Geodesic connectedness of stationary spacetimes with optimal growth

“…Remark 3.2. If the product A(x)C(x) is strictly positive on M 0 , the corresponding Gödel type spacetime reduces to a standard stationary one, and its geodesic connectedness has been deeply studied in previous works (see, for example, [1,6]).…”

Section: Gödel Type Spacetimes and Statement Of The Main Theoremsmentioning

confidence: 99%

A note on geodesic connectedness of Gödel type spacetimes

Bartolo

Candela

Flores

2011

Differential Geometry and its Applications

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“…on Ω 1 (x p , x q ; M 0 ) which is bounded from below and coercive under "good" growth hypotheses on coefficients β and δ in (3.13). Thus, with this extrinsic approach, also in this case the following result can be stated (for all the details, see [4,Theorem 1.2]).…”

Section: 2mentioning

confidence: 90%

Geodesics in semi-Riemannian manifolds: geometric properties and variational tools

Candela¹,

Sánchez²

2008

ESI Lectures in Mathematics and Physics

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Geodesics become an essential element of the geometry of a semi-Riemannian manifold. In fact, their differences and similarities with the (positive definite) Riemannian case, constitute the first step to understand semi-Riemannian Geometry. The progress in the last two decades has become impressive, being especially relevant the systematic introduction of (infinite-dimensional) variational methods.Our purpose is to give an overview, from refinements of classical results to updated variational settings. First, several properties (and especially completeness) of geodesics in some ambient spaces are studied. This includes heuristic constructions of compact incomplete examples, geodesics in warped, GRW or stationary spacetimes, properties in surfaces and spaceforms, or problems on stability of completeness.Then, we study the variational framework, and focus on two fundamental problems of this approach, which regards geodesic connectedness. The first one deals with a variational principle for stationary manifolds, and its recent implementation inside Causality Theory. The second one concerns orthogonal splitting manifolds, and a reasonably selfcontained development is provided, collecting some steps spread in the literature.

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“…These curves have support in J − (q ε ), so the curves (x m , t p ) lie in the compact set J − (q ε ) ∩ (S × {t p }) (recall Proposition 2.4 (ii)), in contradiction with the hypothesis. 2 Even if the core of this proof is essentially contained in [9, Section 5], here we rearrange it for reader's convenience. Although the functional f is defined in C 1 K (p, q), it is natural to consider limits in Ω K (p, q) (cf.…”

Section: Stationary Intrinsic Functional Frameworkmentioning

confidence: 99%