Geodesics dynamics in the Linet–Tian spacetime with Λ<0\Lambda <0

We investigate the geodesics' kinematics and dynamics in the Linet-Tian metric with $\Lambda<0$ and compare with the results for the Levi-Civita metric, when $\Lambda=0$. This is used to derive new stability results about the geodesics' dynamics in static vacuum cylindrically symmetric spacetimes with respect to the introduction of $\Lambda<0$. In particular, we find that increasing $|\Lambda|$ always increases the minimum and maximum radial distances to the axis of any spatially confined planar null geodesic. Furthermore, we show that, in some cases, the inclusion of any $\Lambda<0$ breaks the geodesics' orbit confinement of the $\Lambda=0$ metric, for both planar and non-planar null geodesics, which are therefore unstable. Using the full system of geodesics' equations, we provide numerical examples which illustrate our results.IB and FM thank CMAT, Univ. Minho, for support through the FEDER Funds-COMPETE and FCT Project Est-C/MAT/UI0013/2011. FM is also supported by FCT projects PTDC/MAT/108921/2008 and CERN/FP/123609/2011 and thanks the warm hospitality from Instituto de Fisica, UERJ, Rio de Janeiro, where this work was completed. MFAdaSilva acknowledges the financial support from FAPERJ (Nos. E-26/171.754/2000, E-26/171.533.2002, E-26/170.951/2006, E-26/110.432/2009 and E-26/111.714/2010), Conselho Nacional de Desenvolvimento Cientifico e Tecnologico-CNPq-Brazil (Nos. 450572/2009-9, 301973/2009-1 and 477268/2010-2) and Financiadora de Estudos e Projetos-FINEP-Brazil