Manuscript XX-A@1 · XX-A.tex · section · 3a586110

Compact output smearing is not compact functional support

Observable kernels, products, and spatial images › Compact output smearing is not compact functional support · lines 1179–1199

XX-A@1#eq:instantaneous

The first-order rod operator is instantaneous in clock time. Its scalar Newton kernel has the spacetime form

K_{sp} (τ, x; τ^{'}, y) = - \frac{δ ( τ - τ ^{'} )}{4 π ∣ x - y ∣} .

The actual rod kernels contain its derivatives and the explicitly specified coefficient matrices.

For compact $b$ of nonzero spatial integral,

\partial_{i} G b (x) = \frac{( \int b ) x _{i}}{4 π ∣ x ∣ ^{3}} + O (∣ x ∣^{- 3}) .

Thus the first rod can have a nonzero $1/ r^{2}$ tail.

Displayed formulas in this passage

XX-A@1#eq:instantaneous · line 1183
$K_{sp} (τ, x; τ^{'}, y) = - \frac{δ ( τ - τ ^{'} )}{4 π ∣ x - y ∣} .$
unlabeled · line 1193
$\partial_{i} G b (x) = \frac{( \int b ) x _{i}}{4 π ∣ x ∣ ^{3}} + O (∣ x ∣^{- 3}) .$