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

Proposition: Density derivative

Finite source and density specifications › Quadratic inverse-metric density · lines 644–687 · proof included or linked

Em01-D@2#prop:density, Em01-D@2#eq:Da2

Its directional derivative is

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

If the designated $a ˙_{2} [G; h]$ is identified with this derivative, then

D^{μν, I, J} = D a_{2} [e_{J}] [e_{I}]^{μν} .

For every symmetric $h$ and scalar $ψ$ ,

a_{2} [ψ η] = 0, D a_{2} [ψ η] [h] = 0.

Proof. The inverse metric has coefficients $η^{μν}, - r^{μν}, r^{μ}_{λ} r^{λ ν}$ . The determinant square root has coefficients

1, \frac{1}{2} t_{r}, \frac{1}{8} t_{r}^{2} - \frac{1}{4} r : r .

Their product gives (eq:a2), and ordinary differentiation gives (eq:Da2).

For $G = ψ η$ , the first two terms in (eq:Da2) give $2 ψ h^{μν}$ ; the next terms give $- 2 ψ h^{μν} - ψ t_{h} η^{μν} /2$ , and the last term gives $ψ t_{h} η^{μν} /2$ . The sum is zero. The quadratic identity follows directly.

Displayed formulas in this passage

Em01-D@2#eq:Da2 · line 647
$D a_{2} [G] [h]^{μν} = h^{μ}_{λ} G^{λ ν} + G^{μ}_{λ} h^{λ ν} - \frac{1}{2} (t_{h} G^{μν} + t_{G} h^{μν}) + (\frac{1}{4} t_{G} t_{h} - \frac{1}{2} G : h) η^{μν} .$
unlabeled · line 659
$D^{μν, I, J} = D a_{2} [e_{J}] [e_{I}]^{μν} .$
unlabeled · line 663
$a_{2} [ψ η] = 0, D a_{2} [ψ η] [h] = 0.$
unlabeled · line 675
$1, \frac{1}{2} t_{r}, \frac{1}{8} t_{r}^{2} - \frac{1}{4} r : r .$

Proposition: Density derivative

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