Manuscript VI-B@1 · VI-B-v1.tex · section · 33798723

Normalization and the scalar second variation

The second-order Einstein–scalar coefficient space › Normalization and the scalar second variation · lines 482–539

VI-B@1#eq:action, VI-B@1#eq:equations, VI-B@1#eq:expansion

Use the action convention

S [γ, φ] = \frac{2}{κ ^{2}} \int - γ R [γ] d^{4} x - \frac{1}{2} \int - γ (γ^{μν} \partial_{μ} φ \partial_{ν} φ + m^{2} φ^{2}) d^{4} x .

The Ricci convention is the one yielding (eq:G1). Compactly supported inverse-metric variation gives

G_{μν} (γ) = \frac{κ ^{2}}{4} T_{μν} (γ, φ), (□_{γ} - m^{2}) φ = 0,

where

T_{μν} (γ, φ) = \partial_{μ} φ \partial_{ν} φ - \frac{1}{2} γ_{μν} (γ^{ρ σ} \partial_{ρ} φ \partial_{σ} φ + m^{2} φ^{2}) .

Indeed, the coefficients of the metric variation are $2 G_{μν} / κ^{2} - T_{μν} /2$ . Thus the source factor $1/4$ is fixed, not freely suppressed.

Work modulo $κ^{3}$ , without factorials:

γ = η + κh + κ^{2} j, φ = ϕ_{0} + κ ϕ_{1} + κ^{2} ϕ_{2} .

Define $G^{(2)} [h, h]$ by

G (γ) = κ G^{(1)} (h) + κ^{2} (G^{(1)} (j) + G^{(2)} [h, h]) (mod κ^{3}) .

An explicit differential specification of $G^{(2)}$ appears in Appendix app:G2.

Let

d_{h} = \frac{1}{2} tr_{η} h, e_{h} = \frac{1}{8} (tr_{η} h)^{2} - \frac{1}{4} h_{μν} h^{μν},

A_{h}^{μν} = d_{h} η^{μν} - h^{μν}, C_{h}^{μν} = h^{μ}_{ρ} h^{ρ ν} - d_{h} h^{μν} + e_{h} η^{μν} .

Displayed formulas in this passage

VI-B@1#eq:action · line 485
$S [γ, φ] = \frac{2}{κ ^{2}} \int - γ R [γ] d^{4} x - \frac{1}{2} \int - γ (γ^{μν} \partial_{μ} φ \partial_{ν} φ + m^{2} φ^{2}) d^{4} x .$
VI-B@1#eq:equations · line 495
$G_{μν} (γ) = \frac{κ ^{2}}{4} T_{μν} (γ, φ), (□_{γ} - m^{2}) φ = 0,$
VI-B@1#eq:expansion · line 514
$γ = η + κh + κ^{2} j, φ = ϕ_{0} + κ ϕ_{1} + κ^{2} ϕ_{2} .$
unlabeled · line 502
$T_{μν} (γ, φ) = \partial_{μ} φ \partial_{ν} φ - \frac{1}{2} γ_{μν} (γ^{ρ σ} \partial_{ρ} φ \partial_{σ} φ + m^{2} φ^{2}) .$
unlabeled · line 520
$G (γ) = κ G^{(1)} (h) + κ^{2} (G^{(1)} (j) + G^{(2)} [h, h]) (mod κ^{3}) .$
unlabeled · line 530
$d_{h} = \frac{1}{2} tr_{η} h, e_{h} = \frac{1}{8} (tr_{η} h)^{2} - \frac{1}{4} h_{μν} h^{μν},$
unlabeled · line 534
$A_{h}^{μν} = d_{h} η^{μν} - h^{μν}, C_{h}^{μν} = h^{μ}_{ρ} h^{ρ ν} - d_{h} h^{μν} + e_{h} η^{μν} .$