Let $U^a$ be the affine null tangent and choose another null vector $N^a$ with $U\mathbin\cdot N=-1$. The <screen-space projector>
$$
P^a{}_b=\delta^a_b+U^aN_b+N^aU_b
$$
projects onto the $(D-2)$-dimensional transverse space. The <optical tensor> is
$$
\widehat B_{ab}=P_a{}^cP_b{}^d\nabla_dU_c.
$$
Its irreducible decomposition defines
$$
\boxed{\theta=P^{ab}\widehat B_{ab}},
$$
the <null expansion>,
$$
\boxed{\widehat\sigma_{ab}
=\widehat B_{(ab)}-\frac{\theta}{D-2}P_{ab}},
$$
the <null shear>, and
$$
\boxed{\widehat\omega_{ab}=\widehat B_{[ab]}},
$$
the <null twist>, also called rotation.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 311 / 2 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-311/2/b/solution