Manuscript X-A-conv@1 · X-A-conv.tex · section · c76fffa3

A bilinear notation for the second coordinate coefficient

Relational expansion and retained infinitesimal defects › A bilinear notation for the second coordinate coefficient · lines 1216–1257

X-A-conv@1#eq:a2polar, X-A-conv@1#eq:Vbil, X-A-conv@1#eq:vbil

Let $a_{2} (H_{1}, H_{2})$ denote the symmetric polarization of $a_{2} (H)$ . Explicitly,

a_{2} (H_{1}, H_{2})^{μν} = \frac{1}{2} (H_{1}^{μ}_{ρ} H_{2}^{ρ ν} + H_{2}^{μ}_{ρ} H_{1}^{ρ ν}) - \frac{1}{4} ((tr H_{1}) H_{2}^{μν} + (tr H_{2}) H_{1}^{μν}) + (\frac{1}{8} (tr H_{1}) (tr H_{2}) - \frac{1}{4} H_{1, ρ σ} H_{2}^{ρ σ}) η^{μν} .

Set $W (H) = E_{0} C H$ , and define

V^{μ} (H_{1}, H_{2}) = - E_{0} [\partial_{α} a_{2} (H_{1}, H_{2})^{α μ} + \frac{1}{2} \partial_{α} (a_{1} (H_{1})^{α β} \partial_{β} W (H_{2})^{μ} + a_{1} (H_{2})^{α β} \partial_{β} W (H_{1})^{μ})] .

Then

w = W (H), v = W (J) + V (H, H) .

Displayed formulas in this passage

X-A-conv@1#eq:a2polar · line 1220
$a_{2} (H_{1}, H_{2})^{μν} = \frac{1}{2} (H_{1}^{μ}_{ρ} H_{2}^{ρ ν} + H_{2}^{μ}_{ρ} H_{1}^{ρ ν}) - \frac{1}{4} ((tr H_{1}) H_{2}^{μν} + (tr H_{2}) H_{1}^{μν}) + (\frac{1}{8} (tr H_{1}) (tr H_{2}) - \frac{1}{4} H_{1, ρ σ} H_{2}^{ρ σ}) η^{μν} .$
X-A-conv@1#eq:Vbil · line 1240
$V^{μ} (H_{1}, H_{2}) = - E_{0} [\partial_{α} a_{2} (H_{1}, H_{2})^{α μ} + \frac{1}{2} \partial_{α} (a_{1} (H_{1})^{α β} \partial_{β} W (H_{2})^{μ} + a_{1} (H_{2})^{α β} \partial_{β} W (H_{1})^{μ})] .$
X-A-conv@1#eq:vbil · line 1253
$w = W (H), v = W (J) + V (H, H) .$