Books on Spacetime Hypersurfaces & Foliations of Space Time

[kent davidge]

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

 

[pervect]

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).

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]

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