# I Spacetime Hypersurfaces and Foliations

#### kent davidge

Can anyone recommend me good books on the topics of Spacetime Hypersurfaces and Foliations of Space Time?

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#### pervect

Staff Emeritus
I can't recommend a book, but I can give you the name of the theorem that discusses your question in mathematical detail, "Frobenius theorem", <<wiki link>>. Wiki, however, while it mentions the theorem and gives a terse description, is probably not going to be a great place to learn the details. Looking up the references in the wiki article may be helpful (however I haven't read them personally, so I can't vouch for the specific references quoted).

wiki said:
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, the theorem gives necessary and sufficient conditions for the existence of a foliation by maximal integral manifolds each of whose tangent bundles are spanned by a given family of vector fields (satisfying an integrability condition) in much the same way as an integral curve may be assigned to a single vector field. The theorem is foundational in differential topology and calculus on manifolds.
It seems to me there should be a statement of the theroem involving language that I find more intuitive, but I haven't seen any that really speak to me. What's more intuitive to me are Integral curves of vector fields, and how they wind up describing time-like congruences.

#### George Jones

Staff Emeritus
Gold Member
Can anyone recommend me good books on the topics of Spacetime Hypersurfaces and Foliations of Space Time?
For hypersurfaces, and for the conditions when two spacetimes are joined along a hypersurface, you might want to look at the first three chapters or so of the advanced, but somewhat pedagogical, "A Relativist's Toolkit: The Mathematics of Black-Mechanics" by Eric Poisson,

https://www.amazon.com/dp/0521537800/?tag=pfamazon01-20

"Spacetime Hypersurfaces and Foliations"

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