Causality and Legendrian linking for higher dimensional spacetimes.

Abstract Let ( X m + 1 , g ) be an ( m + 1 ) -dimensional globally hyperbolic spacetime with Cauchy surface M m , and let M ˜ m be the universal cover of the Cauchy surface. Let N X be the contact manifold of all future directed unparameterized light rays in X that we identify with the spherical cotangent bundle S T ∗ M. Jointly with Stefan Nemirovski we showed that when M ˜ m is not a compact manifold, then two points x , y ∈ X are causally related if and only if the Legendrian spheres S x , S y of all light rays through x and y are linked in N X. In this short note we use the contact Bott–Samelson theorem of Frauenfelder, Labrousse and Schlenk to show that the same statement is true for all X for which the integral cohomology ring of a closed M ˜ is not the one of the CROSS (compact rank one symmetric space). If M admits a Riemann metric g ¯ , a point x and a number ℓ > 0 such that all unit speed geodesics starting from x return back to x in time ℓ , then ( M , g ¯ ) is called a Y ℓ x manifold. Jointly with Stefan Nemirovski we observed that causality in ( M × R , g ¯ ⊕ − t 2 ) is not equivalent to Legendrian linking. Every Y ℓ x -Riemann manifold has compact universal cover and its integral cohomology ring is the one of a CROSS. So we conjecture that Legendrian linking is equivalent to causality if and only if one can not put a Y ℓ x Riemann metric on a Cauchy surface M. [ABSTRACT FROM AUTHOR]