Manuscript X-A@1 · X-A.tex · section · dc7f5c0a

Metric jet and density coefficients

Convention sheet › Metric jet and density coefficients · lines 543–560

X-A@1#eq:b, X-A@1#eq:a

Write $h = η^{μν} h_{μν}$ for the trace where no confusion can arise, and set

b_{1} [h] a_{1}^{μν} [h] = \frac{1}{2} h, = \frac{1}{2} h η^{μν} - h^{μν}, b_{2} [h] a_{2}^{μν} [h] = \frac{1}{8} h^{2} - \frac{1}{4} h_{α β} h^{α β}, = h^{μ}_{α} h^{α ν} - \frac{1}{2} h h^{μν} + b_{2} [h] η^{μν} .

The symmetric bilinear polarizations $a_{2} [h, h^{'}]$ and $b_{2} [h, h^{'}]$ are defined by $a_{2} [h, h] = a_{2} [h]$ , $b_{2} [h, h] = b_{2} [h]$ .

Displayed formulas in this passage

X-A@1#eq:b · line 547
$b_{1} [h] a_{1}^{μν} [h] = \frac{1}{2} h, = \frac{1}{2} h η^{μν} - h^{μν}, b_{2} [h] a_{2}^{μν} [h] = \frac{1}{8} h^{2} - \frac{1}{4} h_{α β} h^{α β}, = h^{μ}_{α} h^{α ν} - \frac{1}{2} h h^{μν} + b_{2} [h] η^{μν} .$
X-A@1#eq:a · line 547
$b_{1} [h] a_{1}^{μν} [h] = \frac{1}{2} h, = \frac{1}{2} h η^{μν} - h^{μν}, b_{2} [h] a_{2}^{μν} [h] = \frac{1}{8} h^{2} - \frac{1}{4} h_{α β} h^{α β}, = h^{μ}_{α} h^{α ν} - \frac{1}{2} h h^{μν} + b_{2} [h] η^{μν} .$

Metric jet and density coefficients

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