Closed Timelike Curves in Flat Lorentz Spacetimes

We consider the region of closed timelike curves (CTC’s) in three-dimensional flat Lorentz spacetimes. The interest in this global geometrical feature goes beyond the purely mathematical. Such spacetimes may be considered lower-dimensional toy models of sourceless Einstein gravity or cosmology. In three dimensions all such spacetimes are known: they are quotients of Minkowski space by a suitable group of Poincaré isometries. The presence of CTC’s would indicate the possibility of “time machines”, a region of spacetime where an object can travel along in time and revisit the same event. Such spacetimes also provide a testbed for the chronology protection conjecture, which suggests that quantum back reaction would eliminate CTC’s. In particular, our interest in this note will be to find the set free of CTC’s for E/ < γ >, where E is modeled on Minkowski space and γ is a Poincaré transformation. We describe the set free of CTC’s where γ is hyperbolic, parabolic, and elliptic. Let E denote three-dimensional Minkowski space. This is an affine space with translations in R2,1, the vector space equipped with the standard indefinite bilinear form B(·, ·) of signature (2, 1). (Since E is flat and geodesically complete, the reader may make the identification between E and its set of translations without any major difficulties arising.) We are interested in flat Lorentz manifolds, which are quotients of an open subset X of E by a group Γ of affine Lorentzian isometries that acts properly discontinuously on that subset. Such manifolds X/Γ inherit a local causal structure from E. Let X/Γ be such a flat Lorentz manifold. A timelike vector is a vector v ∈ R2,1 such that B(v, v) < 0. A timelike curve in X/Γ is a C1 path c : [0, 1] → X/Γ whose tangent vectors are all timelike; we say that c is closed if c(0) = c(1). The purpose of this note is to lay the groundwork for understanding regions free of closed timelike curves in a Lorentz spacetime. Date: February 27, 2008. 1 2 VIRGINIE CHARETTE, TODD A. DRUMM, AND DIETER BRILL The CTC region of a spacetime is the set of all points which lie on some closed timelike curve and the CTC-free region is the complement of this space. Suppose Γ acts properly discontinuously on some subset X ⊂ E. Denote by [p] the image of p in X/Γ under projection. We wish to determine the set of all p such that [p] lies in the CTC region of X/Γ. We first note the following basic lemma. Lemma 0.1. Let p ∈ X ⊂ E be a point such that γ(p)− p is a timelike vector, for some γ ∈ Γ, and the line segment starting at p and ending at γ(p) lies entirely in X. Then [p] ∈ X/Γ lies on a smooth closed timelike curve. This lemma is well known, but we provide a proof for completeness. Proof. Let c : R → E be a continuous and piecewise linear path through each point γ(p), defined by the following: c(t) = γ(p) + (t− [t]) (