Tensors: Bar Symbol Over Functions or Indices?

[kent davidge]

When dealing with any tensor quantity, when making a coordinate transformation, we should put a bar (or whatever symbol) over the functions or over the indices? For exemple, should the metric coefficients $g_{\mu \nu}$ be written in another coord sys as $\bar g_{\mu \nu}$ or as $g_{\bar \mu \bar \nu}$?

 

[Pencilvester]

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.

 

[Orodruin]

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.

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 ...

 

[kent davidge]

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.

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 ...

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.

 

[Pencilvester]

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.

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]

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).

Thanks!