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- Thread starter kent davidge
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I fully agree. The symbol itself is the field that does not care about coordinates and what actually is basis dependent are the components. Unfortunately, many people dont agree ...Personally, if I’m referring to the same tensor field simply in different coordinates, it usually makes more sense to me to put the bars over the indices.

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Thank you. For this to be valid, we should consider the same region of the manifold before and after the parametrization, correct? Because otherwise the domain of the functions would be different, what in turn makes the functions different.I fully agree. The symbol itself is the field that does not care about coordinates and what actually is basis dependent are the components. Unfortunately, many people dont agree ...

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It sounds like you’re thinking about overlapping charts on a manifold, in which case, yes, you should be looking at where the domains of the coordinate functions overlap. However, for many problems, the issue is not getting from one patch of spacetime to another, it’s what happens simply when we transform the coordinates we’re using (think of almost all problems in SR, where most reasonable choices of coordinates for inertial observers cover the entire manifold).Thank you. For this to be valid, we should consider the same region of the manifold before and after the parametrization, correct? Because otherwise the domain of the functions would be different, what in turn makes the functions different.

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Thanks!It sounds like you’re thinking about overlapping charts on a manifold, in which case, yes, you should be looking at where the domains of the coordinate functions overlap. However, for many problems, the issue is not getting from one patch of spacetime to another, it’s what happens simply when we transform the coordinates we’re using (think of almost all problems in SR, where most reasonable choices of coordinates for inertial observers cover the entire manifold).

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