Manuscript I-B-G@2 · I-B-G-v2.tex · proposition · 3b4d700f

Proposition: Failure of global hyperbolicity

The prescribed host and its causal geometry › Distance and the causal relation · lines 353–372 · proof included or linked

I-B-G@2#prop:nonGH

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The spacetime $M$ is not globally hyperbolic.

Proof. Fix $r_{*} > 0$ , $φ_{*}$ , $z_{*}$ , and $a > r_{*}$ . The points

p = (- a, r_{*}, φ_{*}, z_{*}), q = (a, r_{*}, φ_{*}, z_{*})

have in their causal diamond the sequence

p_{j} = (0, r_{j}, φ_{*}, z_{*}), r_{j} ↓ 0.

Indeed, the required radial travel length is $r_{*} - r_{j} < a$ . The sequence has no convergent subsequence in $M$ . Thus this causal diamond is not compact.

Displayed formulas in this passage

unlabeled · line 361
$p = (- a, r_{*}, φ_{*}, z_{*}), q = (a, r_{*}, φ_{*}, z_{*})$
unlabeled · line 366
$p_{j} = (0, r_{j}, φ_{*}, z_{*}), r_{j} ↓ 0.$

Nearby in this section

section: Distance and the causal relation
proposition: Proposition: Intrinsic distance and causality
remark: Remark: Local causal certificate versus global proof
section: Distance and the causal relation