Main Question or Discussion Point
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 about the difference between a metric and the metric (as defined on page 22 if anyone has the book, it shouldn't be neccesary however). I understand that a tensor field is a quite general concept. For each point p in a manifold it simply specifies a tensor over the tangent space. I understand that a metric is a non-degenerate, symmetric tensor field of type (0,2) (that is: it takes 2 tangent vectors and turns it into a real number). Thus it defines a pseudo (not neccesarily positive definite) inner product at each tangent space. It is not neccessary to specify this tensor field to be linear, because due to the linearity of tensors, all tensor fields are linear at a given point. How should i interpret the metric however? Is the tensor assigned to all points allowed to change, that is: Given two different tangent spaces, the metric gives two different tensors? Is it constant? What excactly is the metric tensor, is it just the same as the metric?