Manuscript III@2 · III.tex · section · 82dad3e5

Geometry and operators

Conventions, signs, and support › Geometry and operators · lines 361–410

III@2#eq:Lh, III@2#eq:Glin

On $M = R^{1 + 3}$ , use

η = diag (- 1, 1, 1, 1), □_{η} = - \partial_{t}^{2} + Δ, P = □_{η} - m^{2}, m > 0,

where $Δ = \partial_{i} \partial_{i}$ . All fields and gauge parameters are real unless Fourier transforms or complex-valued comparison functions are explicitly used. Indices are raised with $η$ , and symmetrization includes the factor $1/2$ .

For a covector $ξ_{μ}$ , set

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

The same letter denotes its raised vector when used in $X_{ξ}$ .

For a symmetric covariant tensor $h$ , define

(r h)_{μν} = h_{μν} - \frac{1}{2} η_{μν} tr_{η} h,

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

and

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

In four dimensions, $r^{2} = id$ . Our linearized Einstein normalization is

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

Displayed formulas in this passage

III@2#eq:Lh · line 395
$L_{h} u = - h^{μν} \partial_{μ} \partial_{ν} u - (D h)^{ν} \partial_{ν} u .$
III@2#eq:Glin · line 403
$G^{(1)} (h)_{μν} = - \frac{1}{2} □_{η} (r h)_{μν} + \partial_{(μ} (D h)_{ν)} - \frac{1}{2} η_{μν} \partial^{α} (D h)_{α} .$
unlabeled · line 364
$η = diag (- 1, 1, 1, 1), □_{η} = - \partial_{t}^{2} + Δ, P = □_{η} - m^{2}, m > 0,$
unlabeled · line 376
$(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ}, X_{ξ} = ξ^{μ} \partial_{μ} .$
unlabeled · line 385
$(r h)_{μν} = h_{μν} - \frac{1}{2} η_{μν} tr_{η} h,$
unlabeled · line 389
$(D h)_{ν} = \partial^{μ} h_{μν} - \frac{1}{2} \partial_{ν} tr_{η} h = \partial^{μ} (r h)_{μν},$

Geometry and operators

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