Manuscript I-B@1 · I-B-v1.tex · section · 5357688f

A compact rod-detection witness

Optional original-rod lapse attachment › A compact rod-detection witness · lines 1882–1913

I-B@1#eq:rod-witness-metric

The following local calculation reproduces the relevant distinction in XIX-A without using it as an unexamined premise CheungXIX (Theorem thm:rod-detection).

Let $a (t) > 0$ , and assume $H = a ˙ / a \neq = 0$ on a comparison interval. Choose a nonconstant $ψ \in C_{c}^{\infty} (R^{3}; R)$ and a nonzero smooth time bump $b$ supported strictly inside that interval. Set

A = b (t) ψ, D = - b (t) Δ ψ,

and consider the off-shell metric jets

q_{ε} = - (1 + 2 ε A) d t^{2} + a (t)^{2} (1 + 2 ε D) δ_{ij} d x^{i} d x^{j} (mod ε^{3}), T_{ε} = t .

Original rods are harmonic on the $t$ -slices with asymptotic Cartesian boundary condition. Modified rods are harmonic on the slices of

Θ_{ε} = t + ε D / H + O (ε^{2}),

with the same spatial boundary condition. The time label used in both relational lapse observables remains $t$ .

Displayed formulas in this passage

I-B@1#eq:rod-witness-metric · line 1896
$q_{ε} = - (1 + 2 ε A) d t^{2} + a (t)^{2} (1 + 2 ε D) δ_{ij} d x^{i} d x^{j} (mod ε^{3}), T_{ε} = t .$
unlabeled · line 1892
$A = b (t) ψ, D = - b (t) Δ ψ,$
unlabeled · line 1907
$Θ_{ε} = t + ε D / H + O (ε^{2}),$

A compact rod-detection witness

Displayed formulas in this passage

Nearby in this section