We discuss new sufficient conditions under which an affine manifold $(M, abla)$ is geodesically connected. These conditions are shown to be essentially weaker than those discussed in groundbreaking work by Beem and Parker and in recent work by Alexan
der and Karr, with the added advantage that they yield an elementary proof of the main result.
Globally hyperbolic spacetimes with timelike boundary $(overline{M} = M cup partial M, g)$ are the natural class of spacetimes where regular boundary conditions (eventually asymptotic, if $overline{M}$ is obtained by means of a conformal embedding) c
an be posed. $partial M$ represents the naked singularities and can be identified with a part of the intrinsic causal boundary. Apart from general properties of $partial M$, the splitting of any globally hyperbolic $(overline{M},g)$ as an orthogonal product ${mathbb R}times bar{Sigma}$ with Cauchy slices with boundary ${t}times bar{Sigma}$ is proved. This is obtained by constructing a Cauchy temporal function $tau$ with gradient $ abla tau$ tangent to $partial M$ on the boundary. To construct such a $tau$, results on stability of both, global hyperbolicity and Cauchy temporal functions are obtained. Apart from having their own interest, these results allow us to circumvent technical difficulties introduced by $partial M$. As a consequence, the interior $M$ both, splits orthogonally and can be embedded isometrically in ${mathbb L}^N$, extending so properties of globally spacetimes without boundary to a class of causally continuous ones.
Ehlers-Kundt conjecture is a physical assertion about the fundamental role of plane waves for the description of gravitational waves. Mathematically, it becomes equivalent to a problem on the Euclidean plane ${mathbb R}^2$ with a very simple formulat
ion in Classical Mechanics: given a non-necessarily autonomous potential $V(z,u)$, $(z,u)in {mathbb R}^2times {mathbb R}$, harmonic in $z$ (i.e. source-free), the trajectories of its associated dynamical system $ddot{z}(s)=- abla_z V(z(s),s)$ are complete (they live eternally) if and only if $V(z,u)$ is a polynomial in $z$ of degree at most $2$ (so that $V$ is a standard mathematical idealization of vacuum). Here, the conjecture is solved in the significative case that $V$ is bounded polynomially in $z$ for finite values of $uin {mathbb R}$. The mathematical and physical implications of this {em polynomial EK conjecture}, as well as the non-polynomial one, are discussed beyond their original scope.
