Manuscript VII-C@1 · VII-C.tex · section · 8b2ade29

A compact derivation of the boundary-frequency reversal

A compact derivation of the boundary-frequency reversal · lines 1669–1701

The mechanism can be isolated at one massless momentum. For $r > 0$ , the scalar zero-Cauchy inverse is

E_{0} (e^{- i r t}) = - \frac{i t}{2 r} e^{- i r t} + \frac{1}{4 r ^{2}} (e^{i r t} - e^{- i r t}) .

The second term enforces the initial conditions. It introduces an $e^{i r t}$ component although the source has only $e^{- i r t}$ frequency.

For the component selected by $h_{00}$ , the derivative in (eq:CH) multiplies the source by $- i r$ , and the Hadamard spectral measure contributes $1/ (2 r)$ . The reflected mixed covariance therefore has amplitude

- \frac{i}{8 r ^{2}} .

In the mixed triple, two such amplitudes contribute $r^{- 4}$ . The differentiated scalar covariance contributes $r$ , and the remaining radial measure contributes $r^{2} d r$ . Thus the leading radial behavior is

\frac{d r}{r} .

This power count alone is not the proof of divergence: the tensor selection, exact graph identity, nonzero light-ray coefficient, uniform remainder estimate, and topology argument are supplied in Sections 4–7.

Displayed formulas in this passage

unlabeled · line 1673
$E_{0} (e^{- i r t}) = - \frac{i t}{2 r} e^{- i r t} + \frac{1}{4 r ^{2}} (e^{i r t} - e^{- i r t}) .$
unlabeled · line 1687
$- \frac{i}{8 r ^{2}} .$
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$\frac{d r}{r} .$