Dirichlet's theorem and diophantine approximation on manifolds

AbstractWe show that if M ⊂ Rk belongs to a general class of smooth manifolds then, for almost all x ∈ M, Dirichlet's theorem on Diophantine approximation cannot be infinitely improved

Unwrapping Closed Timelike Curves

Closed timelike curves (CTCs) appear in many solutions of the Einstein equation, even with reasonable matter sources. These solutions appear to violate causality and so are considered problematic. Since CTCs reflect the global properties of a spacetime, one can attempt to change its topology, without changing its geometry, in such a way that the former CTCs are no longer closed in the new spacetime. This procedure is informally known as unwrapping. However, changes in global identifications tend to lead to local effects, and unwrapping is no exception, as it introduces a special kind of singularity, called quasi-regular. This "unwrapping" singularity is similar to the string singularities. We give two examples of unwrapping of essentially 2+1 dimensional spacetimes with CTCs, the Gott spacetime and the Godel universe. We show that the unwrapped Gott spacetime, while singular, is at least devoid of CTCs. In contrast, the unwrapped Godel spacetime still contains CTCs through every point. A "multiple unwrapping" procedure is devised to remove the remaining circular CTCs. We conclude that, based on the two spacetimes we investigated, CTCs appearing in the solutions of the Einstein equation are not simply a mathematical artifact of coordinate identifications, but are indeed a necessary consequence of General Relativity, provided only that we demand these solutions do not possess naked quasi-regular singularities.Comment: 29 pages, 9 figure

Invariant differential operators and the Karlhede classification of type N vacuum solutions.

A spacetime calculus based on a single null direction, and which is therefore invariant under null rotations, is employed to show that a type N vacuum solution of Einstein's equations requires the calculation of at most five covariant derivatives of the curvature for its complete Karlhede classification