13.6 Geodesic completeness

The analysis described above concerned only the functions and their averages. In particular, no estimates for the derivatives in the sup norm were derived. However, in order to prove future causal geodesic completeness, it is of interest to have such estimates. According to [75Jump To The Next Citation Point, Proposition 1.8, p. 665]:

Proposition 6 Consider a solution to Equations (11View Equation)–(12View Equation). Then

∥Pt∥ 1 + ∥P𝜃∥ 1 + ∥eP Qt∥ 1 + ∥eP Q𝜃∥ 1 ≤ Kt− 1∕2. C(S ,ℝ) C (S ,ℝ) C (S ,ℝ) C (S ,ℝ)

Using this estimate, it is then possible to prove future causal geodesic completeness; see [75, Theorem 1.9, p. 665].

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