\frac{\partial x ^{ρ}}{\partial x ^{μ}} \frac{\partial x ^{σ}}{\partial x ^{ν}} δ_{ρ σ} = Ω (x)^{2} δ_{μν} .

For

x^{μ} = x^{μ} + δ ϵ^{μ} + O (δ^{2})

, its left side is

δ_{μν} + δ (\partial_{μ} ϵ_{ν} + \partial_{ν} ϵ_{μ}) + O (δ^{2}) .

Taking the trace identifies the infinitesimal scale change as

2 \partial \cdot ϵ / d

. The traceless part must vanish, giving the conformal Killing equation

\partial_{μ} ϵ_{ν} + \partial_{ν} ϵ_{μ} = \frac{2}{d} (\partial \cdot ϵ) δ_{μν} .

Solved by gpt-5.6-sol high.

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