Manuscript V@1 · V.tex · section · e7ddf37d

One response, two notations

Sign, initial-data, and Duhamel audit › One response, two notations · lines 3858–3916

All body formulas use

P = - (\partial_{t}^{2} + Ω_{m}^{2}), L_{h} = - h^{μν} \partial_{μ} \partial_{ν} - D (h)^{ν} \partial_{ν}, P S = - L_{h} ϕ .

L_{h}^{+} = h^{μν} \partial_{μ} \partial_{ν} + D (h)^{ν} \partial_{ν},

then $L_{h}^{+} = - L_{h}$ , so the alternative notation $P S = L_{h}^{+} ϕ$ has exactly the same right-hand side. The two expressions

- \int_{t_{0}}^{t} \frac{sin (( t - s ) Ω _{m} )}{Ω _{m}} L_{h}^{+} ϕ (s) d s, \int_{t_{0}}^{t} \frac{sin (( t - s ) Ω _{m} )}{Ω _{m}} L_{h} ϕ (s) d s

are identical. An extra sign reversal of $J_{raw}$ would be an error.

An actual change from $P S = - L_{h} ϕ$ to $P S = + L_{h} ϕ$ , while keeping $L_{h}$ fixed, would negate the response, the current, and the defect. It would leave the zero conditions $H = 0$ and $E_{1} H = 0$ unchanged.

For $u_{f} = - G_{A} f$ , the raw adjoint identity is

B_{f} (h, ϕ) = \int_{t_{0}}^{T} u_{f} L_{h} ϕ .

Integrating the metric derivative by parts gives

J_{raw}^{μν} = \partial^{(μ} (Θ_{t_{0}} u_{f}) \partial^{ν)} ϕ - \frac{1}{2} η^{μν} \partial_{α} (Θ_{t_{0}} u_{f} \partial^{α} ϕ) .

Its initial-surface components are

j_{Σ}^{00} = \frac{1}{2} u_{0} ϕ ˙_{0}, j_{Σ}^{0 i} = - \frac{1}{2} u_{0} \partial_{i} ϕ_{0}, j_{Σ}^{ij} = \frac{1}{2} δ^{ij} u_{0} ϕ ˙_{0} .

The canonical distribution formula is

P (Θ_{t_{0}} u_{f}) = - f - δ (t - t_{0}) u ˙_{0} - δ^{'} (t - t_{0}) u_{0} .

There is only one $u ˙_{0}$ in the displayed delta coefficient. An intermediate expression with $2 u ˙ δ$ must also expand the product $u (t) δ^{'}$ , which removes one copy.

Figure fig:sign-audit summarizes the conventions audited in this appendix.

Displayed formulas in this passage

unlabeled · line 3861
$P = - (\partial_{t}^{2} + Ω_{m}^{2}), L_{h} = - h^{μν} \partial_{μ} \partial_{ν} - D (h)^{ν} \partial_{ν}, P S = - L_{h} ϕ .$
unlabeled · line 3867
$L_{h}^{+} = h^{μν} \partial_{μ} \partial_{ν} + D (h)^{ν} \partial_{ν},$
unlabeled · line 3873
$- \int_{t_{0}}^{t} \frac{sin (( t - s ) Ω _{m} )}{Ω _{m}} L_{h}^{+} ϕ (s) d s, \int_{t_{0}}^{t} \frac{sin (( t - s ) Ω _{m} )}{Ω _{m}} L_{h} ϕ (s) d s$
unlabeled · line 3887
$B_{f} (h, ϕ) = \int_{t_{0}}^{T} u_{f} L_{h} ϕ .$
unlabeled · line 3891
$J_{raw}^{μν} = \partial^{(μ} (Θ_{t_{0}} u_{f}) \partial^{ν)} ϕ - \frac{1}{2} η^{μν} \partial_{α} (Θ_{t_{0}} u_{f} \partial^{α} ϕ) .$
unlabeled · line 3898
$j_{Σ}^{00} = \frac{1}{2} u_{0} ϕ ˙_{0}, j_{Σ}^{0 i} = - \frac{1}{2} u_{0} \partial_{i} ϕ_{0}, j_{Σ}^{ij} = \frac{1}{2} δ^{ij} u_{0} ϕ ˙_{0} .$
unlabeled · line 3904
$P (Θ_{t_{0}} u_{f}) = - f - δ (t - t_{0}) u ˙_{0} - δ^{'} (t - t_{0}) u_{0} .$