What is the Topological Structure of Hypersurfaces in General Relativity?

[Ravi Mohan]

Hi, I am studying Hypersurfaces and the intrinsic/extrinsic geometry from http://www.blau.itp.unibe.ch/newlecturesGR.pdf with the aim of understanding the Hamiltonian formalism of GR. Although interesting, the notions introduced in these notes lack mathematical rigor.

I am looking for a text which introduces these topics with the full topological structure (written by a mathematician would be better). Thanks.

 

[andrewkirk]

When frustrated by the lack of mathematical rigour in GR texts I turned to John Lee's 'Riemannian Manifolds', which is a mathematical text that is, as expected, rigorous and (in my opinion) reasonably easy to follow. I think it was somebody here on physicsforums that recommended it to me.

There is a prequel by Lee: 'Smooth Manifolds', which may also be useful, depending on what notions you need to use. I have not felt the need to buy it yet but the result is that I am a little undercooked on vector field flows and Lie derivatives.

The downside of Lee's book is that it doesn't specifically address Pseudo-Riemannian manifolds, which is what are used in GR. In many cases the distinction doesn't matter. But sometimes it does and then you have to adapt Lee's proofs to the Pseudo-Riemannian case yourself. Unfortunately I don't know of any mathematical texts (ie not by physicists) that address Pseudo-Riemannian manifolds. Perhaps others can suggest some.

 

[martinbn]

I haven't looked in the notes, what exactly do you fine non-rigorous? Have you looked at O'Neil's book on semi-Riemannian geometry?

 

[Frimus]

Unfortunately I don't know of any mathematical texts (ie not by physicists) that address Pseudo-Riemannian manifolds.

Maybe O'Neills https://www.amazon.com/dp/0125267401/?tag=pfamazon01-20 could make some good

 

[Ravi Mohan]

Thanks andrewkirk, I will look into John Lee's 'Riemannian Manifolds'

I haven't looked in the notes, what exactly do you fine non-rigorous? Have you looked at O'Neil's book on semi-Riemannian geometry?

For instance the author defines an embedding as a map
$$\Phi: \Sigma =\Sigma_n \hookrightarrow M_{n+1}$$
Now this definition raises questions such as

• What is the nature of this map (homeomorphism or diffeomorphism)? Wikipedia gives more rigorous definition of $\Phi$ as a homeomorphism onto its image.
• The author says that this embedding is represented by the parametric equations $\Phi:x^{\alpha}(y^a)$ How are these equations actually composed? I can only think as $y^{\alpha}(\Phi^{-1}(\text{something with }x^a))$ where $y^{\alpha}, x^a$ are charts of $M,\Sigma$ respectively.

I think the author does not care about the structures like topology and even manifold in the notes.

The references mentioned in Wikipedia are https://en.wikipedia.org/wiki/Embedding#References, but I was not sure which one to follow.

I will also look in O'Neill's book on semi-Riemannian geometry. Thanks again.

 

[martinbn]

For the first one, he actually explains what he means by an embedding, at the end of page 308 and the next page. About the notations it should be clear from the examples that he gives.

 

[Ravi Mohan]

For the first one, he actually explains what he means by an embedding, at the end of page 308 and the next page. About the notations it should be clear from the examples that he gives.

I understand what he is explaining and I think it should be enough to serve the purpose (which is understanding the canonical GR). But again it would be nice to have a complete formal definition at one's disposal (like the way Carroll does in his notes).

About the map $\Phi$, the author says

Strictly speaking, such a map is called an (injective) immersion, while an embedding has to satisfy a slightly stronger topological condition, but since we are not concerned with global issues, and since I have not even tried to define what a manifold is (beyond the remarks in section 4.11), it would be ridiculous to worry about such things here and this is more than good enough.

So I was wondering what is the stronger topological condition.

 

[martinbn]

O'Neil might be to your taste, but at times it is about geometry that need not be relevant to relativity. You could perhaps read it selectively.

 

[Cruz Martinez]

So I was wondering what is the stronger topological condition.

The stronger topological condition is that the immersion $f: M \rightarrow N$ is an embedding if $f$ is a homeomorphism of $M$ with $f(M)$ with the subspace topology. This is the case whenever $M$ is compact for example.

 

[George Jones]

You might try "3+1 Formalism in General Relativity: Bases of Numerical Relativity" by Eric Gourgoulhon,

https://www.amazon.com/dp/3642245242/?tag=pfamazon01-20