Manuscript X-G-ST@1 · X-G-ST.tex · section · 3c97567f

Signed operators and tensor types

Frozen operators, switches, and the predecessor interface › Signed operators and tensor types · lines 345–428

X-G-ST@1#eq:Cstar, X-G-ST@1#eq:operators, X-G-ST@1#eq:dividentities, X-G-ST@1#eq:metricGreen, X-G-ST@1#eq:Lie

Set

□ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2} .

The positive equation inverses are

□ E_{0}^{R} = □ E_{0}^{A} = 1, P_{m} E_{m}^{R} = P_{m} E_{m}^{A} = 1.

Their causal support, Cauchy uniqueness, finite propagation, and continuity are used on the completion BGP.

For a covariant symmetric tensor $r$ , put

(P_{tr} r)_{μν} = r_{μν} - \frac{1}{2} η_{μν} tr_{η} r, (C r)^{μ} = \partial_{ν} (P_{tr} r)^{ν μ},

and

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

All component identifications in flat formulas use $η$ and the coordinate volume density.

The formal transpose is bilinear, not a Hilbert-space adjoint. Compact integration by parts gives

C^{*} v = - \frac{1}{2} P_{tr} K v, C K = □, P_{tr}^{2} = 1.

Let

A = \frac{1}{2} G_{lin}, A K = 0, A + \frac{1}{2} C^{*} C = P_{gr} = - \frac{1}{4} P_{tr} □ .

These are the retained X-E/X-F normalization identities XE,XF. They can also be checked by substituting the standard linearized Einstein tensor into (eq:operators). In particular,

\partial_{μ} (A r)^{μν} = 0, \partial_{μ} (C^{*} b)^{μν} = - \frac{1}{2} □ b^{ν} .

The metric Green inverses remain

Δ_{gr}^{R} = - 4 P_{tr} E_{0}^{R}, Δ_{gr}^{A} = - 4 P_{tr} E_{0}^{A} .

The scalar action Hessian is $- P_{m}$ , whose inverses are $- E_{m}^{R}, - E_{m}^{A}$ . It is not the positive equation operator $P_{m}$ .

Metric inputs are covariant tensors. Metric Euler outputs are contravariant tensor densities. Denote their Lie derivatives by $L^{cov}_{ξ}, L^{dens}_{ξ}$ . Thus

(L^{cov}_{ξ} r)_{μν} (L^{dens}_{ξ} A)^{μν} = ξ^{λ} \partial_{λ} r_{μν} + r_{λ ν} \partial_{μ} ξ^{λ} + r_{μ λ} \partial_{ν} ξ^{λ}, = \partial_{λ} (ξ^{λ} A^{μν}) - A^{λ ν} \partial_{λ} ξ^{μ} - A^{μ λ} \partial_{λ} ξ^{ν} .

Their compact pairing satisfies

⟨ r, L^{dens}_{ξ} A ⟩ = - ⟨ L^{cov}_{ξ} r, A ⟩ .

Displayed formulas in this passage

X-G-ST@1#eq:Cstar · line 376
$C^{*} v = - \frac{1}{2} P_{tr} K v, C K = □, P_{tr}^{2} = 1.$
X-G-ST@1#eq:operators · line 382
$A = \frac{1}{2} G_{lin}, A K = 0, A + \frac{1}{2} C^{*} C = P_{gr} = - \frac{1}{4} P_{tr} □ .$
X-G-ST@1#eq:metricGreen · line 400
$Δ_{gr}^{R} = - 4 P_{tr} E_{0}^{R}, Δ_{gr}^{A} = - 4 P_{tr} E_{0}^{A} .$
X-G-ST@1#eq:Lie · line 412
$(L^{cov}_{ξ} r)_{μν} (L^{dens}_{ξ} A)^{μν} = ξ^{λ} \partial_{λ} r_{μν} + r_{λ ν} \partial_{μ} ξ^{λ} + r_{μ λ} \partial_{ν} ξ^{λ}, = \partial_{λ} (ξ^{λ} A^{μν}) - A^{λ ν} \partial_{λ} ξ^{μ} - A^{μ λ} \partial_{λ} ξ^{ν} .$
unlabeled · line 348
$□ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2} .$
unlabeled · line 352
$□ E_{0}^{R} = □ E_{0}^{A} = 1, P_{m} E_{m}^{R} = P_{m} E_{m}^{A} = 1.$
unlabeled · line 361
$(P_{tr} r)_{μν} = r_{μν} - \frac{1}{2} η_{μν} tr_{η} r, (C r)^{μ} = \partial_{ν} (P_{tr} r)^{ν μ},$
unlabeled · line 368
$(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ} .$
X-G-ST@1#eq:dividentities · line 393
$\partial_{μ} (A r)^{μν} = 0, \partial_{μ} (C^{*} b)^{μν} = - \frac{1}{2} □ b^{ν} .$
unlabeled · line 425
$⟨ r, L^{dens}_{ξ} A ⟩ = - ⟨ L^{cov}_{ξ} r, A ⟩ .$