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Two days ago, after running through Func Diff Geometry chapter, spent a while
reading Carroll Spacetime+Geometry, in early chapters. Particularly relevant
was the discussion of 1-forms ("dual vectors") and vectors and tensors. Tensor
language isn't really even used in this book, except in Appendix C. Should
probably work through that appendix in another week or two.
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Starting on chapter 4, about basis fields (basic vector fields and basis
one-form fields), both coordinate and non-coordinate.
What is an example of a non-coordinate basis? Rotations! This will probably
come back up as... Lorentz transforms? Rotations of spacetime, as a change of
frame.
Commutators!
The commutator of two coordinate basis fields is zero. This is one way to tell
if basis fields are coordinate or non-coordinate. Is this sufficient proof?
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