# Foliation of hypersurfaces in General Relativity

As I understand it, it is usual in GR to define a foliation of Spacetime by a family of spacelike hypersurfaces "indexed" by the time variable. Then, in the context of Schwarzschild metric in spherical coordinates, is it correct to define the same way a foliation of these spacelike hypersurfaces by a family of 2-spheres indexed by the radial variable? Can we find a correspondence with a topological ball of r radius? And does this second foliation possess the same properties as the first one? Is it correct to think of the spacelike hypersurfaces and of the 2-spheres as families each consisting of infinite elements empiled on each other along the t and r direction respectively ?

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By the Frobenius theorem you get a foilation if the tangent bundle restricted to that space is closed under the Lie bracket. I.e. you take a look at a basis $L_i$ of the tangent space of interest (with $i=1,...n$, where $n$ is smaller than the dimension of the whole tangent space) and check if all the expressions $[L_i,L_j]$ are again vectors within that space.

For spherical coordinates you'll find $$L_\theta=g_1(\theta,\phi)\partial_\theta+g_2(\theta,\phi)\partial_\phi$$ and $$L_\phi=h_1(\theta,\phi)\partial_\theta+h_2(\theta,\phi)\partial_\phi,$$ in every patch, so you're good. Probably an example is something like $$L_\theta=c_r \partial_\theta,\ \ \ \ L_\phi=c_r f(\theta) \partial_\phi.$$

Here I didn't care about singularities of any kind. The topology of compact spaces is of course different than others. Do you know the hairy ball theorem?

As far as your question goes, the one dimensional case is even trivial so to speak. But there are some situations in numerical general relativity and loop quantum gravity, where the point is really to find nice, spacelike foliations, but that's not what you're asking for. As far as restrictions from Riemann geometry go, that should be all.

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...in the context of Schwarzschild metric in spherical coordinates, is it correct to define the same way a foliation of these spacelike hypersurfaces by a family of 2-spheres indexed by the radial variable?

Only for 2-spheres beneath the event horizon, i.e. those for which r < rhorizon. r = constant describes a timelike hypersurface, if r > rhorizon.

Can we find a correspondence with a topological ball of r radius?

Not sure what you're asking here, but if you're wondering whether the topology of a hypersurface defined by r = constant is S2, the answer is 'no'. Hypersurfaces are hyper surfaces; they are three-dimensional. Their topology in this case is R X S2.

Is it correct to think of the spacelike hypersurfaces and of the 2-spheres as families each consisting of infinite elements empiled on each other along the t and r direction respectively ?

Not quite. The spacelike hypersurfaces are part of a single family, each member of which is a 3-dimensional volume of topology R X S2. These members are indexed by the radius r of the associated 2-sphere, as long as r < rhorizon.

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