Manuscript Em01-B@3 · Em01-B-v3.tex · section · 69189992

Basis pairing and stress polarization

Frozen expression and complete contraction › Basis pairing and stress polarization · lines 560–601

Em01-B@3#eq:stress, Em01-B@3#eq:polarization, Em01-B@3#eq:Hessian, Em01-B@3#eq:C

Choose a basis $e_{I}$ of covariant symmetric tensors and its dual basis. All component contractions below use this basis–dual-basis pairing. Off-diagonal symmetric components therefore introduce no additional ad hoc factor of two.

For signature $+ - - -$ , define

T_{0, μν} [ϕ] T_{μν} (ϕ, ψ) = \partial_{μ} ϕ \partial_{ν} ϕ - \frac{1}{2} η_{μν} (\partial^{λ} ϕ \partial_{λ} ϕ - m^{2} ϕ^{2}), = \partial_{(μ} ϕ \partial_{ν)} ψ - \frac{1}{2} η_{μν} (\partial^{λ} ϕ \partial_{λ} ψ - m^{2} ϕ ψ) .

Then $T_{0} [ϕ] = T (ϕ, ϕ)$ , and differentiation of $T_{0}$ gives $2 T (ϕ, ψ)$ . Hence

V_{g}^{''}_{γ ϕ} [h, ψ] = - \int g (y) h^{μν} (y) T_{μν} (ϕ, ψ) (y) d^{4} y .

The grouped crossed distribution has components

C_{I}^{μν} (x_{1}, x_{2}) = H_{T, γ}^{μν}_{I} (x_{1}) \otimes H_{T, ϕ} (x_{2}) - H_{γ}^{μν}_{I} (x_{1}) \otimes H_{ϕ} (x_{2}) .

Both terms are defined tensor products. Their difference is retained as a single array. No mixed product or alternative covariance is introduced.

Displayed formulas in this passage

Em01-B@3#eq:stress · line 568
$T_{0, μν} [ϕ] T_{μν} (ϕ, ψ) = \partial_{μ} ϕ \partial_{ν} ϕ - \frac{1}{2} η_{μν} (\partial^{λ} ϕ \partial_{λ} ϕ - m^{2} ϕ^{2}), = \partial_{(μ} ϕ \partial_{ν)} ψ - \frac{1}{2} η_{μν} (\partial^{λ} ϕ \partial_{λ} ψ - m^{2} ϕ ψ) .$
Em01-B@3#eq:polarization · line 568
$T_{0, μν} [ϕ] T_{μν} (ϕ, ψ) = \partial_{μ} ϕ \partial_{ν} ϕ - \frac{1}{2} η_{μν} (\partial^{λ} ϕ \partial_{λ} ϕ - m^{2} ϕ^{2}), = \partial_{(μ} ϕ \partial_{ν)} ψ - \frac{1}{2} η_{μν} (\partial^{λ} ϕ \partial_{λ} ψ - m^{2} ϕ ψ) .$
Em01-B@3#eq:Hessian · line 582
$V_{g}^{''}_{γ ϕ} [h, ψ] = - \int g (y) h^{μν} (y) T_{μν} (ϕ, ψ) (y) d^{4} y .$
Em01-B@3#eq:C · line 590
$C_{I}^{μν} (x_{1}, x_{2}) = H_{T, γ}^{μν}_{I} (x_{1}) \otimes H_{T, ϕ} (x_{2}) - H_{γ}^{μν}_{I} (x_{1}) \otimes H_{ϕ} (x_{2}) .$