Global Geodesic Properties of Gödel-type SpaceTimes

Abstract

The aim of this chapter is to review and complete the study of geodesics on Gödel-type spacetimes from a variational viewpoint in the last decade (say, from [10] to [2]). In particular, we prove some new results on geodesic connectedness and geodesic completeness for these spacetimes.

1 Appendix

Taking a connected, finite–dimensional semi–Riemannian manifold $(\mathcal{M},g)$, let ${H}^{1}(I,\mathcal{M})$ be the associated Sobolev space for some auxiliar Riemannian metric on $\mathcal{M}$. Then, ${H}^{1}(I,\mathcal{M})$ is equipped with a structure of infinite–dimensional manifold modelled on the Hilbert space ${H}^{1}(I, {\mathbb{R}}^{n})$. For any $z \in {H}^{1}(I,\mathcal{M})$, the tangent space of ${H}^{1}(I,\mathcal{M})$ at z can be written as follows:

$${T}_{z}{H}^{1}(I,\mathcal{M}) =\{ \zeta \in {H}^{1}(I,T\mathcal{M}) : \zeta (s) \in {T}_{ z(s)}\mathcal{M}\;\text{ for all}s \in I\},$$

where $T\mathcal{M}$ is the tangent bundle of $\mathcal{M}$.

If $\mathcal{M}$ splits globally in the product of two semi–Riemannian manifolds ${\mathcal{M}}_{1}$ and ${\mathcal{M}}_{2}$, i.e. $\mathcal{M} = {\mathcal{M}}_{1} \times {\mathcal{M}}_{2}$, then

$${H}^{1}(I,\mathcal{M}) \equiv {H}^{1}(I,{\mathcal{M}}_{ 1}) \times {H}^{1}(I,{\mathcal{M}}_{ 2})$$

and ${T}_{z}{H}^{1}(I,\mathcal{M}) \equiv {T}_{{z}_{1}}{H}^{1}(I,{\mathcal{M}}_{1}) \times {T}_{{z}_{2}}{H}^{1}(I,{\mathcal{M}}_{2})$ for all $z = ({z}_{1},{z}_{2}) \in \mathcal{M}$.

On the other hand, if $({\mathcal{M}}_{0},\langle \cdot,{\cdot \rangle }_{R})$ is a C ³ complete Riemannian manifold, it can be smoothly and isometrically embedded in a Euclidean space ${\mathbb{R}}^{N}$ (see [24]); moreover such embedding can be chosen closed (see [23]) and this is used in the proof of Lemma 1. Hence, ${H}^{1}(I,{\mathcal{M}}_{0})$ is a closed submanifold of the Hilbert space ${H}^{1}(I, {\mathbb{R}}^{N})$. In this case, we denote by $d(\cdot,\cdot )$ the distance induced on ${\mathcal{M}}_{0}$ by its Riemannian metric $\langle \cdot,{\cdot \rangle }_{R}$, i.e.,

$$d({x}_{p},{x}_{q})\ :=\ \inf \left \{{\int\limits}_{a}^{b}\sqrt{\langle \dot{x},\dot{ {x}\rangle }_{ R}}\;\mathrm{d}s :\; x \in {A}_{{x}_{p},{x}_{q}}\right \},$$

where $x \in {A}_{{x}_{p},{x}_{q}}$ if $x : [a,b] \rightarrow {\mathcal{M}}_{0}$ is any piecewise smooth curve in ${\mathcal{M}}_{0}$ joining ${x}_{p},{x}_{q} \in {\mathcal{M}}_{0}$.

Taking z _p , ${z}_{q} \in \mathcal{M}$, let us consider

$${\it \Omega }^{1}({z}_{ p},{z}_{q}) =\{ z \in {H}^{1}(I,\mathcal{M}) : z(0) = {z}_{ p},\;z(1) = {z}_{q}\},$$

which is a submanifold of ${H}^{1}(I,\mathcal{M})$, complete if $\mathcal{M}$ is complete and having tangent space described as

$${T}_{z}{\it \Omega }^{1}({z}_{ p},{z}_{q}) =\{ \zeta \in {T}_{z}{H}^{1}(I,\mathcal{M}) : \zeta (0) = 0 = \zeta (1)\}\quad \text{ at any}z \in {\it \Omega }^{1}({z}_{ p},{z}_{q}).$$

Moreover, for any l _p , ${l}_{q} \in \mathbb{R}$, let us denote

$$W({l}_{p},{l}_{q}) =\{ l \in {H}^{1}(I, \mathbb{R}) : l(0) = {l}_{ p}\,\;l(1) = {l}_{q}\}.$$

Clearly,

$$W({l}_{p},{l}_{q}) = {H}_{0}^{1}(I, \mathbb{R}) + {l}_{ pq},$$

with ${H}_{0}^{1}(I, \mathbb{R}) =\{ l \in {H}^{1}(I, \mathbb{R}) : l(0) = 0 = l(1)\}$, ${l}_{pq} : s \in I\mapsto (1 - s){l}_{p} + s{l}_{q} \in \mathbb{R}$. Hence, $W({l}_{p},{l}_{q})$ is a closed affine submanifold of the Hilbert space ${H}^{1}(I, \mathbb{R})$ with tangent space

$${T}_{l}W({l}_{p},{l}_{q}) = {H}_{0}^{1}(I, \mathbb{R})\quad \text{ for every}l \in \, W({l}_{ p},{l}_{q}).$$