Causality and Conjugate Points in General Plane Waves

TL;DR

This paper investigates how the asymptotic behavior of the function H(x,u) in general plane wave spacetimes influences their causality properties, establishing conditions for global hyperbolicity, strong causality, and causality.

Contribution

It provides a detailed analysis linking the growth rate of H(x,u) at spatial infinity to the causality classification of pp-wave spacetimes, including new insights into the critical quadratic case.

Findings

Subquadratic H(x,u) implies global hyperbolicity for complete spatial manifolds.

Quadratic H(x,u) ensures strong causality but not necessarily global hyperbolicity.

Certain quadratic growth cases are causal but non-distinguishing, affecting realistic models.

Abstract

Let $M = M_0 \times \R^2$ be a pp--wave type spacetime endowed with the metric $<\cdot,\cdot>_z = <\cdot,\cdot>_x + 2 du dv + H(x,u) du^2$, where $(M_0, <\cdot,\cdot>_x) $ is any Riemannian manifold and $H(x,u)$ an arbitrary function. We show that the behaviour of $H(x,u)$ at spatial infinity determines the causality of $M$, say: (a) if $-H(x,u)$ behaves subquadratically (i.e, essentially $-H(x,u) \leq R_1(u) |x|^{2-\epsilon} $ for some $\epsilon >0$ and large distance $|x|$ to a fixed point) and the spatial part $(M_0, <\cdot,\cdot>_x) $ is complete, then the spacetime $M$ is globally hyperbolic, (b) if $-H(x,u)$ grows at most quadratically (i.e, $-H(x,u) \leq R_1(u) |x|^{2}$ for large $|x|$) then it is strongly causal and (c) $M$ is always causal, but there are non-distinguishing examples (and thus, non-strongly causal), even when $-H(x,u) \leq R_1(u) |x|^{2+\epsilon} $, for small $\epsilon >0$. Therefore, the classical model $M_0 = \R^2$, $H(x,u) = \sum_{i,j} h_{ij}(u) x_i x_j (\not\equiv 0)$, which is known to be strongly causal but not globally hyperbolic, lies in the critical quadratic situation with complete $M_0$. This must be taken into account for realistic applications. In fact, we argue that $-H$ will be subquadratic (and the spacetime globally hyperbolic) if $M$ is asymptotically flat. The relation of these results with the notion of astigmatic conjugacy and the existence of conjugate points is also discussed.