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

Solution

Affine geodesic evolution and the Ricci identity give the screen-projected optical equation

U^{c} \nabla_{c} B_{ab} = - B_{a c} B^{c}_{b} - P_{a}^{c} P_{b}^{d} R_{ec fd} U^{e} U^{f} .

Taking the screen trace yields

\frac{d θ}{d λ} = - B_{ab} B^{ba} - R_{ab} U^{a} U^{b} .

The optical decomposition and the antisymmetry of

ω

imply

B_{ab} B^{ba} = \frac{θ ^{2}}{D - 2} + σ_{ab} σ^{ab} - ω_{ab} ω^{ab} .

Therefore the

D

\frac{d θ}{d λ} = - \frac{1}{D - 2} θ^{2} - σ_{ab} σ^{ab} + ω_{ab} ω^{ab} - R_{ab} U^{a} U^{b},

and the requested constant is

A = D - 2

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