Manuscript V-B@1 · V-B-v1.tex · section · 3fadc1ad

Operators and the fixed-time response

Operators and the fixed-time response · lines 314–347

V-B@1#eq:Einstein, V-B@1#eq:L

Fix $m > 0$ , and set

□ = - \partial_{t}^{2} + Δ, P = □ - m^{2}, Ω_{m} = (- Δ + m^{2})^{1/2} .

For a covector $ξ_{ν}$ , define

(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ}, X_{ξ} = ξ^{μ} \partial_{μ} .

For a symmetric tensor $h$ , write

(r h)_{μν} = h_{μν} - \frac{1}{2} η_{μν} tr_{η} h, (D h)_{ν} = \partial^{μ} (r h)_{μν} .

Trace reversal is an involution in four dimensions.

The linearized Einstein and scalar-response operators are

G^{(1)} (h) L_{h} ϕ = - \frac{1}{2} □ r h + \frac{1}{2} r K D h, = - h^{μν} \partial_{μ} \partial_{ν} ϕ - (D h)^{ν} \partial_{ν} ϕ .

Displayed formulas in this passage

V-B@1#eq:Einstein · line 338
$G^{(1)} (h) L_{h} ϕ = - \frac{1}{2} □ r h + \frac{1}{2} r K D h, = - h^{μν} \partial_{μ} \partial_{ν} ϕ - (D h)^{ν} \partial_{ν} ϕ .$
V-B@1#eq:L · line 338
$G^{(1)} (h) L_{h} ϕ = - \frac{1}{2} □ r h + \frac{1}{2} r K D h, = - h^{μν} \partial_{μ} \partial_{ν} ϕ - (D h)^{ν} \partial_{ν} ϕ .$
unlabeled · line 318
$□ = - \partial_{t}^{2} + Δ, P = □ - m^{2}, Ω_{m} = (- Δ + m^{2})^{1/2} .$
unlabeled · line 324
$(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ}, X_{ξ} = ξ^{μ} \partial_{μ} .$
unlabeled · line 329
$(r h)_{μν} = h_{μν} - \frac{1}{2} η_{μν} tr_{η} h, (D h)_{ν} = \partial^{μ} (r h)_{μν} .$