Understanding Frame Fields in GR: A Beginner's Guide

[joneall]

TL;DR

Rovelli's notation in "General relativity: the essentials" never mentions manifolds, but seems to use them anyway. Why?

I'm having trouble with Rovelli's new book, partly because the info in it is pretty condensed, but also because his subjects are often very different from those in other books on GR like the one by Schutz. For one thing, he never uses the term "manifold", but talks about frame fields, which seem to me to be defined relative to manifolds. He begins with a plane tangent to an arbitrary surface at a point p. Isn't this an example of a local manifold? He supposes arbitrary general coordinates (a term which is the section title but which he has not yet defined) on the surface and takes $$x_p^a$$ as the general coordinate of point p on the surface. (I don't understand why my latex formulas are not showing up correctly, at least not when I click on "Preview". Ooops, they do work, just not on "Preview". Why would that be?)
If the map from this point on the surface to the tangent plane is $$X_p^i (x^a )$$, then the frame field is defined as
$$e_a^i = \frac{\partial X_p^i ( x^a )}{\partial x^a } \bigg|^{ x^a = x_p^a }$$
which is a field on the surface.
I wanted to obtain more info on this subject (Wikipedia is, as usual, unhelpful, being written by experts for other experts, rather than for us poor preterites.), but none of my GR books (Carroll, Schutz, Hartle) mention frame fields.
So am I right about manifolds? And why does he avoid them? And can you recommend any other sources for this subject, which a search of this forum does not find either.

 

[Orodruin]

You cannot do GR without using manifolds at some level. Even if you decide not to go into the details of manifolds or decide to not explicitly call what you are doing calculus on manifolds ...

 

[joneall]

You cannot do GR without using manifolds at some level. Even if you decide not to go into the details of manifolds or decide to not explicitly call what you are doing calculus on manifolds ...

That's what I thought. But Rovelli does not mention them. They are not in the index. Maybe you are suggesting I drop the book...?

 

[PeroK]

none of my GR books (Carroll, Schutz, Hartle) mention frame fields.
So am I right about manifolds? And why does he avoid them? And can you recommend any other sources for this subject, which a search of this forum does not find either.

If you've studied those books you ought to know a lot about GR. What does Rovelli's book add?

 

[vanhees71]

Indeed, and at least in his arxiv lecture notes Carroll treats tetrads, and afaik "frame field" is just another name for tetrads.

 

[dextercioby]

I find this approach interesting, for it seems to emphasize exactly what some other sources are missing. I would expect the QG expert Rovelli to write a book for people who wish to pursue his domain, so he must teach GR from a different perspective. Maybe not mentioning manifolds, but teaching the (for me abstract and demanding) fiber bundle theory is a way to go. I don't have the book, but I'd be happy to find some approach to a known theory for which there are no other 10 to 100 resources available.

And speaking about frame fields, here is a thorough approach:
https://physics.stackexchange.com/questions/667558/how-do-we-mathematically-define-a-vielbein-field

 

[PeterDonis]

afaik "frame field" is just another name for tetrads.

More precisely, a frame field is a field of tetrads--one tetrad for each point in a spacetime (or a region of a spacetime).

 

[pervect]

Summary:: Rovelli's notation in "General relativity: the essentials" never mentions manifolds, but seems to use them anyway. Why?

I'm not familiar with Rovelli's work, sorry, but I have a few other comments.

He begins with a plane tangent to an arbitrary surface at a point p. Isn't this an example of a local manifold?

Most authors call this a tangent space to a manifold. If you are familiar with the notion of the tangent vector to a curve, the tangent space at some point p in a manifold would be the set of tangent vectors at point p to all possible (continuous) curves that pass through point p. More formal treatments might talk about this in terms of derivative operators.

If you're not familiar with the notion of a tangent vector to a curve, want a more formal treatment, or want more discussion of some other sort, we could go into that in more detail. You do seem a bit uncertain, as you ask a question about it.

He supposes arbitrary general coordinates (a term which is the section title but which he has not yet defined) on the surface and takes $$x_p^a$$ as the general coordinate of point p on the surface.

If you replace "surface" as Rovelli uses it with "manifold" here, you should be on-track.