Manuscript II@2 · II.tex · section · 50d64def

Einstein coefficients and differentiation conventions

Locked kinematics, action, and contacts › Einstein coefficients and differentiation conventions · lines 465–506

II@2#eq:EH, II@2#eq:GammaGamma

Fix the curvature sign and normalization by

\frac{2}{κ ^{2}} \int d^{4} x - det (η + κh) R [η + κh] = S_{2} [h] + κ E_{3} [h] + κ^{2} E_{4} [h] + O (κ^{3}),

modulo total derivatives. Equivalently, expand the first-derivative density

\frac{2}{κ ^{2}} - g g^{μν} (Γ^{α}_{μ β} Γ^{β}_{ν α} - Γ^{α}_{μν} Γ^{β}_{α β}), g = η + κh .

For a homogeneous functional $E_{n}$ , put

E_{n}^{[n]} (u_{1}, \dots, u_{n}) = \frac{1}{n !} D^{n} E_{n} [0] (u_{1}, \dots, u_{n}),

v_{n} (u_{1}, \dots, u_{n}) = D^{n} E_{n} [0] (u_{1}, \dots, u_{n}) = n! E_{n}^{[n]} (u_{1}, \dots, u_{n}) .

Thus $v_{n}$ is exactly the coefficient of $z_{1} \dots z_{n}$ in $E_{n} [\sum_{i} z_{i} u_{i}]$ . No further factorial multiplies that coefficient.

Plane-wave formulas mean the coefficient after removing the common momentum-conservation delta. Integration by parts is interpreted algebraically at conserved vertex momentum. No convergent integral of four unlocalized plane waves is required.

Displayed formulas in this passage

II@2#eq:EH · line 468
$\frac{2}{κ ^{2}} \int d^{4} x - det (η + κh) R [η + κh] = S_{2} [h] + κ E_{3} [h] + κ^{2} E_{4} [h] + O (κ^{3}),$
II@2#eq:GammaGamma · line 477
$\frac{2}{κ ^{2}} - g g^{μν} (Γ^{α}_{μ β} Γ^{β}_{ν α} - Γ^{α}_{μν} Γ^{β}_{α β}), g = η + κh .$
unlabeled · line 488
$E_{n}^{[n]} (u_{1}, \dots, u_{n}) = \frac{1}{n !} D^{n} E_{n} [0] (u_{1}, \dots, u_{n}),$
unlabeled · line 492
$v_{n} (u_{1}, \dots, u_{n}) = D^{n} E_{n} [0] (u_{1}, \dots, u_{n}) = n! E_{n}^{[n]} (u_{1}, \dots, u_{n}) .$