Abstract
It has become apparent in recent decades that in discussing fundamental properties of space and time the adequate language is Lorentzian geometry. Correspondingly, we start the book with a brief introduction to that discipline. In the first four sections of this chapter some basic notions—such as convexity, causal simplicity and global hyperbolicity—are defined and some basic facts about them are provided, such as Whitehead’s and Geroch’s theorems, the Gauss lemma, etc. Of course, this recapitulation cannot substitute a systematic presentation, so wherever possible the proofs of those facts are dropped, which is understood as the reference to classical monographs such as (O’Neill, Semi-Riemannian geometry. Academic Press, 1983) [141] or (Hawking and Ellis, The large-scale structure of spacetime. Cambridge University Press, Cambridge, 1973) [76]. Then, in Sect. 5, some new objects—perfectly simple sets—appear and the theorem is proved which states that they exist in arbitrary spacetime. We shall use this theorem in proving Theorem 2 in Chap. 5, but, as it seems, it is interesting also by itself (the perfectly simple sets are ‘as nice as possible’—they are both convex and globally hyperbolic, which makes them exceptionally useful in proving statements).
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Notes
- 1.
For the sake of brevity, a tensor (vector) field is sometimes called a tensor (vector).
- 2.
To avoid confusion, note that ‘extendible’ and ‘extended’ are almost antonyms: an extendible geodesic being that which is not (fully) extended.
- 3.
Not causally simple.
- 4.
For the reasons discussed in great detail in Sect. 2 in Chap. 2.
- 5.
A hypersurface is a submanifold of codimension 1.
- 6.
We have to use this awkward term because the word ‘acausal’ is already taken, see Definition 29.
- 7.
In the tangent bundle T(M), of course.
- 8.
Whenever a coordinate-dependent entities—such as \(\sigma \), position vector or Christoffel symbols (below)—is mentioned in this proof, it is understood that the coordinates \(X^a (p;r)\) are used.
- 9.
In [76], such neighbourhoods are called local causality neighbourhoods and this corollary is accepted without proof. Note in this connection that in [145] by ‘local causality neighbourhoods’ different (not necessarily convex) sets are meant.
- 10.
It would be complete, if Y were empty and \(M'\) were the union of two disjoint spacetimes.
- 11.
There are different definitions of singularity, see [62], for example. In this case, apparently the difference is immaterial.
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Krasnikov, S.V. (2018). Geometrical Introduction. 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_1
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