Geodesics in Static Lorentzian Manifolds with Critical Quadratic Behavior

Abstract

The aim of this paper is t o study the geodesic connectedness of a complete static Lorentzian manifold (M.〈·, ·〉

Geodesic connectedness of stationary spacetimes with optimal growth

“…Observe that the accuracy of the assumption on β is ensured by the family of static spacetimes given in [3,Section 7]. So, in the following example we focus on hypothesis (1.5).…”

“…Remarkably, in the static case some of these properties present a "critical" behavior with respect to a quadratic asymptotic growth of metric coefficient β 1 (a nice survey on this subject can be found in [16]). In fact, in [3] the authors studied the geodesic connectedness of static spacetimes, that is, they asked whether any two points of the spacetime can be connected by a geodesic. They answered positively to this question whenever coefficient β grows (at most) quadratically [3, Theorem 1.1].…”

“…They answered positively to this question whenever coefficient β grows (at most) quadratically [3, Theorem 1.1]. Moreover, this growth is optimal/critical, in the sense that static counterexamples are found with superquadratic β presenting a growth as close as we want to the quadratic one [3,Section 7]. Even more relevant, this analysis is totally fulfilled by using variational techniques, showing that the variational approach is specially well-adapted to this problem.…”

Quadratic Bolza Problems in Static Spacetimes with Critical Asymptotic Behavior

“…If function β in (1.1) is upper bounded then it is easy to prove that functional J 0 in (1.5) is bounded from below and admits at least a minimum for each ∆ t ∈ R; therefore, the corresponding spacetime is geodesically connected (see [6, Theorem 0.1]). Recently, the previous result has been extended in [2] to static manifolds whose coefficient β has at most a quadratic asymptotic behavior with respect to d(·, ·), the distance canonically associated to the Riemannian metric ·, · on M 0 , i.e., (H 2 ) there exist λ ≥ 0, k ∈ R and a pointx ∈ M 0 such that…”

“…Remark that, if M = M 0 ×R is equipped with static metric (1.1) and potential V is independent of time variable t, i.e., (1.6) holds, it can be proved connectedness in M by trajectories under potential V when: (a) both β and V are bounded or at most subquadratic; (b) β is subquadratic while V has a quadratic growth with respect to x (it is enough using the result in [7]); (c) β has a quadratic growth and V is subquadratic (by using the result in [2]). …”