Global hyperbolicity and Palais–Smale condition for action functionals in stationary spacetimes

Abstract

In order to apply variational methods to the action functional for geodesics of a stationary spacetime, some hypotheses, useful to obtain classical Palais-Smale condition, are commonly used: pseudo-coercivity, bounds on certain coefficients of the metric, etc. We prove that these technical assumptions admit a natural interpretation for the conformal structure (causality) of the manifold. As a consequence, any stationary spacetime with a complete timelike Killing vector field and a complete Cauchy hypersurface (thus, globally hyperbolic), is proved to be geodesically connected.

Connectivity by geodesics on globally hyperbolic spacetimes with a lightlike Killing vector field

“…In this paper we are concerned with globally hyperbolic spacetimes admitting a complete causal Killing vector field. The following proposition, which slightly extends [9,Theorem 2.3], provides a precise description of the structure of these spacetimes.…”

Some Remarks on Conformal Symmetries and Bartnik’s Splitting Conjecture

“…Next, consider the following well-known result, obtained in a broader context in Ref. [17], and with a much simplified proof in the particular case of globally hyperbolic spacetimes in [4]: Proposition 2.3. Let (M, g) be a globally hyperbolic spacetime admitting a complete timelike conformal Killing vector field X ∈ X(M).…”

Iteration of closed geodesics in stationary Lorentzian manifolds

“…c(0) ∈ S. N S is a smooth, closed, embedded submanifold of M; let us recall from [7] (see also [9]) the following result: Recall that arc-connected components of M correspond to conjugacy classes of the fundamental group π 1 (M); given one such component * , we will call minimal a closed geodesic γ in * , with γ (0) ∈ S, which is a minimum point for the restriction of f to the arc-connected component of N S containing γ . If M is not simply connected, Proposition 5.5 gives a multiplicity of minimal closed geodesics, however, there is no way of telling whether these geodesics are geometrically distinct.…”