Transforming Vectors and Tensors

[fog37]

TL;DR Summary

Vectors and Tensors types of transformations

Hello,

I was pondering on the following: a vector is a specific entity whose existence is independent of the coordinate system used to describe it.
To start, I guess I need to state that we are describing the vector from the same reference frame using different coordinate systems (Cartesian, Cartesian but rotated, cylindrical, spherical, elliptical, etc.). These coordinate system examples are all "applied" to the flat 3-dimensional Euclidean space (we are not dealing with curved geometries).

Question: in 2D, using the simple Cartesian coordinate system $C_1$, an arbitrary vector $A$ has a description $A= [A_1, A_2]$ where the coordinates are the scaling for the basis vectors. The coordinates $A_1$ and $A_2$ varies if we change coordinate system, for example, using a coordinate system $C_2$ with the same origin $O_1=O_2$ but a rotated version of $C_1$. So far so good.

There are coordinate transformations (2x2 matrices), like rotations, that just change the descriptions of the vector without changing its length or actual direction. What is the category of these types of transformations called? The group of rotations is an example. Any other examples? A translated coordinate system $C_3$ has a different origin than $C_1$ but parallel axes.

There are also transformations that change the length and/or direction of a vector converting it into a totally new vector. How do we call those transformations?

Thank you!

 

[fresh_42]

Linear coordinate transformations in general are given by any regular matrix $GL(V)$, which includes stretches and compressions. If they respect lengths then we will end up with orthogonal transformations, which are all rotations plus possible reflections $O(V)$.

Coordinate transformations which change the origin are no longer linear, but affine linear. This means the can be written as the sum of a linear coordinate transformation plus a translation.

There are still other coordinate systems which do not fit in either of the above categories: polar coordinate systems, e.g. spherical or cylindrical coordinates. They are not Cartesian and have the additional problem, that the origin isn't well defined, since it has no angle of direction.

 

[fog37]

Thank you fresh_42.

Is it correct to state that spherical, elliptical, cylindrical coordinate systems (which are methods to assign anumerical description to points in space), since they have coordinate surfaces that are curved, are examples of curved coordinate systems .
On the other hand, the Cartesian coordinate system is not curve because its coordinate surfaces are planes, correct?

All the coordinate systems mentioned above (curved or not, if that is how we call them) are describing the non-curved, i.e. flat, 3 dimensional Euclidean space.

In the case of relativity (or even in describing the geometry of the spherical planet earth) we are dealing with a four dimensional curved space called spacetime describable using non-Euclidean geometry and the suitable coordinate system type in that case is certainly curved...Is that true? I know that locally, the curved space is flat (allowing us to use Euclidean geometry) since it is a smooth manifold...

 

[fresh_42]

Not sure what you mean by a curved coordinate system. Derived from your example with (hyper)planes in Cartesian coordinate system, namely what happens if we fix one coordinate, then yes, you could say this. On the other hand, fixing the angle in polar coordinates $(r,\varphi)$ give us a straight line with slope $\tan \varphi$, fixing the radius give us a circle. So in this way they are curved.

Coordinate system in general only means that every point can be uniquely described. In polar coordinates we need to add the condition, that the origin has angle zero to be unique, but this is only a technical issue. Yes, they all came from the description of points in $\mathbb{R}^n$ historically. However you cannot say they are restricted to flat spaces. You could as well use them to describe spheres or tori without explicitly relate to their embeddings in $\mathbb{R}^3$. You are right, that curved spaces are usually locally flat and we use those coordinate systems locally then, but this is because we do not automatically have global coordinates. They do not exist necessarily. In such curved cases local coordinate systems are the solution, or coordinate systems which travel along with some point of the space. Altitude and latitude on Earth are an example of non Cartesian coordinates.

Spacetime, i.e. the universe is an example of an object that doesn't allow global coordinates of which kind ever. Locally it is a Cartesian coordinate system. The measurement of distances is a different story as we use the Minkowski metric and not the Euclidean distance.

 

[fog37]

Thanks again, Fresh_42

Sticking with 3D Euclidean space, we can use orthogonal and non-orthogonal coordinate systems to describe the position of points in space. In non orthogonal systems, the basis vectors are not orthogonal to each other, hence not independent of each other.

Basis vectors in (some) non-orthogonal systems are not free vectors: they change direction from point to point (see polar r and $\theta$ vectors). To say that those basis vectors change direction means that their direction varies relative to something...What is that something? Another coordinate system and its axes taken as the standard of comparison, correct? For example, the basis vector $\hat{\theta}$ varies relative to the Cartesian axes...

I am reading about covariant and contravariant and the concepts are explained using, as an example,
a rotation transformation. The same identical vector $A$ in the original coordinate system can change description (not length or direction) in a different rotated coordinate system. The new description is given by a certain transformation $T$. So far so good. The book then talks about the unit basis vectors: the basis vectors of the original coordinate system can also be expressed in the new coordinate system but the transformation is different. Are the original basis vectors rotated to coincide with the new basis vectors and the transformation? IF so. is the required transformation the inverse matrix of $T$? Why do we need to rotate the basis vectors changing them into some new vectors? I am a little confused

 

[fresh_42]

the basis vectors are not orthogonal to each other, hence not independent of each other.

This is not true. $\left\{\begin{bmatrix}1\\0\end{bmatrix}\, , \,\begin{bmatrix}1\\1\end{bmatrix}\right\}$ are not orthogonal, but linearly independent. What you probably mean are polar versus Cartesian coordinates, but the question about orthogonality is a different one.

So far so good.

Congratulations, since this is the most difficult part to understand. Given a vector we have automatically its coordinates in mind, since this is how we describe it. We cannot say: points to the upper right corner. And even this requires a definition about up and down, right and left. So to see vectors independently from the scales we apply to describe it is actually the most difficult part. Simply because we need coordinates to talk about it.

But which scales we set to measure direction and length is up to us. That's why many integrals change between polar and Cartesian coordinates, using whatever is easier to integrate.

I do not see how the transformation is a different one. The only difference is whether the vector is expressed in the new coordinates given the old, or expressed in the old coordinates given the new. One uses the transformation, the other its inverse.

Say you live in Europe and want to rent a car for your US trip. The coordinates of your car maybe mileage and top speed. You are offered cars described by (miles, miles per hour) and you are used to (kilometer, kilometer per hour). Then you need to transform the coordinates, in this case by a proportional stretch. If you live in the US and it's the other way round then you have to use the inverse transformation, a proportional compression.

It is the same thing with more complicated examples. As I don't know what exactly you have read, I can't answer it precisely. Contra- and covariant usually describes whether transformation arrows turn its direction or not. For example if we consider the function "taking the inverse matrix", then $T\, : \,U \longrightarrow V$ turns into $T^{-1}\, : \,V\longrightarrow U$, hence inversion is contravariant. However, physicists use the terms differently, so you should follow the conventions of your book.

It is confusing, and the standard example are vector spaces $V$ and their duals $V^*$ which are all linear functions into the scalar field, say e.g. the reals, $V \longrightarrow \mathbb{R}$. Now we can establish the following correspondence $V \longleftrightarrow V^*$ by $v \longleftrightarrow \langle v,- \rangle$ where $\langle v,- \rangle$ is the inner product by $v$, i.e. the mapping $u \longmapsto \langle v,u \rangle$ in $V^*$. And now it gets messy if we consider basis transformations in $V$, so we have $4$ bases in total, $2$ in $V$ and the corresponding $2$ in $V^*$. One of the most important things to learn in mathematics as well as in physics is to know where you are at which stage of the formulas. E.g. $D_p f$ is a differential $D$ on function spaces at a point $p$ for the function $f$. Now $D$ is a linear transformation, $p$ a point on some surface, $f$ and a function of named surface to somewhere. So at each stage of $D_pf$ we are in a completely different environment. Physicist normally use contra- and covariant to distinguish between $V$ and $V^*$.

 

[fog37]

Ok, thanks!

Rotating the vector versus rotating the coordinate system and keeping the vector the same: a vector $A$ in a 2D coordinate system can be transformed into a different vector by a rotation transformation by an angle $\theta$ while the coordinate system remains the same. The other situation when there are two different coordinate systems, one rotated and one non-rotated but vector $AA$ remains the same (same length and direction), only its two components change in the two coordinate systems.

I think the two transformations are related even if the processes are different.

Covariant/contravariant terminology
the word "transformation" of basis vectors refers to the conversion of the basis vectors in the original (non-rotated) coordinate system to the different basis vectors which point along the coordinate axes in the new rotated system, WHEREAS the "transformation" of vector components refers to the change in the components of the SAME vector referred to two different sets of coordinate systems:

(components of SAME vector in new system) = (inverse transformation matrix) (components of vector in original system)

versus

(new basis vectors) = (direct transformation matrix) (original basis vectors)

We are comparing the transformation of components (but same vector) with the transformation of the basis vectors (new basis vectors).

In the case of certain non-orthogonal coordinate systems, it is possible for the components to transform using a direct matrix, i.e. the same that is used for basis vectors transformation. Hence we talk about covariant components of the vector. The projections of the vector must be made along the dual basis vectors rather than onto the original basis vectors. For orthogonal coordinate systems, the vector components are contravariant, i.e. use a transformation that is inverse of the transformation used to convert basis vectors...

I will surely appreciate later the meaning of all this reasoning...

 

[fog37]

It seems that this business of covariant/contravariant becomes very important when dealing with tensors in different coordinate systems (or reference frames?) and quantities that are invariant between different reference frames...

A reference frame is something equipped with

-- clocks located at every point in space to tell the local time,
and
a way, i.e. a coordinate system of choice, to assign numerical coordinates to each spatial point

correct?

 

[fresh_42]

It seems that this business of covariant/contravariant becomes very important when dealing with tensors in different coordinate systems (or reference frames?) and quantities that are invariant between different reference frames...

A reference frame is something equipped with

-- clocks located at every point in space to tell the local time,
and
a way, i.e. a coordinate system of choice, to assign numerical coordinates to each spatial point

correct?

Yes and yes. Although the part with the clocks is a bit strange. We have one clock measuring everywhere in the reference frame. That makes SR so unintuitive: the change between reference frames isn't a linear transformation anymore.