Manuscript Em02-A@6 · Em02-A-v6.tex · section · cb2f0a9a

Route B: an explicit coincidence-transition calculation

Route B: an explicit coincidence-transition calculation · lines 916–983

Em02-A@6#eq:radial-cone, Em02-A@6#eq:radial-5a, Em02-A@6#eq:radial-5z, Em02-A@6#eq:transition-constants, Em02-A@6#eq:incidence-measures

The coincidence transition is treated in independent cone coordinates rather than by extrapolating the non-tip denominators.

For $T \geq 0$ and $m \in S^{2}$ , write $n_{m} = (1, m)$ . The exact cone measure is

d R (T n_{m}) = \frac{1}{4 π} T d T d m .

For a compact smooth ambient scalar amplitude $H$ ,

⟨ μ_{5 a}, H ⟩ ⟨ μ_{5 z}, H ⟩ = \frac{1}{16 π ^{2}} \int S T H (a, a + S n_{n}, a - T n_{m}, y) d^{4} a d^{4} y d S d T d n d m, = \frac{1}{16 π ^{2}} \int S T H (z - T n_{m}, z + S n_{n}, z, y) d^{4} z d^{4} y d S d T d n d m .

Both radial variables range over $[0, \infty)$ . These formulas include $T = 0$ without introducing a $1/∣ γ ∣$ denominator.

Introduce a second parameter $ϵ > 0$ , independent of $δ$ , and define

θ_{ϵ} (r) = b_{0} (∣ r ∣_{E}^{2} / ϵ^{2}) .

Set

c_{R} = \int_{0}^{\infty} t b_{0} (2 t^{2}) d t = \frac{1}{4} \int_{0}^{\infty} b_{0} (s) d s, c_{47} = \frac{1}{2 2 π} \int_{0}^{\infty} t^{2} b_{0} (t^{2}) d t .

Both constants are finite and strictly positive.

In the characteristic chart define the incidence measure $ν_{0}$ by

⟨ ν_{0}, H ⟩ = \int H (a, b, a, y)_{u = 0} d τ d v d^{2} ξ .

Thus, locally,

d μ_{0} = \frac{β}{2 π} d ν_{0} .

Displayed formulas in this passage

Em02-A@6#eq:radial-cone · line 925
$d R (T n_{m}) = \frac{1}{4 π} T d T d m .$
Em02-A@6#eq:radial-5a · line 932
$⟨ μ_{5 a}, H ⟩ ⟨ μ_{5 z}, H ⟩ = \frac{1}{16 π ^{2}} \int S T H (a, a + S n_{n}, a - T n_{m}, y) d^{4} a d^{4} y d S d T d n d m, = \frac{1}{16 π ^{2}} \int S T H (z - T n_{m}, z + S n_{n}, z, y) d^{4} z d^{4} y d S d T d n d m .$
Em02-A@6#eq:radial-5z · line 932
$⟨ μ_{5 a}, H ⟩ ⟨ μ_{5 z}, H ⟩ = \frac{1}{16 π ^{2}} \int S T H (a, a + S n_{n}, a - T n_{m}, y) d^{4} a d^{4} y d S d T d n d m, = \frac{1}{16 π ^{2}} \int S T H (z - T n_{m}, z + S n_{n}, z, y) d^{4} z d^{4} y d S d T d n d m .$
Em02-A@6#eq:transition-constants · line 960
$c_{R} = \int_{0}^{\infty} t b_{0} (2 t^{2}) d t = \frac{1}{4} \int_{0}^{\infty} b_{0} (s) d s, c_{47} = \frac{1}{2 2 π} \int_{0}^{\infty} t^{2} b_{0} (t^{2}) d t .$
Em02-A@6#eq:incidence-measures · line 979
$d μ_{0} = \frac{β}{2 π} d ν_{0} .$
unlabeled · line 956
$θ_{ϵ} (r) = b_{0} (∣ r ∣_{E}^{2} / ϵ^{2}) .$
unlabeled · line 972
$⟨ ν_{0}, H ⟩ = \int H (a, b, a, y)_{u = 0} d τ d v d^{2} ξ .$