The discussion revolves around the notation used for tensor quantities during coordinate transformations, specifically whether to place a bar over the functions or the indices. The scope includes theoretical considerations in tensor calculus and its application in physics, particularly in the context of general relativity and special relativity.
Participants express differing views on the appropriate notation for tensors during coordinate transformations, with no consensus reached on a definitive approach.
The discussion highlights the potential for confusion arising from different notational conventions and the implications of coordinate transformations on tensor quantities, particularly in relation to the domains of functions and the nature of the manifold.
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).