Manuscript X-B-ST@1 · X-B-ST.tex · section · d03c150c

Density and coordinate coefficients

The frozen classical handoff › Density and coordinate coefficients · lines 390–482

X-B-ST@1#eq:densities, X-B-ST@1#eq:coordinates, X-B-ST@1#eq:coordinatelaw, X-B-ST@1#eq:preflight, X-B-ST@1#eq:Lambda

Indices are raised with $η$ . Set $H = tr_{η} h$ and

b_{1} [h] a_{1}^{μν} [h] = \frac{1}{2} H, = \frac{1}{2} H η^{μν} - h^{μν}, b_{2} [h] a_{2}^{μν} [h] = \frac{1}{8} H^{2} - \frac{1}{4} h_{α β} h^{α β}, = h^{μ}_{λ} h^{λ ν} - \frac{1}{2} H h^{μν} + b_{2} [h] η^{μν} .

Thus

- det q q^{μν} = η^{μν} + κ a_{1}^{μν} [h] + κ^{2} A_{2}^{μν} [h, k],

where

A_{2} [h, k] = a_{1} [k] + a_{2} [h], B_{2} [h, k] = b_{1} [k] + b_{2} [h] .

The supplied coordinate coefficients are

w^{μ} [h] v^{μ} [h, k] = - E_{0, R} \partial_{α} (σ a_{1}^{α μ} [h]), = - E_{0, R} \partial_{α} [σ A_{2}^{α μ} [h, k] + σ a_{1}^{α β} [h] \partial_{β} w^{μ} [h]] .

The retarded massless inverse on these smooth past-compact sources is understood by finite propagation, equivalently by the zero-data prescription at $t = 0$ . No spatial decay is assumed on $h, k$ .

These are the coefficients of the divergence-form density prescription

A_{ε} [q] = η^{- 1} + σ (A_{q} - η^{- 1}) .

They are not coefficients of a convolution regulator or of a wave equation obtained by multiplying a different source equation by $σ$ .

The handoff includes smoothness, fixed-width continuity, retained-past dependence, and the laws

w [h^{'}] v [h^{'}, k^{'}] = w [h] + ξ, = v [h, k] + X_{ξ} w [h] + ζ + \frac{1}{2} X_{ξ} ξ .

In particular,

w [K ξ] = ξ, ξ \in g_{p} .

These are supplied classical chart data XA,XAST, not a quantum multiplication theorem.

For later kernel arguments the linear coefficient has the explicit distributional representation

w^{μ} [h] (x) Λ_{ε}^{μ, α β} (x, z) a^{ν μ, α β} = \int Λ_{ε}^{μ, α β} (x, z) h_{α β} (z) d^{4} z, = σ (z^{0}) a^{ν μ, α β} \partial_{z^{ν}} G_{0, R} (x - z), = \frac{1}{2} η^{ν μ} η^{α β} - \frac{1}{2} (η^{ν α} η^{μ β} + η^{ν β} η^{μα}) .

The unsmeared kernel is distributional. Smoothness of its observation-smeared first derivatives does not mean that $Λ_{ε} (x, z)$ is a smooth two-point function.

Displayed formulas in this passage

X-B-ST@1#eq:densities · line 393
$b_{1} [h] a_{1}^{μν} [h] = \frac{1}{2} H, = \frac{1}{2} H η^{μν} - h^{μν}, b_{2} [h] a_{2}^{μν} [h] = \frac{1}{8} H^{2} - \frac{1}{4} h_{α β} h^{α β}, = h^{μ}_{λ} h^{λ ν} - \frac{1}{2} H h^{μν} + b_{2} [h] η^{μν} .$
X-B-ST@1#eq:coordinates · line 418
$w^{μ} [h] v^{μ} [h, k] = - E_{0, R} \partial_{α} (σ a_{1}^{α μ} [h]), = - E_{0, R} \partial_{α} [σ A_{2}^{α μ} [h, k] + σ a_{1}^{α β} [h] \partial_{β} w^{μ} [h]] .$
X-B-ST@1#eq:coordinatelaw · line 447
$w [h^{'}] v [h^{'}, k^{'}] = w [h] + ξ, = v [h, k] + X_{ξ} w [h] + ζ + \frac{1}{2} X_{ξ} ξ .$
X-B-ST@1#eq:preflight · line 454
$w [K ξ] = ξ, ξ \in g_{p} .$
X-B-ST@1#eq:Lambda · line 463
$w^{μ} [h] (x) Λ_{ε}^{μ, α β} (x, z) a^{ν μ, α β} = \int Λ_{ε}^{μ, α β} (x, z) h_{α β} (z) d^{4} z, = σ (z^{0}) a^{ν μ, α β} \partial_{z^{ν}} G_{0, R} (x - z), = \frac{1}{2} η^{ν μ} η^{α β} - \frac{1}{2} (η^{ν α} η^{μ β} + η^{ν β} η^{μα}) .$
unlabeled · line 406
$- det q q^{μν} = η^{μν} + κ a_{1}^{μν} [h] + κ^{2} A_{2}^{μν} [h, k],$
unlabeled · line 412
$A_{2} [h, k] = a_{1} [k] + a_{2} [h], B_{2} [h, k] = b_{1} [k] + b_{2} [h] .$
unlabeled · line 437
$A_{ε} [q] = η^{- 1} + σ (A_{q} - η^{- 1}) .$