Manuscript X-D-ST@1 · X-D-ST.tex · section · e0cf8fdb

Coordinate equations

Frozen data, completed tests, and the topology sheet › Coordinate equations · lines 382–425

X-D-ST@1#eq:coordinate-prescription, X-D-ST@1#eq:coordinates

Let

X_{ε}^{ρ} = x^{ρ} + κ w_{ε}^{ρ} + κ^{2} v_{ε}^{ρ} .

The inherited divergence-form prescription is

\partial_{μ} ([η^{μν} + σ_{ε} (κ a^{μν} + κ^{2} A^{μν})] \partial_{ν} X_{ε}^{ρ}) = 0 (mod κ^{3}),

with flat coordinate Cauchy data at $t = 0$ . Equivalently,

□ w_{ε}^{ρ} □ v_{ε}^{ρ} w_{ε} (0) = - \partial_{μ} (σ_{ε} a^{μ ρ}), = - \partial_{μ} [σ_{ε} (A^{μ ρ} + a^{μν} \partial_{ν} w_{ε}^{ρ})], = \partial_{t} w_{ε} (0) = 0, v_{ε} (0) = \partial_{t} v_{ε} (0) = 0.

Here

□ = \partial_{t}^{2} - Δ_{x} .

Existence and uniqueness follow successively from the smooth flat Cauchy problem, locally by finite propagation when the backgrounds do not decay BGP. The switch $g$ does not occur in (eq:coordinates).

Displayed formulas in this passage

X-D-ST@1#eq:coordinate-prescription · line 391
$\partial_{μ} ([η^{μν} + σ_{ε} (κ a^{μν} + κ^{2} A^{μν})] \partial_{ν} X_{ε}^{ρ}) = 0 (mod κ^{3}),$
X-D-ST@1#eq:coordinates · line 402
$□ w_{ε}^{ρ} □ v_{ε}^{ρ} w_{ε} (0) = - \partial_{μ} (σ_{ε} a^{μ ρ}), = - \partial_{μ} [σ_{ε} (A^{μ ρ} + a^{μν} \partial_{ν} w_{ε}^{ρ})], = \partial_{t} w_{ε} (0) = 0, v_{ε} (0) = \partial_{t} v_{ε} (0) = 0.$
unlabeled · line 385
$X_{ε}^{ρ} = x^{ρ} + κ w_{ε}^{ρ} + κ^{2} v_{ε}^{ρ} .$
unlabeled · line 417
$□ = \partial_{t}^{2} - Δ_{x} .$