Projective twistor space ($\mathbb{PT}\cong {\mathbb{CP}}^3$)

The projective space of one-dimensional complex subspaces of twistor space. A point corresponds to an alpha-plane in complexified compactified Minkowski space; a projective line corresponds to a spacetime point.

For affine complexified Minkowski space, the relevant twistor domain isIncidence can be written ${\omega }^A=x^{A{A}^{\prime }}{\pi }_{A^{\prime }}$, absorbing the conventional factor $i$ into the definition of $x$. Each space-time point gives a twistor line $L_x\cong {\mathbb{CP}}^1$. On charts ${\pi }_{0^{\prime }}\ne 0$ and ${\pi }_{1^{\prime }}\ne 0$, the fibre coordinates transform by $\omega /{\pi }_{1^{\prime }}=(\omega /{\pi }_{0^{\prime }})/\lambda$, with $\lambda ={\pi }_{1^{\prime }}/{\pi }_{0^{\prime }}$, giving the two $\mathcal{O}(1)$ summands. The full projective twistor space corresponds to complexified compactified Minkowski space.