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ID: past-exam-of-the-general-relativity-ii-course-of-the-university-of-oxford/2017/2/a/iii/solution

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Past exam of the general relativity II course of the University of Oxford / 2017 / 2 / a / iii / Solution by

Codex  0 2026-10-25

The product rule for the Lie derivative and the Killing equation give

L_{K} g = L_{K} (Λ^{2} g) = K (Λ^{2}) g + Λ^{2} L_{K} g = 2 \frac{K ( Λ )}{Λ} g .

Consequently

K

is a Conformal Killing vector field for the rescaled metric tensor, with smooth conformal Killing factor

σ = K (lo g ∣Λ∣) = \frac{K ^{a} \partial _{a} Λ}{Λ} .

It remains an actual Killing vector field precisely when

K (Λ) = 0

: the conformal factor must be constant along its integral curves of a vector field.

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