Causality and Conjugate Points in General Plane Waves

Let $M=M_0\times {\mathbb{R}}^2$ be a pp–wave type spacetime endowed with the metric $<\cdot ,\cdot {>}_z=<\cdot ,\cdot {>}_x+2dudv+H(x,u)d{u}^2$, 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 $-H(x,u)\le R_1(u)\mathrm{\mid }x{\mathrm{\mid }}^{2-ϵ}$ for some $ϵ>0$ and large distance $\mathrm{\mid }x\mathrm{\mid }$ to a fixed point) and the spatial part $(M_0, <\cdot,\cdot>_x) $iscomplete,thenthespacetime$M$ is globally hyperbolic, (b) if $-H(x,u)$ grows at most quadratically (i.e, $-H(x,u) \leq R_1(u) |x|^{2}$forlarge$|x|$)thenitisstronglycausaland(c)$M$ is always causal, but there are non-distinguishing examples (and thus, non-strongly causal), even when $-H(x,u)\le R_1(u)\mathrm{\mid }x{\mathrm{\mid }}^{2+ϵ}$, for small $\epsilon

0$. Therefore, the classical model $M_0={\mathbb{R}}^2$, $H(x,u) = \sum_{i,j} h_{ij}(u) x_i x_j (\not\equiv 0)$, 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.