# Does spacetime position not form a four-vector?

You are correct.

Position is a vector when you are working in a vector space, since, well, it is a vector space. Even then, if you use a nonlinear coordinate system, the coordinates of a point expressed in that coordinate system will not behave as a vector, since a nonlinear coordinate system is basically a nonlinear map from the vector space to $\mathbb{R}^n$, and nonlinear maps do not preserve the linear structure.

On a manifold, there is no sense in attempting to "vectorize" points. A point is a point, an element of the manifold, a vector is a vector, element of a tangent space at a point. Of course you can map points into $n$-tuples, that is part of the definition of a topological manifold, but there is no reason why the inverse of this map should carry the linear structure over to the manifold.

And now, for a purely personal opinion: While Carroll's book is really good, the physicist's way of attempting to categorize **everything** by "transformation properties" is extremely counterproductive, and leads to such misunderstandings as you have overcome here. If one learns proper manifold theory, this is clear from the start...

Great reasoning: as in Uldreth's fantastic answer but I would add one more thing that may help cement your good understanding in place.

Co-ordinates are absolutely not vectors, they are *labels* on charts and are no more vectors than your street address is a vector. Almost certainly the reason people make the implication that you have correctly identified as wrong is this: in *flat* space (*i.e.* Euclidean, Minkowski or generally signatured spaces), *affine* co-ordinates for positions can have *two* roles: they are *both* labels *and* (once one has chosen an origin) superposition weights that combine linear basis *tangents* to the Euclidean (Minkowski ...) manifold linearly to yield a general tangent to the manifold. If you think about it, what I have just said is a slightly different take on Uldreth's second paragraph that begins "Position is a vector ...".

It's worth saying that I definitely recall the following learning sequence as a teenager. When beginning high school at about age 11, I was first shown co-ordinates (Cartesian of course) as *labels*. I suspect that this is how they are introduced to all children. I distinctly recall the idea that only two years later was the notion (that only works for Cartesian and generally affine co-ordinates) of a point's co-ordinates as a position *vector* introduced. Before that I had a very clear idea of a vector as a displacement or link between *two* points, an idea that, through the appropriate limit, leads to the tangent idea in a general manifold. On reading your question, I laugh when I recall the teacher's implying that the second role of co-ordinates as position vectors was a "new and advanced" way to look at vectors, whereas on the contrary it is a way of thinking that you correctly understand to be very limited and only workable in the affine case.

Here is a bare bones easy way to see that coordinate tuples are not 4-vectors.

Start in an inertial coordinate system in flat spacetime. Change the coordinate system with a constant translation:

$x' = x + A $

$y' = y$

$z' = z$

$t' = t$

Even in this idealistic case, 4-vectors and coordinate tuples transform differently. The components of the 4-vectors don't change at all in this case, while the coordinate tuples do.