On the existence of closed timelike geodesics in compact spacetimes

Abstract

Abstract. Let M be a compact spacetime which admits a regular globally hyperbolic covering, and C a nontrivial free timelike homotopy class of closed timelike curves in M. We prove that C contains a longest curve (which must be a closed timelike geodesic) if and only if the timelike injectivity radius of C is finite; i.e., C has a bounded length. As a consequence among others, we deduce that for a compact static spacetime there exists a closed timelike geodesic within every nontrivial free timelike homotopy class having a finite timelike injectivity radius.

Ricci-flat left-invariant Lorentzian metrics on 2-step nilpotent Lie groups

“…It has been shown in [14] that a 2-step nilpotent Lie group N admits a flat left-invariant Lorentzian metric if and only if N is a trivial central extension of the three-dimensional Heisenberg group H 3 . At this point, a natural question arises:…”

“…As we have seen in Example 13, Case 2, any left-invariant Lorentzian metric , on H 3 for which the center is degenerate is flat (see also [14] or [20]). Conversely, by Theorem 15, if H 2n+1 admits a Ricci-flat left-invariant Lorentzian metric, then its Lie algebra H 2n+1 has a pseudo-orthonormal basis {b, z 1 , .…”

“…We recall here that, although they are close to being abelian, 2-step nilpotent Lie groups possess a very rich geometry (see [3], [4], [5], [7], [8], [9], [10], [12], [13], [14], [15], [16], [17], [18]). …”

“…In [14], it has been shown that a 2-step nilpotent Lie group N admits a flat left-invariant Lorentzian metric if and only if N is a trivial central extension of the three-dimensional Heisenberg group H 3 . Ricci-flat left-invariant pseudo-Riemannian metrics on 2-step nilpotent Lie groups have been studied in [5], using a different method from ours.…”

The causal hierarchy of spacetimes

“…Remark 3.90. The compactness of S cannot be removed (Guediri's counterexample, see [24] and references therein). Nevertheless, it can be replaced by the existence of a class of conjugacy C of the fundamental group which contains a timelike curve and satisfies one of the following two conditions (see [47]):…”