That the general spacetime reduces to Minkowski spacetime in a small neighbourhood can be proven, quite easy actually, by introducing Riemann normal coordinates.
Indeed it should be possible in much the same way as SR (Special Relativity) allows us to prove the Principle of Relativity (that all physical laws are the same in inertial systems). I haven't made a proof so far, but I have at least found an argument for it:
Quite generally GR (General...
It seems as if Wald doesn't mention anywhere that the metric of a curved spacetime must reduce to the metric of flat spacetime in a sufficiently small section of spacetime. I know this is one of the formulations of the Equivalence principle, but not the one that Wald uses. Wald uses the weak...
Before I started learning GR I made a thorough research about the different books available. I also found the following 2 links which I found quite helpful:
http://arxiv.org/abs/0806.2316
http://math.ucr.edu/home/baez/physics/Administrivia/rel_booklist.html#intro_gr
I think I got it now...
I just realized my post before was a mess, so I have edited it to make it more clear what I mean.
I understand the abstract index notation, though I still have some difficulty in interpreting the subscript on the covariant derivative. The subscript mirrors the "extra" covector associated with...
I'm having some trouble understanding Wald's definition of a derivative operator (also called the covariant derivative). When Wald attaches a lower index to the derivative operator what does that mean? How should I interpret his requirement 4 and 5: t(f)=t^a\nabla_a f and \nabla_a \nabla_b...
Ah yes the Hartle book is also very good, I had the book in an introductory course to GR. Thank you for your recommendation though. While I find it very well written it has a major flaw: It doesn't teach you how to solve Einsteins equation. It postulates the solutions. This is the price you have...
Great, thank you for taking time to write such a detailed answer.
I have a book by John M. Lee on Smooth Manifolds that I have studied a bit. I remember seeing a chapter about the tangent bundle, though I haven't read it. Maybe I should spend some time studying this book some in detail. It is...
Thank you, your post really helped a lot. It seems to me that Wald isn't completely consistent with his own definitions of a metric and a metric tensor since he use the 2 words as synonyms.
I am not quite sure what you mean when you say that the tensor field F takes a point to the pair...
Thank you for answer. I think there might be some insight hidden in some mathematical precision, so let me see if I get this straight:
Let T(k,l) be the set of all tensors of type (k,l). Let M be a manifold. A tensor field of type (k,l) is then a map, F : M \rightarrow T(k,l) (where the...
I have started to teach myself General Relativity and I have been pointed to a book by Robert M. Wald called General Relativity. I really like it actually, I like how it doesn't skip the math behind the theory. It makes it appear more beautiful to me. However I think the book is quite vague...