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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 311

Affine geodesic evolution and the <Ricci identity> give the screen-projected optical equation
$$
U^c\nabla_c\widehat B_{ab}
=-\widehat B_{ac}\widehat B^c{}_b
-P_a{}^cP_b{}^dR_{ecfd}U^eU^f.
$$
Taking the screen trace yields
$$
\frac{d\theta}{d\lambda}
=-\widehat B_{ab}\widehat B^{ba}-R_{ab}U^aU^b.
$$
The optical decomposition and the antisymmetry of $\widehat\omega$ imply
$$
\widehat B_{ab}\widehat B^{ba}
=\frac{\theta^2}{D-2}
+\widehat\sigma_{ab}\widehat\sigma^{ab}
-\widehat\omega_{ab}\widehat\omega^{ab}.
$$
Therefore the $D$-dimensional <Null Raychaudhuri equation> is
$$
\boxed{
\frac{d\theta}{d\lambda}
=-\frac1{D-2}\theta^2
-\widehat\sigma_{ab}\widehat\sigma^{ab}
+\widehat\omega_{ab}\widehat\omega^{ab}
-R_{ab}U^aU^b},
$$
and the requested constant is $\boxed{A=D-2}$.

Solution

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

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

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

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