Manuscript Em01-D@2 · Em01-D-v2.tex · section · d3b49151

Quadratic inverse-metric density

Finite source and density specifications › Quadratic inverse-metric density · lines 625–643

Em01-D@2#eq:a2

Write $t_{r} = tr_{η} r$ and $G : h = G_{μν} h^{μν}$ . For the declared density

- det (η + sr) (η + sr)^{μν} = η^{μν} + s a_{1}^{μν} [r] + s^{2} a_{2}^{μν} [r] + O (s^{3}),

the quadratic coefficient is

a_{2}^{μν} [r] = r^{μ}_{λ} r^{λ ν} - \frac{1}{2} t_{r} r^{μν} + (\frac{1}{8} t_{r}^{2} - \frac{1}{4} r : r) η^{μν} .

Displayed formulas in this passage

Em01-D@2#eq:a2 · line 635
$a_{2}^{μν} [r] = r^{μ}_{λ} r^{λ ν} - \frac{1}{2} t_{r} r^{μν} + (\frac{1}{8} t_{r}^{2} - \frac{1}{4} r : r) η^{μν} .$
unlabeled · line 629
$- det (η + sr) (η + sr)^{μν} = η^{μν} + s a_{1}^{μν} [r] + s^{2} a_{2}^{μν} [r] + O (s^{3}),$