Manuscript VII-F@1 · VII-F.tex · proposition · 2d81c03f

Proposition: Particle slice defects

Intrinsic-angle audit · lines 1916–1959 · proof included or linked

VII-F@1#prop:particle-defects

The particle's intrinsic slice defects are

κ_{P}^{-} = 2 π - 3 arccos (- \frac{4}{5}) (s < 0),

and

κ_{P}^{+} = 2 π - 6 arctan (\frac{4}{5}) (s > 0) .

Neither formula identifies the particle's slice defect with the fixed spacetime deficit $μ = π /2$ .

Proof. On an incoming $V_{i}$ face, the induced metric in sector coordinates is

G_{h} = I - (2/3 2/3) (2/3 2/3) .

For the two ray directions,

cos θ_{h} = \frac{- 4/9}{( 5 /3 ) ^{2}} = - \frac{4}{5} .

Three such angles meet at the particle.

On $U_{i}^{-}$ , the induced metric is

G_{k} = diag (1, 16/25) .

The angle between $(1, 0)$ and $(1, 1)$ is $arctan (4/5)$ . The same angle occurs on $U_{i}^{+}$ . There are six such angles.

Displayed formulas in this passage

unlabeled · line 1919
$κ_{P}^{-} = 2 π - 3 arccos (- \frac{4}{5}) (s < 0),$
unlabeled · line 1925
$κ_{P}^{+} = 2 π - 6 arctan (\frac{4}{5}) (s > 0) .$
unlabeled · line 1936
$G_{h} = I - (2/3 2/3) (2/3 2/3) .$
unlabeled · line 1943
$cos θ_{h} = \frac{- 4/9}{( 5 /3 ) ^{2}} = - \frac{4}{5} .$
unlabeled · line 1953
$G_{k} = diag (1, 16/25) .$