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So I have read that components of vectors transform linearly between frames like $$ A'^\mu = \frac{\partial x'^\mu}{\partial x^\nu} A^\nu .$$

I have also read that coordinates in general do not transform linearly between coordinate systems, however their differentials do:

$$ dx'^\mu = \frac{\partial x'^\mu}{\partial x^\nu} dx^\nu .$$

What I cannot grasp is how on earth can vector components transform between coordinate systems linearly when the coordinates do not? What if the vector in question is a position vector, then the components are the coordinates and then we come to a contradiction? At this point I become confused.

Also, in deriving the vector component transformation, I have always seen the example of transforming from a Cartesian $x$,$y$ plane to another one rotated at an angle $\theta$ relative to the first. Well in this example the components of the vector transform linearly and I can completely see this, but so do the coordinates!

Any help would be much appreciated!

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  • $\begingroup$ What do you mean by coordinates do not transform linearly? The transformation between spacetime coordinates is just a Lorentz transformation $\Lambda_{\mu}^{\nu}$. $\endgroup$ – InertialObserver Jan 2 '17 at 19:42
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    $\begingroup$ Transforming from Cartesian to spherical polar is not linear $\endgroup$ – Matt0410 Jan 2 '17 at 19:44
  • $\begingroup$ Oh, so you are saying the transformation between different coordinates in that sense, not in the sense of "different reference frame". $\endgroup$ – InertialObserver Jan 2 '17 at 19:45
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    $\begingroup$ Do you mind my asking where you heard that components of a vector transform linearly? That is not quite correct. $\endgroup$ – InertialObserver Jan 2 '17 at 20:01
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    $\begingroup$ Well the first equation above is in all textbooks I have read on General Relativity. Doesn't that imply a linear sum of the unprimed components? $\endgroup$ – Matt0410 Jan 2 '17 at 20:11
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In a general curved spacetime, vectors are only defined in the tangent space of individual points, and are by definition part of a vector space. This is the only thing that makes sense since one cannot assign a consistent rule to map vectors at one point to another in a curved spacetime. This is related to the fact that there is a non-trivial holonomy when there is non-zero curvature. Coordinates on the other hand are defined on larger patches of the spacetime.

General transformations of coordinates could be non-linear, but they would always induce a local transformation on vectors within their own tangent spaces. This would then result in a linear transformation on the vectors.

The example of the Cartesian plane is confusing because it has zero curvature. Try this with some other non-trivial manifold like a sphere.

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For clarification you have to distinguish between coordinate systems and sets of base vectors. The linearity in transforming the vector components is a consequence of the properties of vector spaces. Coordinate transformations are somewhat different from base changes.

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