The discussion centers on the notation used for tensor quantities during coordinate transformations, specifically whether to place a bar over the functions or the indices. Participants agree that the tensor symbol itself remains invariant, while the components are basis-dependent. The consensus leans towards placing bars over the indices when referring to the same tensor field in different coordinates. Additionally, the importance of considering overlapping charts on a manifold is emphasized for valid transformations.
PREREQUISITESMathematicians, physicists, and students studying differential geometry, particularly those interested in tensor analysis and coordinate transformations in the context of General and Special Relativity.
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 don't agree ...Pencilvester said: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.
Pencilvester said:I think this is more of a notational preference. There might be a clearer choice depending on the type of problem you’re working, though. 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.
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.Orodruin said: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 don't agree ...
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).kent davidge said: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.
Thanks!Pencilvester said: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).