Choose an auxiliary null vector $N^a$ with $U\mathbin\cdot N=-1$. The <screen-space projector> is
$$
P^a{}_b=\delta^a_b+U^aN_b+N^aU_b,
$$
and the <optical tensor> is $\widehat B_{ab}=P_a{}^cP_b{}^dB_{cd}$ for $B_{ab}=\nabla_bU_a$. Because the screen is two-dimensional, its irreducible decomposition is
$$
\widehat B_{ab}=\frac12\theta P_{ab}+\widehat\sigma_{ab}+\widehat\omega_{ab},
$$
where
$$
\boxed{\theta=P^{ab}\widehat B_{ab}},
\qquad
\boxed{\widehat\sigma_{ab}=\widehat B_{(ab)}-\frac12\theta P_{ab}},
\qquad
\boxed{\widehat\omega_{ab}=\widehat B_{[ab]}}.
$$
These are respectively the <null expansion>, <null shear>, and <null twist>, also called rotation.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 311 / 2 / a / Solution

 Solution

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