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 ${\tilde{M}}^m$ be the universal cover of the Cauchy surface. Let ${\mathcal{N}}_X$ be the contact manifold of all future directed unparameterized light rays in $X$ that we identify with the spherical cotangent bundle $S{T}^{\ast }M.$ Jointly with Stefan Nemirovski we showed when ${\tilde{M}}^m$ is {\bf not\/} a compact manifold, then two points $x,y\in X$ are causally related if and only if the Legendrian spheres ${\mathfrak{S}}_x,{\mathfrak{S}}_y$ of all light rays through $x$ and $y$ are linked in ${\mathcal{N}}_X.$ 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 $\tilde{M}$ is {\bf not} the one of the CROSS (compact rank one symmetric space). If $M$ admits a Riemann metric $\overline{g}$, a point $x$ and a number $\ell >0$ such that all unit speed geodesics starting from $x$ return back to $x$ in time $\ell$, then $(M,\overline{g})$ is called a $Y_{\ell }^x$ manifold. Jointly with Stefan Nemirovski we observed that causality in $(M\times \mathbb{R},\overline{g}\oplus -t^2)$ is {\bf not} equivalent to Legendrian linking. Every $Y_{\ell }^x$-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 {\bf not} put a $Y_{\ell }^x$ Riemann metric on a Cauchy surface $M.$

6 pages, exposition is a bit changed

References in corpus (5)

• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Globally hyperbolic spacetimes can be defined as "causal" instead of "strongly causal"
• Linking and causality in globally hyperbolic spacetimes
• Redshift and contact forms
• The Contact Structure in the Space of Light Rays

Cited by in corpus (4)

• Khovanov homology and causality in spacetimes
• The space of light rays: Causality and $L$-boundary
• Topological consequences of null-geodesic refocusing and applications to $Z^x$ manifolds
• Detecting Causality with Symplectic Quandles