Abstract
In this chapter we start with the introduction, in Sect. 1, of the concept of appearing time machine and give a simple example thereof. In the following section, a few less trivial examples are considered. The first of them is the Misner spacetime. It is one of the oldest and presumably the most important time machines: indeed, being just a flat cylinder it is in a sense the time machine in its pure form. And anyway, with its rich and counter-intuitive structure the Misner space deserves studying at least as a wonderful source of counter-examples [133]. Its simplest generalizations to the non-flat case are considered too. Then, in Sect. 3, we briefly discuss the process of evolution of a wormhole into a time machine, a widely known and popular process. Its importance lies in the fact that it is one of the most ‘realistic’ scenarios of how the universe might lose its global hyperbolicity. In Sect. 4 we introduce the notions of compactly generated and compactly determined Cauchy horizons. All time machines with such horizons are shown to have some important common properties. However, none of those properties are compulsory for a general time machine. We show that by example in Sect. 5.
Everyone knows that dragons don’t exist. But while this simplistic formulation may satisfy the layman, it does not suffice for the scientific mind. [There were] three distinct kinds of dragon: the mythical, the chimerical, and the purely hypothetical. They were all, one might say, nonexistent, but each nonexisted in an entirely different way.
The Third Sally, or The Dragons of Probability [112]
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Notes
- 1.
True, within such a scenario the arrow of time looks mysterious [120]. Then again, it looks so in any scenario.
- 2.
In contrast to eternal ones mentioned above.
- 3.
This spacetime is, basically, a variant of the time machine proposed as an example, in [74].
- 4.
The geodesics of the latter kind have self-intersections, in M. This, however, does not violate causality, all such geodesics being spacelike.
- 5.
Likewise, they appear in a spacetime where two wormholes—both with a constant distance between the mouths—move with respect to each other.
- 6.
This question is quite old, see, for example [64].
- 7.
These are, in particular, all time machines considered in this book so far, excluding the DP space.
- 8.
It is worth mentioning, therefore, that slightly different categories were considered too. In [88] the set which is required to be compact is the entire
, while in [142] it is a subset \({{\mathcal {Q}}}\) of a Cauchy surface of \(M^{\mathrm {r}}\) such that first CTCs appear on the boundary of \({{\mathcal {D}}}({{\mathcal {Q}}})\). - 9.
In the topology of the horizon, of course.
- 10.
- 11.
By the example of Misner space, one sees that in a non-globally hyperbolic \(M^e\) even this may not be the case.
- 12.
For the sake of simplicity, in discussing the boundedness of tensor components we shall assume that the whole
is covered by a single coordinate system. The generalization to the case when such a system does not exist is straightforward, because 
always can be covered by a finite number of compact sets 
each of which is covered by a single chart. - 13.
As before, we consider geodesic generators of \(\mathcal {H}^+\) to be future directed.
- 14.
Strictly speaking, \(\upmu _t\) is not defined in points in which a few generators meet, so one has to show that such points form a set of measure zero.
- 15.
The “past” part is trivial, because such are all generators by Corollary 7.
- 16.
And even the dominant energy condition. On the other hand, the strong energy condition never holds in spacetimes like \(M_w\) [124].
, while in [142] it is a subset 
is covered by a single coordinate system. The generalization to the case when such a system does not exist is straightforward, because 
always can be covered by a finite number of compact sets 
each of which is covered by a single chart.Author information
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Krasnikov, S.V. (2018). Time Machines. In: Back-in-Time and Faster-than-Light Travel in General Relativity. Fundamental Theories of Physics, vol 193. Springer, Cham. https://doi.org/10.1007/978-3-319-72754-7_4
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