Manuscript En08-A@1 · En08-A.tex · section · c688d76d

Conditional quadratic signature

Signature and Lorentz-cone selection › Conditional quadratic signature · lines 1223–1242

The full negative region is not convex in the Lorentzian case: one first chooses a time-oriented component. Likewise, the two closed sheets of ${Q \leq 0}$ meet at zero, so a future cone must not be defined as a connected component of that closed set.

A higher-index quadratic form can contain smaller pointed cones. For example,

Q (t_{1}, t_{2}, x) = - t_{1}^{2} - t_{2}^{2} + ∥ x ∥_{2}^{2}

is nonpositive on

{t_{1} \geq t_{2}^{2} + ∥ x ∥_{2}^{2}} .

The theorem concerns a full component, not an arbitrarily chosen subcone.

Displayed formulas in this passage

unlabeled · line 1233
$Q (t_{1}, t_{2}, x) = - t_{1}^{2} - t_{2}^{2} + ∥ x ∥_{2}^{2}$
unlabeled · line 1237
${t_{1} \geq t_{2}^{2} + ∥ x ∥_{2}^{2}} .$

Nearby in this section

theorem: Theorem: Quadratic component classification