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

Lemma: The background scalar clock is not a harmonic time coordinate

The scalar-clock lapse and the original spatial rods › The cosmological comparison background · lines 546–562 · proof included or linked

XIX-A@1#lem:clock-not-harmonic

On (eq:background),

□_{q_{0}} T_{0} = - T ¨_{0} - 3 H T ˙_{0} = - \frac{( 3 p - 1 ) 2 p M _{Pl}}{t ^{2}} \neq = 0.

In particular, the inflaton clock is not a spacetime-harmonic coordinate component on this background.

Proof. For a homogeneous scalar, $□_{q_{0}} u = - u ¨ - 3 H u ˙$ . Substitution of (eq:background) gives the formula.

Displayed formulas in this passage

unlabeled · line 549
$□_{q_{0}} T_{0} = - T ¨_{0} - 3 H T ˙_{0} = - \frac{( 3 p - 1 ) 2 p M _{Pl}}{t ^{2}} \neq = 0.$