The problem is discussed of whether a traveller can reach a remote object and return back sooner than a photon would when taken into account that the traveller can partly control the geometry of his world. It is argued that under some reasonable assumptions in globally hyperbolic spacetimes the traveller cannot hasten reaching the destination. Nevertheless, it is perhaps possible for him to make an arbitrarily long round-trip within an arbitrarily short (from the point of view of a terrestrial observer) time.

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We define the time travel paradox in physical terms and prove its existence by constructing an explicit example. We argue further that in theories -such as general relativity -where the spacetime geometry is subject to nothing but differential equations and initial data no paradoxes arise.

Theorem. 1) Any spacetime U has a maximal extension M max such that all closed causal curves in M max (if they exist there) are confined to the chronological past of U.2) The assertion remains true, even if the definition of spacetime is complemented by an arbitrary local geometric condition C.Obviously the meaning and the validity of the second part of the theorem depends crucially on what is understood by "local" and "geometric". The latter term is transparent-a condition is called geometric, iff it holds in a spacetime M, when and only when it holds in any spacetime isometric to M-but the situation with locality is more subtle. In [1] the following inadequate definition crept into the text:1. Definition. We call a condition C local if the following is true: C holds in a spacetime M if and only if it holds in any U which is isometric to an open subset of M. 2. Definition. The condition (property) C is local, if for any open covering {V α } of an arbitrary spacetime M the following equivalence is true ‡The two definitions are not equivalent. For example, the former is satisfied by thenon-local according to definition 2-property "to have timelike diameter § not greater than 1". Definition 2 is closer to the intuitive notion of locality: in particular, it makes the implication ( * ) true, which solves all the problems mentioned above. Thus, the only correction needed by [1] is the replacement of definition 1 by definition 2. ‡ Cf. "particularly interesting conditions" imposed on properties in Appendix B of [4].§ See [5] for definition.

In proving inequality (49) for m 2 m ; m (see the paragraph around Eq. (50)) an error is made: the term Ew 2 ÿ in fact cannot be neglected. This error invalidates the whole proof and thus the main result of the paper. So, at the moment one has to admit that there is no evidence that a static [1] macroscopic wormhole can exist without some matter more exotic than the quantized electromagnetic or scalar fields.[1] For traversable, but non-static wormholes such evidence does exist, see S. Krasnikov Phys. Rev. D 73, 084006 (2006).