= Affine projective twistor space
{title2=$PT_{\rm aff}=\mathbb{CP}^3\setminus\mathbb{CP}^1_\infty$}
For affine <complexified Minkowski space>, the relevant twistor domain is
$$
PT_{\rm aff}=\{[\omega^A,\pi_{A'}]\in\mathbb{CP}^3:\pi\ne0\}
=\mathbb{CP}^3\setminus\mathbb{CP}^1_\infty
\cong\operatorname{Tot}(\mathcal O(1)\oplus\mathcal O(1)).
$$
Incidence can be written $\omega^A=x^{AA'}\pi_{A'}$, 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 $\pi_{0'}\ne0$ and $\pi_{1'}\ne0$, the fibre coordinates transform by $\omega/\pi_{1'}=(\omega/\pi_{0'})/\lambda$, with $\lambda=\pi_{1'}/\pi_{0'}$, giving the two $\mathcal O(1)$ summands. The full <projective twistor space> corresponds to <complexified compactified Minkowski space>.
Back to article page
