Causality and Legendrian linking for higher dimensional spacetimes

JOURNAL OF GEOMETRY AND PHYSICS

卷:133

Article

Chernov, Vladimir¹ 

[1] Dartmouth Coll, Dept Math, 6188 Kemeny Hall, Hanover, NH 03755 USA

关键词: Globally hyperbolic spacetime;    Light ray;    Contact structure;    Legendrian linking;    Causality;    Legendrian Low conjecture;   

DOI  :  10.1016/j.geomphys.2018.06.018

来源: Elsevier

【 摘 要 】

Let (Xm+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 ST*M. Jointly with Stefan Nemirovski we showed that when (M) over tilde (m) is not a compact manifold, then two points x, y is an element of X are causally related if and only if the Legendrian spheres (sic)(x), (sic)(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) over tilde is not the one of the CROSS (compact rank one symmetric space). If M admits a Riemann metric (g) over bar, a point x and a number l > 0 such that all unit speed geodesics starting from x return back to x in time f, then (M, (g) over bar) is called a Y-l(x) manifold. Jointly with Stefan Nemirovski we observed that causality in (M x R, (g) over bar circle plus -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. (C) 2018 Elsevier B.V. All rights reserved.

【 授权许可】

Free