Learn Physics with Eigenchris: Relativity & Tensors

[robphy]

This set of videos by eigenchris (separate playlists on Relativity and on Tensors) also looks interesting
and can help anyone interested in learning about these topics.

A while back I watched some of them and thought they could be helpful.
I like his presentation of one-forms.
(I've been interested in Visualizing Tensors for a while... especially in directly applying them to physics.
Some his presentations gave me ideas on how to introduce one-forms to my classes in Thermodynamics
and Electromagnetism.)

https://www.youtube.com/channel/UCN8wTUlSAroLslWyf87E2pw

 

[Ibix]

Nice videos. I'd seen the co-vectors as a set of planes before, but the notion of rank 2+ tensors being represented in matrix notation as rows-of-rows or even rows-of-columns-of-rows was new to me. I think I agree with him that it rapidly gets out of hand, but it might have its place in computational work.

 

[Rene Dekker]

In the "Tensors for beginners" video series, he has the vector (contra-variant) indices at the bottom, and the the co-vector (co-variant) indices at the top. In another course that I have been following from Alex Flourney, it is just the other way around. What is the standard way of doing this?

 

[ergospherical]

Usually vector up and co-vector down, but so long as you're consistent it doesn't make a difference. There was a comment about Feynman's gravitational lectures which I thought was vaguely amusing:

 

[vanhees71]

Well, I've never encountered any textbook nor paper, where it was other than that lower indices indicate covariant and upper indices contravariant indices. I think I'd refuse to read any source that does it the other way. It would drive me nuts ;-)). It's already bad enough that there are so many sign conventions around, starting from the harmless one between east- and west-coast convention for the pseudometric, i.e., using a signature (3,1) (dominantly by GR people but also some HEP people, including Weinberg) or (1,3). The really dangerous sign convention occurs when it comes to the Levi-Civita symbol, i.e,. whether you say $\epsilon^{0123}=+1$ or $\epsilon_{0123}=+1$ and not clearly telling somewhere which convention is followed.

Fortunately it seems to be common convention to have the lower indices indicating the covariant and upper indices the contravariant tensor components. I think one should boycot any writing that's not following this standard, because otherwise we'll have even one more confusing convention in the literature.

It's bad enough that mathematicians destroy the intuitive way of Dirac's bra-ket formalism by having the scalar product of a Hilbert space linear in the first and semilinear in the second argument rather than using the physicists' convention thanks to Dirac's ingenious bra-ket notation, where it is the other way around (well, that's however another story...).

 

[Ibix]

In the "Tensors for beginners" video series, he has the vector (contra-variant) indices at the bottom, and the the co-vector (co-variant) indices at the top. In another course that I have been following from Alex Flourney, it is just the other way around. What is the standard way of doing this?

I think he only does this for for a short while in #3 where he labels everything with lower indices (the lazy works-for-Cartesian-coordinates way), but he introduces upper and lower index notation at the end of that video (edit: about here, if I've got the link to a time stamp to work).

He always labels the basis vectors with lower indices, but (other than the introductory exception) vector components have upper indices. Basis covectors and covector components are the other way around. I think that's standard, isn't it?

 

[vanhees71]

Of course if you have a Euclidean vector space and work with Cartesian coordinates you can put all indices as lower ones. That's common practice in physics when just working with good old 3D vector calculus.

If you work with general bases (maybe in a vector space without any fundamental form) you have to distinguish between bases of the vector space (usually written with a lower index) and its dual space. If you have a basis $\vec{b}_k$ of the vector space the corresponding dual basis of the dual space (co-vector space) is defined by $\underline{b}^j(b_k) \equiv b^j b_k=\delta_k^j$.

A vector's components wrt. to the $\vec{b}_k$ are denoted with upper indices,
$$\vec{v}=v^k \vec{b}_k.$$
A co-vector's components wrt. the dual basis $\underline{b}^j$ are labeled with lower indices,
$$\underline{L}=L_j \underline{b}^j.$$
If you now define new basis vectors
$$\vec{b}_l'={T^k}_l \vec{b}_k$$
and define the inverse matrix as ${U^l}_k$,
$$\vec{b}_k = {U^l}_k \vec{b}_l',$$
then the dual basis is given by
$$\underline{b}^{\prime m} \vec{b}_l'=\delta_l^m={T^k}_l \underline{b}^{\prime m} \vec{b}_k \; \Rightarrow \; \underline{b}^{\prime m} \vec{b}_k={U^m}_k.$$
From this one gets
$$\underline{b}^{\prime m}={U^m}_n \underline{b}^n,$$
because
$${U^m}_n \underline{b}^n \vec{b}_k = {U^m}_n \delta_k^n = {U^m}_k =\underline{b}^{\prime m} \vec{b}_k.$$
The dual basis thus transforms contragrediently to the basis. Usually one says the bases transform covariantly and the corresponding dual bases contravariantly.

From this you have of course
$$\underline{b}^n={T^n}_m \underline{b}^{\prime m}.$$
Then it's clear that this holds for the components of vectors (having upper indices) and transforming contravariantly,
$$\vec{v}=v^k \vec{b}_k=v^k {U^l}_k \vec{b}_l \; \Rightarrow\; v^{\prime l}={U^l}_k v^k,$$
and the components of a linear form (having lower indices) covariantly, and indeed
$$\underline{L}=L_j \underline{b}^j =L_j {T^j}_k \underline{b}^{\prime k} \; \Rightarrow \; L_k'=L_j {T^j}_k.$$

 

[robphy]

Well, I've never encountered any textbook nor paper, where it was other than that lower indices indicate covariant and upper indices contravariant indices. I think I'd refuse to read any source that does it the other way. It would drive me nuts ;-)).

I think he only does this for for a short while in #3 where he labels everything with lower indices (the lazy works-for-Cartesian-coordinates way), but he introduces upper and lower index notation at the end of that video (edit: about here, if I've got the link to a time stamp to work).

I think eigenchris does a good job bringing along viewers from "beginner level" to "intermediate level" and beyond. Ideas and notations are being gradually developed. (Is there any intro-level math methods book that uses the index-placement of abstract index notation from the beginning?)

I vaguely remember having to remind beginning students (learning the notation)
that $x^1$ is "$x$" but $x^2$ is "$y$".

Along these lines, I taught a math methods class that primarily had physics students...
but there was a math major among them. It took a few minutes for me to realize
that she interprets $E_x$, not as the $x$-component of $\vec E$, but as $\frac{\partial E}{\partial x}$.

 

[Ibix]

I think the videos start with $\vec v=v_i\vec b_i$ to avoid the comments being full of "$v^i$? What does raising a vector to the $i$th power mean?" Once he's established the component and basis transformation rules and wants to start talking about covectors, he sharpens up his notation.

 

[checksix]

I found the videos extremely helpful. I'm just a novice and struggled with MTW, etc. on my own without much success. I appreciated the way he (eigenchris) used simplified notation in the beginning then added more details as needed, rather than bombarding the reader with everything all at once. I finally "got" what a covector is, then everything started to fall into place.

 

[robphy]

I start off with column vectors (as vectors [the arrows]) and row vectors (a co-vectors [later, the planes])
$$\left[ \begin{array}{c}1\\2\\3\end{array} \right] \mbox{ and } \begin{array}{ccc}\left[1\right.&2&\left. 3 \right]\\ \vphantom{\frac12}\\ \vphantom{\frac12} \end{array}$$
Then I use arrows as $$\vec V \mbox{ and } \underset{\rightarrow}{A},$$ then eventually
$$V^a\mbox{ and } A_a$$
...somehow getting them to think of the abstract index---and not as "something to sum over".

 

[vanhees71]

I think eigenchris does a good job bringing along viewers from "beginner level" to "intermediate level" and beyond. Ideas and notations are being gradually developed. (Is there any intro-level math methods book that uses the index-placement of abstract index notation from the beginning?)

I vaguely remember having to remind beginning students (learning the notation)
that $x^1$ is "$x$" but $x^2$ is "$y$".

Along these lines, I taught a math methods class that primarily had physics students...
but there was a math major among them. It took a few minutes for me to realize
that she interprets $E_x$, not as the $x$-component of $\vec E$, but as $\frac{\partial E}{\partial x}$.

I've nothing against using the all-index-down-notation for Cartesian Ricci calculus. I'd also not bother First-semester physics students with co- and contravariant indices, but I'd never introduce a notation, which used nowhere else. Objects with lower (upper) indices everywhere in the physics literature transform covariantly (contravariantly), which of coarse is a convention, but why should one introduce the opposite convention, which nobody else uses? To confuse the students?

 

[Rene Dekker]

I think he only does this for for a short while in #3 where he labels everything with lower indices (the lazy works-for-Cartesian-coordinates way), but he introduces upper and lower index notation at the end of that video (edit: about here, if I've got the link to a time stamp to work).

He always labels the basis vectors with lower indices, but (other than the introductory exception) vector components have upper indices. Basis covectors and covector components are the other way around. I think that's standard, isn't it?

I think you are actually correct. I thought that he did it the other way around in the later videos as well, but it seems to be that it was I who was confused.