Manuscript I-B-VP-2@1 · I-B-VP-2-v1.tex · section · f4bd2b11

Why a null-set convention cannot control a wavefront

Two checks on the transition argument › Why a null-set convention cannot control a wavefront · lines 1968–2011

At $δ \in B_{α}$ , the geometric image set changes and the diffraction weight has its corresponding angular transition. Choosing a pointwise value on that angular set does not determine a distributional extension: distributions supported there are not ruled out by agreement off the set.

The proof instead uses (eq:Wclosed), already identified with the spectral covariance before taking a boundary value. At $Q = - 1$ , every possible wavefront covector is constrained to

λ d Q + ν d δ .

Since

Q + 1 = \frac{F _{ap}}{2 r r ^{'}},

its endpoint components are

κ = - \frac{λ T}{r r ^{'}} d t + \frac{λ ( r + r ^{'} )}{r r ^{'}} d r + ν d φ + \frac{λ Z}{r r ^{'}} d z,

κ^{'} = \frac{λ T}{r r ^{'}} d t^{'} + \frac{λ ( r + r ^{'} )}{r r ^{'}} d r^{'} - ν d φ^{'} - \frac{λ Z}{r r ^{'}} d z^{'} .

Their characteristic equations are

p (X, κ) = \frac{ν ^{2}}{α ^{2} r ^{2}}, p (X^{'}, κ^{'}) = \frac{ν ^{2}}{α ^{2} r ^{'2}} .

Thus both force $ν = 0$ . This calculation works when one or two angular transition representatives coincide, and in particular when $1/ α$ is an integer. No separate endpoint prescription is needed.

Displayed formulas in this passage

unlabeled · line 1979
$λ d Q + ν d δ .$
unlabeled · line 1983
$Q + 1 = \frac{F _{ap}}{2 r r ^{'}},$
unlabeled · line 1987
$κ = - \frac{λ T}{r r ^{'}} d t + \frac{λ ( r + r ^{'} )}{r r ^{'}} d r + ν d φ + \frac{λ Z}{r r ^{'}} d z,$
unlabeled · line 1994
$κ^{'} = \frac{λ T}{r r ^{'}} d t^{'} + \frac{λ ( r + r ^{'} )}{r r ^{'}} d r^{'} - ν d φ^{'} - \frac{λ Z}{r r ^{'}} d z^{'} .$
unlabeled · line 2002
$p (X, κ) = \frac{ν ^{2}}{α ^{2} r ^{2}}, p (X^{'}, κ^{'}) = \frac{ν ^{2}}{α ^{2} r ^{'2}} .$