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joneall

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- 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.

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.