Manuscript XII-D@1 · XII-D.tex · section · 308a2080

D0: the frozen sheet and the source convention

D0: the frozen sheet and the source convention · lines 368–469

XII-D@1#eq:source-coefficients, XII-D@1#eq:response

Use

M = R^{4}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x}, P_{m} = □ + m^{2}, m \geq 0,

M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, 0 < ε < T .

Fields are ambient smooth configurations. No spatial decay is required. All finite-slab constructions depend only on their restrictions to a neighborhood of the slab.

Retain the fixed ramp

σ_{ε} (t) = χ (t / ε), χ (s) = ⎩ ⎨ ⎧ 0, \frac{e ^{- 1/ s}}{e ^{- 1/ s} + e ^{- 1/ (1 - s)}}, 1, s \leq 0, 0 < s < 1, s \geq 1,

and

g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1]), U \subset int {g = 1} \cap {ε < t < T}, g_{U} = C_{c}^{\infty} (U; T M) .

We assume $U \neq = \emptyset$ , as in the plateau handoff. Neither external function is varied.

Trace reversal and divergence are

P r = r - \frac{1}{2} η tr_{η} r, D^{ν} r = \partial_{μ} (P r)^{μν} .

Parentheses include $\frac{1}{2}$ .

For completeness, the coefficients in (eq:source-input) have the finite prescription

u_{0}^{μν} = η^{μν}, u_{1}^{μν} = - γ^{μν}, u_{2}^{μν} = γ^{μ}_{λ} γ^{λ ν},

b_{0} = 1, b_{1} = \frac{1}{2} tr_{η} γ, b_{2} = \frac{1}{8} (tr_{η} γ)^{2} - \frac{1}{4} γ_{μν} γ^{μν},

C_{r}^{μν α β} Γ_{n}^{ρ}_{α β} M_{2}^{μν ρ} = i + j + k = r i, j, k \geq 0 \sum b_{i} (u_{j}^{μν} u_{k}^{α β} - u_{j}^{μ (α} u_{k}^{β) ν}), r = 0, 1, 2, = \frac{1}{2} u_{n - 1}^{ρ λ} (\partial_{α} γ_{β λ} + \partial_{β} γ_{α λ} - \partial_{λ} γ_{α β}), n = 1, 2, = C_{0}^{μν α β} Γ_{2}^{ρ}_{α β} + C_{1}^{μν α β} Γ_{1}^{ρ}_{α β} .

Only the displayed coefficients occur in the sums. The original $J_{3}$ is the quadratic coefficient $2 [κ^{2}] (- q G^{μν})$ , with $q = η + κγ$ and XII-C's curvature convention.

Thus $J_{D}$ is a quadratic differential polynomial of total field-derivative order at most two, with smooth coefficients built from $σ_{ε}, σ_{ε}^{'}, σ_{ε}^{''}$ . Its polarization is

J_{D} (γ_{1}, γ_{2}) = \frac{1}{2} (J_{D} [γ_{1} + γ_{2}] - J_{D} [γ_{1}] - J_{D} [γ_{2}]) .

It vanishes for $t \leq 0$ and is flat there. For $t \geq ε$ , it equals $J_{3}$ .

The prescribed response is

u_{D} [γ] = - P E_{0}^{R} (J_{D} [γ]) .

This is not the response to $σ_{ε} J_{3}$ alone.

Displayed formulas in this passage

XII-D@1#eq:source-coefficients · line 425
$C_{r}^{μν α β} Γ_{n}^{ρ}_{α β} M_{2}^{μν ρ} = i + j + k = r i, j, k \geq 0 \sum b_{i} (u_{j}^{μν} u_{k}^{α β} - u_{j}^{μ (α} u_{k}^{β) ν}), r = 0, 1, 2, = \frac{1}{2} u_{n - 1}^{ρ λ} (\partial_{α} γ_{β λ} + \partial_{β} γ_{α λ} - \partial_{λ} γ_{α β}), n = 1, 2, = C_{0}^{μν α β} Γ_{2}^{ρ}_{α β} + C_{1}^{μν α β} Γ_{1}^{ρ}_{α β} .$
XII-D@1#eq:response · line 464
$u_{D} [γ] = - P E_{0}^{R} (J_{D} [γ]) .$
unlabeled · line 372
$M = R^{4}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x}, P_{m} = □ + m^{2}, m \geq 0,$
unlabeled · line 378
$M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, 0 < ε < T .$
unlabeled · line 387
$σ_{ε} (t) = χ (t / ε), χ (s) = ⎩ ⎨ ⎧ 0, \frac{e ^{- 1/ s}}{e ^{- 1/ s} + e ^{- 1/ (1 - s)}}, 1, s \leq 0, 0 < s < 1, s \geq 1,$
unlabeled · line 398
$g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1]), U \subset int {g = 1} \cap {ε < t < T}, g_{U} = C_{c}^{\infty} (U; T M) .$
unlabeled · line 407
$P r = r - \frac{1}{2} η tr_{η} r, D^{ν} r = \partial_{μ} (P r)^{μν} .$
unlabeled · line 415
$u_{0}^{μν} = η^{μν}, u_{1}^{μν} = - γ^{μν}, u_{2}^{μν} = γ^{μ}_{λ} γ^{λ ν},$
unlabeled · line 420
$b_{0} = 1, b_{1} = \frac{1}{2} tr_{η} γ, b_{2} = \frac{1}{8} (tr_{η} γ)^{2} - \frac{1}{4} γ_{μν} γ^{μν},$
unlabeled · line 454
$J_{D} (γ_{1}, γ_{2}) = \frac{1}{2} (J_{D} [γ_{1} + γ_{2}] - J_{D} [γ_{1}] - J_{D} [γ_{2}]) .$

Nearby in this section

remark: Remark: Forcing sign versus action coefficient