Manuscript XX-B@1 · XX-B.tex · section · 56d708a5

The complete second-order classical field equations

The complete second-order classical field equations · lines 577–672

XX-B@1#eq:J, XX-B@1#eq:C, XX-B@1#eq:R2, XX-B@1#eq:Rscalar2, XX-B@1#eq:G2, XX-B@1#eq:U, XX-B@1#eq:T2, XX-B@1#eq:A, XX-B@1#eq:scalar2

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This section supplies a finite coefficient algorithm for the quadratic Einstein–scalar terms. It also fixes what “through quadratic order” means. No nonlinear differential operator is being assigned a linear Green inverse.

Define

L (h)_{σ μν} = \nabla_{μ} h_{ν σ} + \nabla_{ν} h_{μ σ} - \nabla_{σ} h_{μν} .

For the jet (eq:jet), the inverse metric coefficients are

J_{0}^{μν} = g ˉ^{μν}, J_{1}^{μν} = - h^{μν}, J_{2}^{μν} = 2 h^{μ}_{ρ} h^{ρ ν} - k^{μν} .

The connection difference coefficients are

C_{1}^{ρ}_{μν} = \frac{1}{2} g ˉ^{ρ σ} L (h)_{σ μν}, C_{2}^{ρ}_{μν} = \frac{1}{2} g ˉ^{ρ σ} L (k)_{σ μν} - h^{ρ σ} L (h)_{σ μν} .

Thus

(R_{2})_{μν} R_{2} (G_{2})_{μν} = \nabla_{ρ} C_{2}^{ρ}_{μν} - \nabla_{ν} C_{2}^{ρ}_{μ ρ} + 2 C_{1}^{ρ}_{ρ λ} C_{1}^{λ}_{μν} - 2 C_{1}^{ρ}_{ν λ} C_{1}^{λ}_{μ ρ}, = J_{2}^{μν} R ˉ_{μν} + 2 J_{1}^{μν} (R_{1})_{μν} + J_{0}^{μν} (R_{2})_{μν}, = (R_{2})_{μν} - \frac{1}{2} k_{μν} R ˉ - h_{μν} R_{1} - \frac{1}{2} g ˉ_{μν} R_{2} .

Put $a_{μ} = \partial_{μ} φ$ , $b_{μ} = \partial_{μ} ψ$ . For $U_{s} = g_{s}^{- 1} (d ϕ_{s}, d ϕ_{s})$ ,

U_{0} U_{1} U_{2} = J_{0}^{μν} v_{μ} v_{ν}, = J_{1}^{μν} v_{μ} v_{ν} + 2 J_{0}^{μν} v_{μ} a_{ν}, = J_{2}^{μν} v_{μ} v_{ν} + 4 J_{1}^{μν} v_{μ} a_{ν} + 2 J_{0}^{μν} a_{μ} a_{ν} + 2 J_{0}^{μν} v_{μ} b_{ν} .

Write $V_{0} = V (ϕ ˉ)$ , $V_{1} = V^{'} φ$ , and $V_{2} = V^{'} ψ + V^{''} φ^{2}$ . Then

(T_{2})_{μν} = v_{μ} b_{ν} + b_{μ} v_{ν} + 2 a_{μ} a_{ν} - \frac{1}{2} k_{μν} (U_{0} + 2 V_{0}) - h_{μν} (U_{1} + 2 V_{1}) - \frac{1}{2} g ˉ_{μν} (U_{2} + 2 V_{2}) .

For the scalar equation, define

A_{0, μν} A_{1, μν} A_{2, μν} = \nabla_{μ} v_{ν}, = \nabla_{μ} a_{ν} - C_{1}^{ρ}_{μν} v_{ρ}, = \nabla_{μ} b_{ν} - C_{2}^{ρ}_{μν} v_{ρ} - 2 C_{1}^{ρ}_{μν} a_{ρ} .

Its complete second coefficient is

(E_{ϕ})_{2} = J_{2}^{μν} A_{0, μν} + 2 J_{1}^{μν} A_{1, μν} + J_{0}^{μν} A_{2, μν} - V^{''} ψ - V^{'''} φ^{2} .

Displayed formulas in this passage

XX-B@1#eq:J · line 593
$J_{0}^{μν} = g ˉ^{μν}, J_{1}^{μν} = - h^{μν}, J_{2}^{μν} = 2 h^{μ}_{ρ} h^{ρ ν} - k^{μν} .$
XX-B@1#eq:C · line 600
$C_{1}^{ρ}_{μν} = \frac{1}{2} g ˉ^{ρ σ} L (h)_{σ μν}, C_{2}^{ρ}_{μν} = \frac{1}{2} g ˉ^{ρ σ} L (k)_{σ μν} - h^{ρ σ} L (h)_{σ μν} .$
XX-B@1#eq:R2 · line 610
$(R_{2})_{μν} R_{2} (G_{2})_{μν} = \nabla_{ρ} C_{2}^{ρ}_{μν} - \nabla_{ν} C_{2}^{ρ}_{μ ρ} + 2 C_{1}^{ρ}_{ρ λ} C_{1}^{λ}_{μν} - 2 C_{1}^{ρ}_{ν λ} C_{1}^{λ}_{μ ρ}, = J_{2}^{μν} R ˉ_{μν} + 2 J_{1}^{μν} (R_{1})_{μν} + J_{0}^{μν} (R_{2})_{μν}, = (R_{2})_{μν} - \frac{1}{2} k_{μν} R ˉ - h_{μν} R_{1} - \frac{1}{2} g ˉ_{μν} R_{2} .$
XX-B@1#eq:Rscalar2 · line 610
$(R_{2})_{μν} R_{2} (G_{2})_{μν} = \nabla_{ρ} C_{2}^{ρ}_{μν} - \nabla_{ν} C_{2}^{ρ}_{μ ρ} + 2 C_{1}^{ρ}_{ρ λ} C_{1}^{λ}_{μν} - 2 C_{1}^{ρ}_{ν λ} C_{1}^{λ}_{μ ρ}, = J_{2}^{μν} R ˉ_{μν} + 2 J_{1}^{μν} (R_{1})_{μν} + J_{0}^{μν} (R_{2})_{μν}, = (R_{2})_{μν} - \frac{1}{2} k_{μν} R ˉ - h_{μν} R_{1} - \frac{1}{2} g ˉ_{μν} R_{2} .$
XX-B@1#eq:G2 · line 610
$(R_{2})_{μν} R_{2} (G_{2})_{μν} = \nabla_{ρ} C_{2}^{ρ}_{μν} - \nabla_{ν} C_{2}^{ρ}_{μ ρ} + 2 C_{1}^{ρ}_{ρ λ} C_{1}^{λ}_{μν} - 2 C_{1}^{ρ}_{ν λ} C_{1}^{λ}_{μ ρ}, = J_{2}^{μν} R ˉ_{μν} + 2 J_{1}^{μν} (R_{1})_{μν} + J_{0}^{μν} (R_{2})_{μν}, = (R_{2})_{μν} - \frac{1}{2} k_{μν} R ˉ - h_{μν} R_{1} - \frac{1}{2} g ˉ_{μν} R_{2} .$
XX-B@1#eq:U · line 633
$U_{0} U_{1} U_{2} = J_{0}^{μν} v_{μ} v_{ν}, = J_{1}^{μν} v_{μ} v_{ν} + 2 J_{0}^{μν} v_{μ} a_{ν}, = J_{2}^{μν} v_{μ} v_{ν} + 4 J_{1}^{μν} v_{μ} a_{ν} + 2 J_{0}^{μν} a_{μ} a_{ν} + 2 J_{0}^{μν} v_{μ} b_{ν} .$
XX-B@1#eq:T2 · line 645
$(T_{2})_{μν} = v_{μ} b_{ν} + b_{μ} v_{ν} + 2 a_{μ} a_{ν} - \frac{1}{2} k_{μν} (U_{0} + 2 V_{0}) - h_{μν} (U_{1} + 2 V_{1}) - \frac{1}{2} g ˉ_{μν} (U_{2} + 2 V_{2}) .$
XX-B@1#eq:A · line 655
$A_{0, μν} A_{1, μν} A_{2, μν} = \nabla_{μ} v_{ν}, = \nabla_{μ} a_{ν} - C_{1}^{ρ}_{μν} v_{ρ}, = \nabla_{μ} b_{ν} - C_{2}^{ρ}_{μν} v_{ρ} - 2 C_{1}^{ρ}_{μν} a_{ρ} .$
XX-B@1#eq:scalar2 · line 664
$(E_{ϕ})_{2} = J_{2}^{μν} A_{0, μν} + 2 J_{1}^{μν} A_{1, μν} + J_{0}^{μν} A_{2, μν} - V^{''} ψ - V^{'''} φ^{2} .$
unlabeled · line 586
$L (h)_{σ μν} = \nabla_{μ} h_{ν σ} + \nabla_{ν} h_{μ σ} - \nabla_{σ} h_{μν} .$

The complete second-order classical field equations

Displayed formulas in this passage

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