 Projective Minkowski compactification

The Lorentzian real form of the Klein quadric is the projectivized real null cone of signature $(2,4)$. Scaling a nonzero null vector makes its positive two-dimensional and negative four-dimensional parts both unit vectors. Real projectivization then identifies $(u,v)$ with $(-u,-v)$, yielding the displayed quotient. Its universal cover is $\mathbb{R}\times S^3$, and its fundamental group is infinite cyclic. An affine Minkowski chart omits a null cone at infinity. This projective compact space is distinct from an unwrapped causal cover used for a Penrose diagram.