Causality and Conjugate Points in General Plane Waves

Details Causality and Conjugate Points in General Plane Waves Causality and Conjugate Points in General Plane Waves

2002-11-26

arxiv.org/abs/gr-qc/0211086v2

Class.Quant.Grav. 20 (2003) 2275-2292

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

Version improved with further discussions and a new section; 21 pages, Latex

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Causality and Conjugate Points in General Plane Waves does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Causality and Conjugate Points in General Plane Waves, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Causality and Conjugate Points in General Plane Waves will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-172530