Causality and conjugate points in general plane waves

Abstract

Let ℳ = ℳ0 × ℝ2 be a pp-wave-type spacetime endowed with the metric ⟨·, ·⟩z = ⟨·, ·⟩x + 2du dv + H(x, u) du2, where (ℳ0, ⟨·, ·⟩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 ℳ, say: (a) if −H(x, u) behaves subquadratically (i.e, essentially −H(x, u) ⩽ R1(u)|x|2−ϵ for some ϵ > 0 and large distance |x| to a fixed point) and the spatial part (ℳ0, ⟨·, ·⟩x) is complete, then the spacetime ℳ 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) ℳ is always causal, but there are non-distinguishing examples (and thus, not strongly causal), even when −H(x, u) ⩽ R1(u)|x|2+ϵ, for small ϵ > 0.Therefore, the classical model ℳ0 = ℝ2, 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 ℳ0. 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.