Manuscript VIII-A@3 · VIII-A-v3.tex · lemma · 2398a747

Lemma: Spinless affine matching

Spinless affine geometry and realization › Affine conventions · lines 637–661 · proof included or linked

VIII-A@3#A:lem:spinless

The map $H_{Π}$ fixes $W$ pointwise, its Lorentz part fixes $u$ , and its oriented rotation on $u^{⊥}$ has angle $μ$ . These conditions determine $H_{Π}$ uniquely from $μ, W, u$ and the declared orientation.

Proof. For $z = A (t e_{0})$ ,

H_{Π} z = A J (μ) (t e_{0}) = A (t e_{0}) = z .

The Lorentz part fixes $C e_{0}$ and acts on its oriented orthogonal plane by the conjugate spatial rotation.

Conversely, a proper future-preserving Lorentz map fixing $u$ restricts to an orientation-preserving Euclidean rotation on $u^{⊥}$ . The prescribed oriented angle determines its Lorentz part $L$ . Pointwise fixation of $a + R u$ then requires $b = (I - L) a$ . Replacing $a$ by $a + t_{0} u$ does not change this translation.

Displayed formulas in this passage

unlabeled · line 648
$H_{Π} z = A J (μ) (t e_{0}) = A (t e_{0}) = z .$

Lemma: Spinless affine matching

Displayed formulas in this passage

Nearby in this section