Manuscript X-C-ST@1 · X-C-ST.tex · proposition · b679954e

Proposition: Conformal diagnostic

Diagnostics of the completed coefficient › Massless conformal check · lines 1982–2053 · proof included or linked

X-C-ST@1#prop:conformal, X-C-ST@1#eq:conformal-coeff, X-C-ST@1#eq:conformal-value

Let $m = 0$ , $h = ψ η$ , $k = 0$ . Then

a_{1} [h] = ψ η^{- 1}, b_{1} [h] = 2 ψ, b_{2} [h] = ψ^{2}, a_{2} [h] = 0,

and therefore $L_{2, g} [h, 0] = 0$ .

Suppose $ψ = ψ (t)$ and $g = g_{0} (t)$ near the complete contributing causal compact, with spatial cutoffs outside that compact. Define

Ψ (t) Γ (t) = \int_{0}^{t} σ (s) ψ (s) d s, = \int_{0}^{t} g_{0} (s) ψ (s) d s, I_{σ} (t) I_{g} (t) = \int_{0}^{t} (σ (s) ψ (s))^{2} d s, = \int_{0}^{t} (g_{0} (s) ψ (s))^{2} d s .

On the smooth free solution $u = t$ ,

w^{0} = - Ψ, v^{0} = I_{σ}, ρ_{1, g} = - Γ, ρ_{2, g} = I_{g} .

The corresponding classical evaluations are

⟨ e_{1}, t ⟩ ⟨ e_{2}, t ⟩ = \int f (Ψ - Γ) d^{4} x, = \int f [I_{g} - I_{σ} + Ψ ψ (σ - g_{0})] d^{4} x .

Proof. In four dimensions the conformal metric satisfies

- det ((1 + κ ψ) η) = (1 + κ ψ)^{2}, - det q q^{- 1} = (1 + κ ψ) η^{- 1} .

Thus the kinetic density has no quadratic term, and the quadratic mass density does not contribute when $m = 0$ .

The coordinate first equation gives $w^{0} = - Ψ$ , with zero spatial components on the retained calculation. The second equation gives

\partial_{t}^{2} v^{0} = - \partial_{t} (σ ψ \partial_{t} w^{0}) = \partial_{t} (σ ψ)^{2},

hence $v^{0} = I_{σ}$ . The scalar equations similarly give

ρ_{1, g} = - Γ, P_{0} ρ_{2, g} = \partial_{t} (g_{0} ψ)^{2},

and hence $ρ_{2, g} = I_{g}$ .

For $u = t$ , the bare second term is $Ψ σ ψ - I_{σ}$ , the mixed term is $- Ψ g_{0} ψ$ , and the new iteration is $I_{g}$ . Adding gives (eq:conformal-value).

Displayed formulas in this passage

unlabeled · line 1996
$Ψ (t) Γ (t) = \int_{0}^{t} σ (s) ψ (s) d s, = \int_{0}^{t} g_{0} (s) ψ (s) d s, I_{σ} (t) I_{g} (t) = \int_{0}^{t} (σ (s) ψ (s))^{2} d s, = \int_{0}^{t} (g_{0} (s) ψ (s))^{2} d s .$
X-C-ST@1#eq:conformal-coeff · line 2005
$w^{0} = - Ψ, v^{0} = I_{σ}, ρ_{1, g} = - Γ, ρ_{2, g} = I_{g} .$
X-C-ST@1#eq:conformal-value · line 2011
$⟨ e_{1}, t ⟩ ⟨ e_{2}, t ⟩ = \int f (Ψ - Γ) d^{4} x, = \int f [I_{g} - I_{σ} + Ψ ψ (σ - g_{0})] d^{4} x .$
unlabeled · line 1986
$a_{1} [h] = ψ η^{- 1}, b_{1} [h] = 2 ψ, b_{2} [h] = ψ^{2}, a_{2} [h] = 0,$
unlabeled · line 2023
$- det ((1 + κ ψ) η) = (1 + κ ψ)^{2}, - det q q^{- 1} = (1 + κ ψ) η^{- 1} .$
unlabeled · line 2035
$\partial_{t}^{2} v^{0} = - \partial_{t} (σ ψ \partial_{t} w^{0}) = \partial_{t} (σ ψ)^{2},$
unlabeled · line 2042
$ρ_{1, g} = - Γ, P_{0} ρ_{2, g} = \partial_{t} (g_{0} ψ)^{2},$

Nearby in this section

section: Massless conformal check