Manuscript III-B@1 · III-B-v1.tex · section · d2c98f62

Operators, signs, and smooth evolution

Operators, signs, and smooth evolution · lines 477–513

III-B@1#eq:Lh, III-B@1#eq:Glin

Fix $m > 0$ , and define on the auxiliary slab

□ = - \partial_{s}^{2} + Δ, P = □ - m^{2} .

Here $m$ has the inverse-length normalization appropriate to $s = c (T - T_{0})$ .

For a covector $ξ_{μ}$ , set

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

For symmetric $h$ , put

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

L_{h} u = - h^{μν} \partial_{μ} \partial_{ν} u - (D h)^{ν} \partial_{ν} u,

and

G^{(1)} (h)_{μν} = - \frac{1}{2} □ (r h)_{μν} + \partial_{(μ} (D h)_{ν)} - \frac{1}{2} η_{μν} \partial^{α} (D h)_{α} .

Symmetrization includes the factor $1/2$ .

Displayed formulas in this passage

III-B@1#eq:Lh · line 499
$L_{h} u = - h^{μν} \partial_{μ} \partial_{ν} u - (D h)^{ν} \partial_{ν} u,$
III-B@1#eq:Glin · line 505
$G^{(1)} (h)_{μν} = - \frac{1}{2} □ (r h)_{μν} + \partial_{(μ} (D h)_{ν)} - \frac{1}{2} η_{μν} \partial^{α} (D h)_{α} .$
unlabeled · line 481
$□ = - \partial_{s}^{2} + Δ, P = □ - m^{2} .$
unlabeled · line 488
$(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ}, X_{ξ} = ξ^{μ} \partial_{μ} .$
unlabeled · line 493
$(r h)_{μν} = h_{μν} - \frac{1}{2} η_{μν} tr_{η} h, (D h)_{ν} = \partial^{μ} (r h)_{μν},$