Manuscript XIX-A@1 · XIX-A.tex · proposition · e30d8062

Proposition: A prescribed foliation can be encoded by an added scalar

Fourth refusal: incidence parameter and branch-normal speed › An important equality that does not erase the separation · lines 1350–1373 · proof included or linked

XIX-A@1#prop:foliation-clock

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Suppose $\partial_{s} F > 0$ on a regular patch. Define the local scalar $S_{F} (U, x)$ implicitly by $U = F (S_{F} (U, x), x)$ . Then its raw scalar-clock lapse equals the branch-normal speed:

N [q_{cone}, S_{F}] = L_{s}

at $U = F (s, x)$ .

Proof. Implicit differentiation gives

d S_{F} = \frac{d U - d _{x} F}{\partial _{s} F} .

Therefore

- q_{cone}^{- 1} (d S_{F}, d S_{F}) = \frac{1 - ∣ \nabla F ∣ ^{2}}{( \partial _{s} F ) ^{2}} .

Taking the positive inverse square root proves the equality.

Displayed formulas in this passage

unlabeled · line 1356
$N [q_{cone}, S_{F}] = L_{s}$
unlabeled · line 1364
$d S_{F} = \frac{d U - d _{x} F}{\partial _{s} F} .$
unlabeled · line 1368
$- q_{cone}^{- 1} (d S_{F}, d S_{F}) = \frac{1 - ∣ \nabla F ∣ ^{2}}{( \partial _{s} F ) ^{2}} .$

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section: An important equality that does not erase the separation