= Projective Minkowski compactification
{c}
{title2=$\overline M_L=(S^1\times S^3)/\mathbb Z_2$}
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>.
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