Manuscript XI-A@1 · XI-A.tex · section · b8211ecc

Exact metric derivative and its expansion

Metric Euler derivative through coupling order two › Exact metric derivative and its expansion · lines 709–726

For $ϕ = ϕ_{0} + κ ϕ_{1} + \dots$ , the order-one scalar source is exactly $- \frac{1}{2} g T_{0} [ϕ_{0}]$ . At order two, replace $J_{1} [γ, ϕ]$ in (eq:eexp) by

J_{1} [γ, ϕ_{0}] + T ˙_{0} [ϕ_{0}; ϕ_{1}],

where

T_{0} ˙^{μν} = p_{0}^{μ} p_{1}^{ν} + p_{1}^{μ} p_{0}^{ν} - η^{μν} (p_{0 α} p_{1}^{α} - m^{2} ϕ_{0} ϕ_{1}) .

This distinguishes the metric variation of the stress density from the correction due to the scalar solution.

Displayed formulas in this passage

unlabeled · line 714
$J_{1} [γ, ϕ_{0}] + T ˙_{0} [ϕ_{0}; ϕ_{1}],$
unlabeled · line 718
$T_{0} ˙^{μν} = p_{0}^{μ} p_{1}^{ν} + p_{1}^{μ} p_{0}^{ν} - η^{μν} (p_{0 α} p_{1}^{α} - m^{2} ϕ_{0} ϕ_{1}) .$

Nearby in this section

section: Exact metric derivative and its expansion
theorem: Theorem: Metric Euler coefficients