José Luis Flores

Causality and Conjugate Points in General Plane Waves

Let = 0 × 2 be a pp-wave-type spacetime endowed with the metric , z = , x + 2du dv + H(x, u) du2, where (0, , x) is any Riemannian manifold and H(x, u) is an arbitrary function. We show that the behaviour of H(x, u) at spatial infinity determines the causality of , say: (a) if −H(x, u) behaves subquadratically (i.e, essentially −H(x, u) ≤ R1(u)|x|2− for some > 0 and large distance |x| to a fixed point) and the spatial part (0, , x) is complete, then the spacetime is globally hyperbolic, (b) if −H(x, u) grows at most quadratically (i.e, −H(x, u) ≤ R1(u)|x|2 for large |x|) then it is strongly causal and (c) is always causal, but there are non-distinguishing examples (and thus, not strongly causal), even when −H(x, u) ≤ R1(u)|x|2+, for small > 0. Therefore, the classical model 0 = 2, H(x, u) = ∑i, j hij(u)xixj(≢ 0), which is known to be strongly causal but not globally hyperbolic, lies in the critical quadratic situation with complete 0. This must be taken into account for realistic applications. In fact, we argue that −H will be subquadratic (and the spacetime globally hyperbolic) if is asymptotically flat. The relation of these results with the notion of astigmatic conjugacy and the existence of conjugate points is also discussed.

The Causal Boundary of Spacetimes Revisited

We present a new development of the causal boundary of spacetimes, originally introduced by Geroch, Kronheimer and Penrose. Given a strongly causal spacetime (or, more generally, a chronological set), we reconsider the GKP ideas to construct a family of completions with a chronology and topology extending the original ones. Many of these completions present undesirable features, like those which appeared in previous approaches by other authors. However, we show that all these deficiencies are due to the attachment of an “excessively big” boundary. In fact, a notion of “completion with minimal boundary” is then introduced in our family such that, when we restrict to these minimal completions, which always exist, all previous objections disappear. The optimal character of our construction is illustrated by a number of satisfactory properties and examples.

On General Plane Fronted Waves. Geodesics

AbstractA general class of Lorentzian metrics,

The causal boundary of wave-type spacetimes

\mathcal{M}_0 \times \mathbb{R}^2

On the geometry of pp-wave type spacetimes

,

The causal boundary of product spacetimes

\langle \cdot ,\; \cdot \rangle _z = \langle \cdot ,\; \cdot \rangle _x + 2\;\;du\;\;dv + H(x,u)\;du^2