than-light travel and time travel are discussed. After introducing tachyons in special relativity as a warm-up, exotic spacetime geometries in general relativity such as warp drives and wormholes are discussed and analyzed, including their limitations. Time travel paradoxes are also discussed in detail, including some of their proposed resolutions.
Some known relativistic paradoxes are reconsidered for closed spaces, using a simple geometric model. For two twins in a closed space, a real paradox seems to emerge when the traveling twin is moving uniformly along a geodesic and returns to the starting point without turning back. Accordingly, the reference frames (RF) of both twins seem to be equivalent, which makes the twin paradox irresolvable: each twin can claim to be at rest and therefore to have aged more than the partner upon their reunion. In reality, the paradox has the resolution in this case as well. Apart from distinction between the two RF with respect to actual forces in play, they can be distinguished by clock synchronization. A closed space singles out a truly stationary RF with single-valued global time; in all other frames, time is not a single-valued parameter. This implies that even uniform motion along a spatial geodesic in a compact space is not truly inertial, and there is an effective force on an object in such motion. Therefore, the traveling twin will age less upon circumnavigation than the stationary one, just as in flat space-time. Ironically, Relativity in this case emerges free of paradoxes at the price of bringing back the pre-Galilean concept of absolute rest. An example showing the absence of paradoxes is also considered for a more realistic case of a time-evolving closed space.
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.
In this paper we review the derivation of light bending obtained before the discovery of General Relativity (GR). It is intended for students learning GR or specialist that will find new lights and connexions on these historic derivations. Since 1915, it is well known that the observed light bending stems from two contributions : the first one is directly deduced from the equivalence principle alone and was obtained by Einstein in 1911; the second one comes from the spatial curvature of spacetime. In GR, those two components are equal, but other relativistic theories of gravitation can give different values to those contributions. In this paper, we give a simple explanation, based on the wave-particle picture of why the first term, which relies on the equivalence principle, is identical to the one obtained by a purely Newtonian analysis. In this context of wave analysis, we emphasize that the dependency of the velocity of light with the gravitational potential, as deduced by Einstein concerns the phase velocity. Then, we wonder whether Einstein could have envisaged already in 1911 the second contribution, and therefore the correct result. We argue that considering a length contraction in the radial direction, along with the time dilation implied by the equivalence principle, could have led Einstein to the correct result.
Recently, we introduced in arXiv:1105.2434 a model for product adoption in social networks with multiple products, where the agents, influenced by their neighbours, can adopt one out of several alternatives. We identify and analyze here four types of paradoxes that can arise in these networks. To this end, we use social network games that we recently introduced in arxiv:1202.2209. These paradoxes shed light on possible inefficiencies arising when one modifies the sets of products available to the agents forming a social network. One of the paradoxes corresponds to the well-known Braess paradox in congestion games and shows that by adding more choices to a node, the network may end up in a situation that is worse for everybody. We exhibit a dual version of this, where removing available choices from someone can eventually make everybody better off. The other paradoxes that we identify show that by adding or removing a product from the choice set of some node may lead to permanent instability. Finally, we also identify conditions under which some of these paradoxes cannot arise.
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