Linking and causality in globally hyperbolic spacetimes
| dc.creator | Chernov, Vladimir | |
| dc.creator | Rudyak, Yuli B. | |
| dc.date | 2006-11-05 | |
| dc.date | 2007-03-28 | |
| dc.date.accessioned | 2026-07-07T11:23:48Z | |
| dc.date.available | 2026-07-07T11:23:48Z | |
| dc.description | The linking number $lk$ is defined if link components are zero homologous. Our affine linking invariant $alk$ generalizes $lk$ to the case of linked submanifolds with arbitrary homology classes. We apply $alk$ to the study of causality in Lorentz manifolds. Let $M^m$ be a spacelike Cauchy surface in a globally hyperbolic spacetime $(X^{m+1}, g)$. The spherical cotangent bundle $ST^*M$ is identified with the space $N$ of all null geodesics in $(X,g).$ Hence the set of null geodesics passing through a point $x\in X$ gives an embedded $(m-1)$-sphere $S_x$ in $N=ST^*M$ called the sky of $x.$ Low observed that if the link $(S_x, S_y)$ is nontrivial, then $x,y\in X$ are causally related. This motivated the problem (communicated by Penrose) on the Arnold's 1998 problem list to apply link theory to the study of causality. The spheres $S_x$ are isotopic to fibers of $(ST^*M)^{2m-1}\to M^m.$ They are nonzero homologous and $lk(S_x,S_y)$ is undefined when $M$ is closed, while $alk(S_x, S_y)$ is well defined. Moreover, $alk(S_x, S_y)\in Z$ if $M$ is not an odd-dimensional rational homology sphere. We give a formula for the increment of $\alk$ under passages through Arnold dangerous tangencies. If $(X,g)$ is such that $alk$ takes values in $\Z$ and $g$ is conformal to $g'$ having all the timelike sectional curvatures nonnegative, then $x, y\in X$ are causally related if and only if $alk(S_x,S_y)\neq 0$. We show that $x,y$ in nonrefocussing $(X, g)$ are causally unrelated iff $(S_x, S_y)$ can be deformed to a pair of $S^{m-1}$-fibers of $ST^*M\to M$ by an isotopy through skies. Low showed that if $(ß, g)$ is refocussing, then $M$ is compact. We show that the universal cover of $M$ is also compact. | |
| dc.description | We added: Theorem 11.5 saying that a Cauchy surface in a refocussing space time has finite pi_1; changed Theorem 7.5 to be in terms of conformal classes of Lorentz metrics and did a few more changes. 45 pages, 3 figures. A part of the paper (several results of sections 4,5,6,9,10) is an extension and development of our work math.GT/0207219 in the context of Lorentzian geometry. The results of sections 7,8,11,12 and Appendix B are new | |
| dc.identifier | https://arxiv.org/abs/math/0611111 | |
| dc.identifier | http://arxiv.org/abs/math/0611111 | |
| dc.identifier | Commun.Math.Phys.279:309-354,2008 | |
| dc.identifier | doi:10.1007/s00220-008-0414-8 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/195016 | |
| dc.subject | Geometric Topology | |
| dc.subject | General Relativity and Quantum Cosmology | |
| dc.subject | Mathematical Physics | |
| dc.subject | Algebraic Topology | |
| dc.subject | Primary 57Q45, 53C50; Secondary 53C80, 57R17, 58D10, 83C75 | |
| dc.title | Linking and causality in globally hyperbolic spacetimes | |
| dc.type | text |