Screen space of a null congruence ($k^{\perp }/\mathrm{span}\{k\}$)

The screen space of a null congruence describes directions transverse to a bundle of lightlike trajectories. For a nonzero null vector $k$, its orthogonal space $k^{\perp }$ contains $k$ itself, so the induced metric there is degenerate. Removing this physically longitudinal direction gives the quotient vector spaceIn four-dimensional Lorentzian manifold it is two-dimensional and has a positive-definite induced metric under the $(-,+,+,+)$ convention.

Choosing an auxiliary null direction lets one represent this quotient by an ordinary transverse plane and a screen projector. That choice is useful for calculation, but the quotient is the intrinsic object. The optical matrix of a null congruence then describes expansion, shear and rotation of transverse separations rather than a spurious change along the rays themselves.