An exact quadratic identity for the invisible directions

The rank calculation has a concrete second-order form.

Fix one point of ${\mathcal{N}}^{\circ }$ with parameters $\alpha ,\beta ,\mathbf{n}$. Choose an orthonormal spatial basis transverse to $\mathbf{n}$. Let $v,w\in {\mathbb{R}}^2$ be transverse spatial displacements, and set

Here $(0,v)$ and $(0,w)$ have no time or longitudinal spatial component.