Manuscript X-D-ST@1 · X-D-ST.tex · section · e0cf8fdb

Conformal temporal diagnostic

Diagnostics and sharpness of the rate › Conformal temporal diagnostic · lines 1753–1828

X-D-ST@1#eq:diagnostic, X-D-ST@1#eq:diagnostic-limit

Let $m = 0$ , $h = ψ (t) η$ , and $k = 0$ near a causal cylinder containing the retained calculation. Spatial cutoffs outside that cylinder give admissible localized backgrounds without changing the result. Then

a^{μν} = ψ η^{μν}, A^{μν} = 0.

The massless second scalar vertex vanishes.

Suppose $g = g_{0} (t)$ on that cylinder, and define

Ψ_{ε} (t) Γ (t) = \int_{0}^{t} σ_{ε} (s) ψ (s) d s, = \int_{0}^{t} g_{0} (s) ψ (s) d s, J_{ε} (t) J_{g} (t) = \int_{0}^{t} (σ_{ε} (s) ψ (s))^{2} d s, = \int_{0}^{t} (g_{0} (s) ψ (s))^{2} d s .

Direct integration of the coordinate and scalar equations gives

w_{ε}^{0} = - Ψ_{ε}, v_{ε}^{0} = J_{ε}, ρ_{1, g} [t] = - Γ, ρ_{2, g} [t] = J_{g} .

All relevant spatial components vanish on this calculation.

On the smooth massless solution $u = t$ ,

⟨ d_{1, ε}, t ⟩ ⟨ d_{2, ε}, t ⟩ = \int f (Ψ_{ε} - Γ) d^{4} x, = \int f [J_{g} - J_{ε} + Ψ_{ε} ψ (σ_{ε} - g_{0})] d^{4} x .

For instance, the bare second term is $Ψ_{ε} σ_{ε} ψ - J_{ε}$ , the mixed term is $- Ψ_{ε} g_{0} ψ$ , and the iterated scalar response is $J_{g}$ . This verifies their relative signs and the counting.

Let

Ψ_{0} (t) = \int_{0}^{t} ψ (s) d s, J_{0} (t) = \int_{0}^{t} ψ (s)^{2} d s .

On the observation support the limiting evaluations are

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

The layer differences in $Ψ_{ε}, J_{ε}$ are $O (ε)$ , exactly as the theorem predicts. The right-hand initial velocities are

\partial_{t} w_{0}^{0} (0) = - ψ (0), \partial_{t} v_{0}^{0} (0) = ψ (0)^{2} .

The multiplied-equation first coordinate instead has zero initial velocity.

Displayed formulas in this passage

unlabeled · line 1766
$Ψ_{ε} (t) Γ (t) = \int_{0}^{t} σ_{ε} (s) ψ (s) d s, = \int_{0}^{t} g_{0} (s) ψ (s) d s, J_{ε} (t) J_{g} (t) = \int_{0}^{t} (σ_{ε} (s) ψ (s))^{2} d s, = \int_{0}^{t} (g_{0} (s) ψ (s))^{2} d s .$
X-D-ST@1#eq:diagnostic · line 1786
$⟨ d_{1, ε}, t ⟩ ⟨ d_{2, ε}, t ⟩ = \int f (Ψ_{ε} - Γ) d^{4} x, = \int f [J_{g} - J_{ε} + Ψ_{ε} ψ (σ_{ε} - g_{0})] d^{4} x .$
X-D-ST@1#eq:diagnostic-limit · line 1810
$⟨ d_{1, 0}, t ⟩ ⟨ d_{2, 0}, t ⟩ = \int f (Ψ_{0} - Γ) d^{4} x, = \int f [J_{g} - J_{0} + Ψ_{0} ψ (1 - g_{0})] d^{4} x .$
unlabeled · line 1760
$a^{μν} = ψ η^{μν}, A^{μν} = 0.$
unlabeled · line 1777
$w_{ε}^{0} = - Ψ_{ε}, v_{ε}^{0} = J_{ε}, ρ_{1, g} [t] = - Γ, ρ_{2, g} [t] = J_{g} .$
unlabeled · line 1805
$Ψ_{0} (t) = \int_{0}^{t} ψ (s) d s, J_{0} (t) = \int_{0}^{t} ψ (s)^{2} d s .$
unlabeled · line 1822
$\partial_{t} w_{0}^{0} (0) = - ψ (0), \partial_{t} v_{0}^{0} (0) = ψ (0)^{2} .$

Conformal temporal diagnostic

Displayed formulas in this passage

Nearby in this section