Kerr spacetime admits no conformally flat slicing of the form t = F(r, θ, a), with the obstruction appearing at fifth order in spin, and Kerr-de Sitter admits no analytic spatially flat slicing beyond linear order.
Eliminating all but Θ- dependencies in the angular equations gives 1 2ρ2 ∂ϑ ( Oϑϑ − Oϕϕ sin2 ϑ ) + ∂ϕ ( Oϑϕ ρ2 sin2 ϑ ) = ∂ϑ [ sin ϑ ∂ϑ ( Θ sin ϑ )] + 1 sin2 ϑ ∂2 ϕ Θ = 0
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On the absence of conformally flat slicings of the Kerr spacetime
Kerr spacetime admits no conformally flat slicing of the form t = F(r, θ, a), with the obstruction appearing at fifth order in spin, and Kerr-de Sitter admits no analytic spatially flat slicing beyond linear order.