Components and basis transform differently

[Ratzinger]

Why do components and basis vectors transform under Lorentz transformation differently (inverse Lorentz for basis), whereas for orthogonal transformation components and basis are transformed by same matrix?

Thank you

 

[dextercioby]

Are u talking about the group theoretical approach to relativity,or differential geometry ?I think you're mixing them.Components & basis vectors would imply geometry,while Lorentz transformation & orthogonal transformation would imply $\mbox{O(3)/SO(3)}$ or $SO(3,1)$...?

So which one would u prefer?

Daniel.

 

[Ich]

Why do components and basis vectors transform under Lorentz transformation differently (inverse Lorentz for basis), whereas for orthogonal transformation components and basis are transformed by same matrix?

Thank you

I´m quite surprised that you want follow the dictatorship of relativism by tranforming even the basis. I expected you to rather conserve absolute values.

 

[dextercioby]

Well,that's the correct way to do it.Mathematcally.Physicists are rather sloppy,here,i must say,because are usually interested in how the commponents of (pseudo)tensors behave when a change of noninertial reference frames is done...They have no use for the transformation laws of the basis (co)vectors.

Daniel.

 

[pervect]

Why do components and basis vectors transform under Lorentz transformation differently (inverse Lorentz for basis), whereas for orthogonal transformation components and basis are transformed by same matrix?

Thank you

The basis and the components ony transform in the same manner for a very special set of coordinates - orthonormal coordinates to be precise.

Probably the very simplest way to see why coordinates and vectors transform differently is to look at what happens when you double the length of the basis vector. It should be easy to see that if you double the basis vector, you halve the coordinate value, and vica-versa. That is why coordinates and basis vectors transform differently (oppositely).

Note that when you double one of the basis vectors, you are no longer in an orthonormal basis (you are orthogonal, but not normal).

So there is nothing special about the Lorentz transform - in general ALL coordinates transform differently than basis vectors. (Orthonormal coordinate systems are the exception - of course, they are a very common and important exception).