Manuscript XV-F-patch-ii@1 · XV-F-patch-ii.tex · lemma · 9674ac67

Lemma: The conditional value of

Patch 3: coefficient instantiation (Q3) › The quadratic coefficient, a 2 , and · lines 1180–1215 · proof included or linked

XV-F-patch-ii@1#lem:dformula, XV-F-patch-ii@1#eq:a2dot, XV-F-patch-ii@1#eq:dtensor

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If XIV-H's $a_{2}$ is (eq:a2), then

a_{2} ˙^{μν} [G; h] = h^{μ}_{α} G^{α ν} + G^{μ}_{α} h^{α ν} - \frac{1}{2} (tr_{η} h) G^{μν} - \frac{1}{2} (tr_{η} G) h^{μν} + (\frac{1}{4} (tr_{η} G) (tr_{η} h) - \frac{1}{2} G_{α β} h^{α β}) η^{μν},

and by (eq:Dfrom), with $I = (c d)$ for $h$ and $J = (ab)$ for $G$ ,

D^{μν, (c d), (ab)} = d^{μν, ab, c d} := η^{μ (c} η^{d) (a} η^{b) ν} + η^{μ (a} η^{b) (c} η^{d) ν} - \frac{1}{2} η^{c d} η^{μ (a} η^{b) ν} - \frac{1}{2} η^{ab} η^{μ (c} η^{d) ν} + (\frac{1}{4} η^{ab} η^{c d} - \frac{1}{2} η^{a (c} η^{d) b}) η^{μν},

symmetric in $(μν)$ , in $(ab)$ , in $(c d)$ , and under $(ab) \leftrightarrow (c d)$ .

Proof. Differentiate (eq:a2) along $G + s h$ ; the derivative of $b_{2}$ is $\frac{1}{4} tr G tr h - \frac{1}{2} G_{α β} h^{α β}$ . Writing each term of (eq:a2dot) as a contraction of $G_{ab} h_{c d}$ with constant $η$ 's and symmetrizing over the index pairs, which is harmless because $G$ , $h$ are symmetric, gives (eq:dtensor); the exchange symmetry is Lemma lem:quadratic.

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XV-F-patch-ii@1#eq:a2dot · line 1183
$a_{2} ˙^{μν} [G; h] = h^{μ}_{α} G^{α ν} + G^{μ}_{α} h^{α ν} - \frac{1}{2} (tr_{η} h) G^{μν} - \frac{1}{2} (tr_{η} G) h^{μν} + (\frac{1}{4} (tr_{η} G) (tr_{η} h) - \frac{1}{2} G_{α β} h^{α β}) η^{μν},$
XV-F-patch-ii@1#eq:dtensor · line 1194
$D^{μν, (c d), (ab)} = d^{μν, ab, c d} := η^{μ (c} η^{d) (a} η^{b) ν} + η^{μ (a} η^{b) (c} η^{d) ν} - \frac{1}{2} η^{c d} η^{μ (a} η^{b) ν} - \frac{1}{2} η^{ab} η^{μ (c} η^{d) ν} + (\frac{1}{4} η^{ab} η^{c d} - \frac{1}{2} η^{a (c} η^{d) b}) η^{μν},$

Lemma: The conditional value of

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