- Summary
- Looking for fundamental reason why we only look at independent coordinates in GR

I can see that by the tensor transformation law of the Kronecker delta that

##\frac{\partial x^a}{\partial x^b}=\delta^a_b##

And thus coordinates must be independent of eachother.

But is there a more straightforward and fundamental reason why we don’t consider dependent coordinates? Is it just built-in to the transformation law?

Obviously if we could have dependent coordinates, then for a metric like this plane wave one:

##ds^2=-2dudv+f(u)g(u)dy^2+f(u)/g(u)dz^2##

...I could rescale my y and z coordinates by:

##dy’^2=f(u)g(u)dy^2## and ##dz’^2=f(u)/g(u)dz^2##

And have

##ds^2=-2dudv+dy’^2+dz’^2##

And thus turn the metric (or any metric by such a rescaling) to flat spacetime. So I see why it’s good to have linearly independent coordinates to forbid transformations like the above.

But why? Is this requirement the very motivation for the tensor transformation laws? What’s the underlying reason?

Thanks.

##\frac{\partial x^a}{\partial x^b}=\delta^a_b##

And thus coordinates must be independent of eachother.

But is there a more straightforward and fundamental reason why we don’t consider dependent coordinates? Is it just built-in to the transformation law?

Obviously if we could have dependent coordinates, then for a metric like this plane wave one:

##ds^2=-2dudv+f(u)g(u)dy^2+f(u)/g(u)dz^2##

...I could rescale my y and z coordinates by:

##dy’^2=f(u)g(u)dy^2## and ##dz’^2=f(u)/g(u)dz^2##

And have

##ds^2=-2dudv+dy’^2+dz’^2##

And thus turn the metric (or any metric by such a rescaling) to flat spacetime. So I see why it’s good to have linearly independent coordinates to forbid transformations like the above.

But why? Is this requirement the very motivation for the tensor transformation laws? What’s the underlying reason?

Thanks.