Manuscript I-B-VP-2@1 · I-B-VP-2-v1.tex · proposition · f4bd2b11

Proposition: The regular host is not globally hyperbolic

The regular host and the spectral covariance › Geometry, bundle, and non-global-hyperbolicity · lines 296–318 · proof included or linked

I-B-VP-2@1#prop:nonGH

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The function $t$ is temporal, but $M$ is not globally hyperbolic.

Proof. The inverse metric gives $g^{- 1} (d t, d t) = - 1$ . Choose $R > 0$ , $T_{0} > R$ , and fixed $φ_{0}, z_{0}$ . The points

p = (- T_{0}, R, φ_{0}, z_{0}), q = (T_{0}, R, φ_{0}, z_{0})

have a causal diamond containing

x_{m} = (0, R / m, φ_{0}, z_{0}), m \geq 2 :

the radial spatial travel length from either endpoint is less than $R < T_{0}$ . This sequence has no convergent subsequence in $M$ . Thus the diamond is not compact.

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unlabeled · line 306
$p = (- T_{0}, R, φ_{0}, z_{0}), q = (T_{0}, R, φ_{0}, z_{0})$
unlabeled · line 311
$x_{m} = (0, R / m, φ_{0}, z_{0}), m \geq 2 :$

Proposition: The regular host is not globally hyperbolic

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