cianfa72
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- Spatial homogeneity and isotropy provide a spacetime foliation by spacelike 3d hypersurfaces
From this lecture at minute 15:00 onwards, the conditions for spacetime spatially homogenous and isotropic imply the existence of 6 ##\mathbb R##-linear independent spacelike Killing Vector Fields (KVFs) w.r.t. the metric tensor ##g##.
The lecturer (Dr. Schuller) claims that such 6 independent KVFs define spacelike hypersurfaces foliating the spacetime. Those KVFs close w.r.t. Lie bracket commutator, therefore by Frobenius's theorem they are integrable.
My question is: how one can conclude that the integral of such (integrable) subbundle actually provides a foliation by hypersurfaces of dimension 3 ?
The lecturer (Dr. Schuller) claims that such 6 independent KVFs define spacelike hypersurfaces foliating the spacetime. Those KVFs close w.r.t. Lie bracket commutator, therefore by Frobenius's theorem they are integrable.
My question is: how one can conclude that the integral of such (integrable) subbundle actually provides a foliation by hypersurfaces of dimension 3 ?
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