Manuscript Em01-F@3 · Em01-F-v3.tex · theorem · 225e59af

Theorem: Evaluated mixed source and coefficient rule

Explicit cubic density and source audit › Mixed curvature calculation · lines 1367–1431 · proof included or linked

Em01-F@3#thm:source, Em01-F@3#eq:Qevaluated, Em01-F@3#eq:qnormalform, Em01-F@3#eq:q-extraction

The mixed source is

Q (h, G)_{μν} = 2 [E (h, G)_{μν} + \frac{1}{2} (tr_{η} h) G_{1} (G)_{μν} + \frac{1}{2} (tr_{η} G) G_{1} (h)_{μν} - h_{μ}^{α} G_{1} (G)_{α ν} - h_{ν}^{α} G_{1} (G)_{μα} - G_{μ}^{α} G_{1} (h)_{α ν} - G_{ν}^{α} G_{1} (h)_{μα}] .

It is symmetric and has total derivative order at most two.

Every constant coefficient in

(Q (h, G))^{L} = α, β, I, J \sum q_{α β}^{L, I, J} (\partial^{α} G_{J}) (\partial^{β} h_{I})

is determined by

q_{α β}^{L, I, J} = [p^{α} k^{β}] (e^{L})^{μν} e^{- (k + p) \cdot x} Q (e_{I} e^{k \cdot x}, e_{J} e^{p \cdot x})_{μν} .

The dot in this formal generating formula is the coordinate covector pairing. The expression is independent of $x$ . The coefficients vanish for $∣ α ∣ + ∣ β ∣ > 2$ .

Proof. Expand the connection of $η + s h + tG$ . Its linear terms are $s C (h) + tC (G)$ , and its mixed coefficient is $s t D (h, G)$ . Its mixed Ricci, scalar-curvature, and Einstein coefficients are exactly (eq:mixedRicci)–(eq:mixedEinstein).

The determinant supplies the two trace terms, and the two inverse metrics in the lowered Euler density supply the four index conversion terms in (eq:Qevaluated). The Euler prefactor $2/ λ$ supplies the factor two.

Alternatively, differentiate (eq:J2evaluated) at $G$ in direction $h$ . Its $V_{2}$ derivative is $E (h, G)$ ; the derivatives of $b_{1} V_{1}$ and of the index-conversion terms are the remaining six terms. The density calculation and the mixed curvature calculation therefore agree.

All resulting terms are products of jets with total derivative order at most two. Exponential substitution converts them into the independent monomials $p^{α} k^{β}$ , so coefficient extraction determines every coefficient uniquely.

Displayed formulas in this passage

Em01-F@3#eq:Qevaluated · line 1370
$Q (h, G)_{μν} = 2 [E (h, G)_{μν} + \frac{1}{2} (tr_{η} h) G_{1} (G)_{μν} + \frac{1}{2} (tr_{η} G) G_{1} (h)_{μν} - h_{μ}^{α} G_{1} (G)_{α ν} - h_{ν}^{α} G_{1} (G)_{μα} - G_{μ}^{α} G_{1} (h)_{α ν} - G_{ν}^{α} G_{1} (h)_{μα}] .$
Em01-F@3#eq:qnormalform · line 1387
$(Q (h, G))^{L} = α, β, I, J \sum q_{α β}^{L, I, J} (\partial^{α} G_{J}) (\partial^{β} h_{I})$
Em01-F@3#eq:q-extraction · line 1396
$q_{α β}^{L, I, J} = [p^{α} k^{β}] (e^{L})^{μν} e^{- (k + p) \cdot x} Q (e_{I} e^{k \cdot x}, e_{J} e^{p \cdot x})_{μν} .$

Theorem: Evaluated mixed source and coefficient rule

Displayed formulas in this passage

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