Manuscript VII-A@1 · VII-A.tex · section · 11da3fe6

Coupling expansion versus a scalar background

Restriction to a named Einstein–scalar coupling jet › Coupling expansion versus a scalar background · lines 1382–1436

VII-A@1#eq:ESaction, VII-A@1#eq:Einstein, VII-A@1#eq:Gexp, VII-A@1#eq:T0

For definiteness, normalize the unswitched Einstein–scalar action as

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

Its metric equation is

G_{μν} [g] = κ^{2} T_{μν} [g, ϕ] .

This fixes the normalization of $κ$ for this section.

The expansion is a coupling expansion. A nonzero free field $ϕ_{0}$ is allowed at order $κ^{0}$ because its gravitational source enters (eq:Einstein) at order $κ^{2}$ . It is not being identified with a nonzero background expectation value in a linearization about a different Einstein–scalar background.

Define the Einstein-tensor coefficients by

G_{μν} [η + κh + κ^{2} k] = κ G_{μν}^{(1)} [h] + κ^{2} (G_{μν}^{(1)} [k] + G_{μν}^{(2)} [h, h]) + O (κ^{3}) .

An explicit coefficient definition is given in Appendix app:curvature.

The leading stress tensor is

T_{μν}^{(0)} [ϕ_{0}] = \partial_{μ} ϕ_{0} \partial_{ν} ϕ_{0} - \frac{1}{2} η_{μν} (\partial_{ρ} ϕ_{0} \partial^{ρ} ϕ_{0} + m^{2} ϕ_{0}^{2}) .

Displayed formulas in this passage

VII-A@1#eq:ESaction · line 1385
$S_{ES} [g, ϕ] = \frac{1}{2 κ ^{2}} \int ρ_{g} R [g] d^{4} x - \frac{1}{2} \int ρ_{g} (g^{μν} \partial_{μ} ϕ \partial_{ν} ϕ + m^{2} ϕ^{2}) d^{4} x .$
VII-A@1#eq:Einstein · line 1396
$G_{μν} [g] = κ^{2} T_{μν} [g, ϕ] .$
VII-A@1#eq:Gexp · line 1412
$G_{μν} [η + κh + κ^{2} k] = κ G_{μν}^{(1)} [h] + κ^{2} (G_{μν}^{(1)} [k] + G_{μν}^{(2)} [h, h]) + O (κ^{3}) .$
VII-A@1#eq:T0 · line 1426
$T_{μν}^{(0)} [ϕ_{0}] = \partial_{μ} ϕ_{0} \partial_{ν} ϕ_{0} - \frac{1}{2} η_{μν} (\partial_{ρ} ϕ_{0} \partial^{ρ} ϕ_{0} + m^{2} ϕ_{0}^{2}) .$

Coupling expansion versus a scalar background

Displayed formulas in this passage

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