Undergrad Is this answer about Lorentz transforms correct?

[kent davidge]

User Jack Fraser says that both $X$ and $\Lambda$ on his post are tensors. Is this right?

Post: https://www.quora.com/While-derivin...u-know-that-the-transformation-will-be-linear

 

[Orodruin]

No.

 

[Ibix]

$X$ is, $\Lambda$ is not.

Thinking in terms of components, the components of a tensor multiply the outer product of the basis vectors or one-forms, and the sum of the components times those basis entities forms the geometric object that is the tensor. The Lorentz transform is just a matrix of numbers that relate tensor components in one coordinate system to tensor components in another.

 

[haushofer]

How would a coordinate transformation transform under a coordinate transformation?

 

[kent davidge]

$X$ is, $\Lambda$ is not.

I think neither of them are tensors. The referred user talks about coordinate transformation, so his $X$ are coordinates.

How would a coordinate transformation transform under a coordinate transformation?

no way

 

[Ibix]

I think neither of them are tensors. The referred user talks about coordinate transformation, so his $X$ are coordinates.

But coordinates do not, in general, transform by $\Lambda^{\mu'}{}_\mu X^\mu$. He can get away with it in Einstein coordinates on flat spacetime because there's a trivial relationship between the coordinates and a set of "position vectors" defined in the tangent space associated with the origin.

So what he's doing is not - strictly - transforming coordinates. He's defining a set of vectors, associating them with the coordinates, transforming the vectors, and associating them with the transformed coordinates.

Treating $X$ as any generic vector is fine and my previous answer stands. If you want to interpret $X$ as coordinates then you need to make additional assumptions.