Manuscript Em01-G@5 · Em01-G-v5.tex · lemma · 3fbfe951

Lemma: Source identities

Supported algebraic coefficients · lines 567–625 · proof included or linked

Em01-G@5#lem:source, Em01-G@5#eq:constant-slot, Em01-G@5#eq:mixed-divergence, Em01-G@5#eq:gauge-slot

The printed source obeys

q_{ν} Q (H, G; 0, q)^{μν} (p + q)_{ν} Q (H, G; p, q)^{μν} = 0, = - C_{H} (q)_{ν ρ}^{μ} T_{G} (p)^{ν ρ} - C_{G} (p)_{ν ρ}^{μ} T_{H} (q)^{ν ρ} .

Also

Q (H, G; 0, q) = 0 if H = q \otimes t + t \otimes q .

Proof. Consider the raised Einstein density

J (g)^{μν} = 2 - det g g^{μ ρ} g^{ν σ} (R (g)_{ρ σ} - \frac{1}{2} g_{ρ σ} g^{α β} R (g)_{α β}) .

The mixed coefficient of $J (η + sH + tG)$ , lowered with $η$ , is (eq:source). The connection and Ricci mixed coefficients are $D, U$ ; the scalar and Einstein coefficients are $s, E$ . The determinant derivative supplies the trace terms, and the two inverse-metric derivatives supply the four index-conversion terms.

The density Bianchi identity

\partial_{ν} J (g)^{μν} + Γ (g)_{ν ρ}^{μ} J (g)^{ν ρ} = 0

gives (eq:mixed-divergence) at mixed order. At $p = 0$ , its right side vanishes.

For a direct constant-background check, the linearized Ricci symbol at $g_{t} = η + tG$ is

\frac{1}{2} [q_{μ} q_{g_{t}}^{α} H_{α ν} + q_{ν} q_{g_{t}}^{α} H_{α μ} - q_{g_{t}}^{2} H_{μν} - q_{μ} q_{ν} tr_{g_{t}} H] .

Its Einstein contraction is transverse. For $H = q \otimes t + t \otimes q$ , the symbol is zero at every $g_{t}$ , as is seen by substitution; the corresponding vector amplitude is $g_{t}^{- 1} t$ . Differentiate the raised density in $t$ to obtain (eq:constant-slot) and (eq:gauge-slot).

Displayed formulas in this passage

Em01-G@5#eq:constant-slot · line 570
$q_{ν} Q (H, G; 0, q)^{μν} (p + q)_{ν} Q (H, G; p, q)^{μν} = 0, = - C_{H} (q)_{ν ρ}^{μ} T_{G} (p)^{ν ρ} - C_{G} (p)_{ν ρ}^{μ} T_{H} (q)^{ν ρ} .$
Em01-G@5#eq:mixed-divergence · line 570
$q_{ν} Q (H, G; 0, q)^{μν} (p + q)_{ν} Q (H, G; p, q)^{μν} = 0, = - C_{H} (q)_{ν ρ}^{μ} T_{G} (p)^{ν ρ} - C_{G} (p)_{ν ρ}^{μ} T_{H} (q)^{ν ρ} .$
Em01-G@5#eq:gauge-slot · line 579
$Q (H, G; 0, q) = 0 if H = q \otimes t + t \otimes q .$
unlabeled · line 588
$J (g)^{μν} = 2 - det g g^{μ ρ} g^{ν σ} (R (g)_{ρ σ} - \frac{1}{2} g_{ρ σ} g^{α β} R (g)_{α β}) .$
unlabeled · line 603
$\partial_{ν} J (g)^{μν} + Γ (g)_{ν ρ}^{μ} J (g)^{ν ρ} = 0$
unlabeled · line 612
$\frac{1}{2} [q_{μ} q_{g_{t}}^{α} H_{α ν} + q_{ν} q_{g_{t}}^{α} H_{α μ} - q_{g_{t}}^{2} H_{μν} - q_{μ} q_{ν} tr_{g_{t}} H] .$

Lemma: Source identities

Displayed formulas in this passage

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