Manuscript X-A-conv-ST@1 · X-A-conv-ST.tex · section · 8ca47088

The action and its scalar solution

Strict positivity and the evaluated classical obstruction › The action and its scalar solution · lines 781–870

X-A-conv-ST@1#eq:action, X-A-conv-ST@1#eq:scalarequation, X-A-conv-ST@1#eq:rho1, X-A-conv-ST@1#eq:rho2

For completeness, retain the parent's first-derivative gravitational density representative

ℓ_{gr} (κ, h) = - \frac{2}{κ ^{2}} A (q)^{μν} (Γ^{ρ}_{μ σ} Γ^{σ}_{ν ρ} - Γ^{ρ}_{μν} Γ^{σ}_{ρ σ}), A (q)^{μν} = - det q q^{μν},

and

ℓ_{sc} (κ, h, ϕ) = \frac{1}{2} A (q)^{μν} \partial_{μ} ϕ \partial_{ν} ϕ .

The Christoffel symbols and curvature use the convention

R_{μν} = \partial_{ρ} Γ^{ρ}_{μν} - \partial_{ν} Γ^{ρ}_{μ ρ} + Γ^{ρ}_{ρ λ} Γ^{λ}_{μν} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ ρ} .

Write

ℓ_{gr} + ℓ_{sc} = ℓ_{0} + κ ℓ_{1} + κ^{2} ℓ_{2} + \dots

using $q = η + κh$ , and freeze

S_{g}^{[2]} = \int (ℓ_{0} + κ g ℓ_{1} + κ^{2} g ℓ_{2}) d^{4} x .

The free integral is interpreted variationally with compact variations. The chosen density is not replaced by another representative after switching XConv.

For a metric jet, its scalar equation through the retained order is

D_{g} [q] ρ = \partial_{μ} [(η^{μν} + κ g a_{1} (h)^{μν} + κ^{2} g (a_{1} (k)^{μν} + a_{2} (h)^{μν})) \partial_{ν} ρ] = 0.

Thus the second vertex has one factor $g$ , not $g^{2}$ .

Define

V_{j, g} [h] u = \partial_{μ} (g a_{j} (h)^{μν} \partial_{ν} u), W_{g} [h, k] = V_{1, g} [k] + V_{2, g} [h] .

With free input $ρ_{0}$ and zero positive-order Cauchy data, the selected scalar is

ρ_{g} = ρ_{0} + κ ρ_{1, g} + κ^{2} ρ_{2, g},

where

ρ_{1, g} ρ_{2, g} = - E_{0} V_{1, g} [h] ρ_{0}, = E_{0} V_{1, g} [h] E_{0} V_{1, g} [h] ρ_{0} - E_{0} W_{g} [h, k] ρ_{0} .

These formulas follow by expanding (eq:scalarequation) and applying Lemma lem:Cauchy. In particular,

V_{j, g} [h] u = g \partial_{μ} (a_{j} (h)^{μν} \partial_{ν} u) + (\partial_{μ} g) a_{j} (h)^{μν} \partial_{ν} u .

No switch derivative has been discarded.

Displayed formulas in this passage

X-A-conv-ST@1#eq:action · line 815
$S_{g}^{[2]} = \int (ℓ_{0} + κ g ℓ_{1} + κ^{2} g ℓ_{2}) d^{4} x .$
X-A-conv-ST@1#eq:scalarequation · line 827
$D_{g} [q] ρ = \partial_{μ} [(η^{μν} + κ g a_{1} (h)^{μν} + κ^{2} g (a_{1} (k)^{μν} + a_{2} (h)^{μν})) \partial_{ν} ρ] = 0.$
X-A-conv-ST@1#eq:rho1 · line 851
$ρ_{1, g} ρ_{2, g} = - E_{0} V_{1, g} [h] ρ_{0}, = E_{0} V_{1, g} [h] E_{0} V_{1, g} [h] ρ_{0} - E_{0} W_{g} [h, k] ρ_{0} .$
X-A-conv-ST@1#eq:rho2 · line 851
$ρ_{1, g} ρ_{2, g} = - E_{0} V_{1, g} [h] ρ_{0}, = E_{0} V_{1, g} [h] E_{0} V_{1, g} [h] ρ_{0} - E_{0} W_{g} [h, k] ρ_{0} .$
unlabeled · line 785
$ℓ_{gr} (κ, h) = - \frac{2}{κ ^{2}} A (q)^{μν} (Γ^{ρ}_{μ σ} Γ^{σ}_{ν ρ} - Γ^{ρ}_{μν} Γ^{σ}_{ρ σ}), A (q)^{μν} = - det q q^{μν},$
unlabeled · line 796
$ℓ_{sc} (κ, h, ϕ) = \frac{1}{2} A (q)^{μν} \partial_{μ} ϕ \partial_{ν} ϕ .$
unlabeled · line 802
$R_{μν} = \partial_{ρ} Γ^{ρ}_{μν} - \partial_{ν} Γ^{ρ}_{μ ρ} + Γ^{ρ}_{ρ λ} Γ^{λ}_{μν} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ ρ} .$
unlabeled · line 810
$ℓ_{gr} + ℓ_{sc} = ℓ_{0} + κ ℓ_{1} + κ^{2} ℓ_{2} + \dots$
unlabeled · line 839
$V_{j, g} [h] u = \partial_{μ} (g a_{j} (h)^{μν} \partial_{ν} u), W_{g} [h, k] = V_{1, g} [k] + V_{2, g} [h] .$
unlabeled · line 847
$ρ_{g} = ρ_{0} + κ ρ_{1, g} + κ^{2} ρ_{2, g},$
unlabeled · line 864
$V_{j, g} [h] u = g \partial_{μ} (a_{j} (h)^{μν} \partial_{ν} u) + (\partial_{μ} g) a_{j} (h)^{μν} \partial_{ν} u .$

The action and its scalar solution

Displayed formulas in this passage

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