Tensor Calculus Questions

[Markus Hanke]

I have some questions here, which I’m hoping someone might be able to help with. I’ve spent the last few years focussing on other things in my life, so I’m afraid I’ve lost touch with the some of the basics - I’ve recently attempted to once again put pen to paper and actually work out some GR tensor calculus practice problems from scratch by hand, and…let’s just say it didn’t go so well 😕

1. Notational question - assume we are working in the context of GR, ie we are on a semi-Riemannian manifold endowed with the Levi-Civita connection and a metric. What is the actual significance of the vertical alignment (or lack thereof) of indices on tensors and spinors? In other words, what is the actual difference between the following three notations (let B be a rank-2 tensor), if any at all?

\[B_{\nu }^{\mu } \ vs\ B{_{\nu }}^{\mu } \ vs\ B{^{\mu}}_{\nu}\]

2. I need to really revise and - above all - practice my tensor calculus index gymnastics, but I’m having trouble finding a suitable text that actually focuses on the mechanics of index manipulation, rather than abstract definitions and proofs (which is what you most often get in GR texts). Does anyone here have recommendations? What I am specifically after is something not too high-level that goes through the various concepts in tensor index manipulation, provides worked examples, and then gives exercises to work through. The relevant chapters of MTW actually are good in that regard (they’re on a level I can follow easily enough), but I think the material is presented too concisely and quickly - I’m looking for something that introduces it more slowly and in more detail, including worked examples, and gives many more exercises of varying levels of difficulty to do. I understand the concepts involved reasonably well if I see them written down in an equation, I just need much more practice in actually using them in a pen-on-paper kind of way - which is an entirely different skill set. So I’m after something that really drills home the mechanics of index manipulation through worked examples and exercises. Any suggestions, anyone?

TIA.

[Lorentz Jr]

This is all WAY over my head, but have you ever tried using geometric algebra? It's supposed to be simpler and more versatile.

Geometric Algebra Techniques for General Relativity

GRAVITY, GAUGE THEORIES AND GEOMETRIC ALGEBRA

Geometric Calculus in Gravity Theory

Geometric Algebra, Spacetime Physics and Gravitation

[Markus Hanke]

2 hours ago, Lorentz Jr said:

This is all WAY over my head, but have you ever tried using geometric algebra? It's supposed to be simpler and more versatile.

Yes, I have indeed - but only to the extent of ordinary exterior calculus in the context of GR, as presented in (eg) MTW. I’d definitely like to deepen my understanding and skills in the area of geometric algebra, since it is a very powerful formalism; many aspects of physics can be cast into that language. But that’s a future project - at present I need to recap my tensor calculus. But thank you for the links, I really do appreciate that  +1

PS. I think I need to add some clarification to question (1) in my OP (which, for some reason, it won’t let me edit?). I’m good with raising and lowering indices, thus the relationship between the latter two notations

\[B{^{\mu}}{_{\nu}}=g^{\mu \alpha} g_{\nu \beta}B{_{\alpha}}{^{\beta}}\]

isn’t the problem. What I’m wondering about is specifically the notation where two upper and lower indices are vertically aligned; thus I’m wondering how the above relates to \(B^{\mu}_{\nu}\).

Edited February 25, 2023 by Markus Hanke

[joigus]

2 hours ago, Markus Hanke said:

1. Notational question - assume we are working in the context of GR, ie we are on a semi-Riemannian manifold endowed with the Levi-Civita connection and a metric. What is the actual significance of the vertical alignment (or lack thereof) of indices on tensors and spinors? In other words, what is the actual difference between the following three notations (let B be a rank-2 tensor), if any at all?

 

Bμν vs Bνμ vs Bμν

 

This is because the first and the second indices generally act on different indices irrespective of whether they are covariant --they transform with the same matrix as the basis members-- or contravariant --the transform with the inverse matrix. For the Euclidean case, this is of no importance, but as you well know, for Minkowski, it matters. Consider the Lorentz transformations,

1) Boost in the t-x plane:

\[ B=\left(\begin{array}{cccc} \gamma & \beta\gamma & 0 & 0\\ \beta\gamma & \gamma & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array}\right) \]

\[ \left.B_{\mu}\right.^{\nu}=\left.B_{\nu}\right.^{\mu} \]

2) Rotation in the x-y plane

\[ R=\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & \cos\theta & -\sin\theta & 0\\ 0 & \sin\theta & \cos\theta & 0\\ 0 & 0 & 0 & 1 \end{array}\right) \]

But,

\[ \left.R_{\mu}\right.^{\nu}\neq\left.R_{\nu}\right.^{\mu} \]

The way I do it in LaTeX is,

\left.B_{\mu}\right.^{\nu}

 

2 hours ago, Markus Hanke said:

2. I need to really revise and - above all - practice my tensor calculus index gymnastics, but I’m having trouble finding a suitable text that actually focuses on the mechanics of index manipulation, rather than abstract definitions and proofs (which is what you most often get in GR texts). Does anyone here have recommendations? What I am specifically after is something not too high-level that goes through the various concepts in tensor index manipulation, provides worked examples, and then gives exercises to work through. The relevant chapters of MTW actually are good in that regard (they’re on a level I can follow easily enough), but I think the material is presented too concisely and quickly - I’m looking for something that introduces it more slowly and in more detail, including worked examples, and gives many more exercises of varying levels of difficulty to do. I understand the concepts involved reasonably well if I see them written down in an equation, I just need much more practice in actually using them in a pen-on-paper kind of way - which is an entirely different skill set. So I’m after something that really drills home the mechanics of index manipulation through worked examples and exercises. Any suggestions, anyone?

TIA.

The Schaum series perhaps?

PS: Sorry, Markus. I think I swapped co- and contravariant indices in my answer. Let me fix it.

I meant,

\[ \left.B_{\mu}\right.^{\nu}=\left.B^{\nu}\right._{\mu} \]

for the boost, and,

\[ \left.R_{\mu}\right.^{\nu}\neq\left.R^{\nu}\right._{\mu} \]

for the rotation.

I hope that helps.

[Markus Hanke]

11 minutes ago, joigus said:

I hope that helps.

Thank you joigus.
Unfortunately though this isn’t quite what I meant, but the fault is entirely mine - I didn’t make my question clear enough, and the additional clarification I just added came too late for your post, and somehow got tacked onto Lorentz Jr’s post instead of the OP. It’s all gone a bit messy here. Apologies for causing you lots of typesetting work. 

My question was actually about vertical alignment of indices - how does \(B^{\mu}_{\nu}\) formally relate to (eg) \(B{^{\mu}}{_{\nu}}\), with the emphasis on vertical alignment or lack thereof of tensor indices? Or is there perhaps no difference at all?

Edited February 25, 2023 by Markus Hanke

[joigus]

2 minutes ago, Markus Hanke said:

Thank you joigus.
Unfortunately though this isn’t quite what I meant, but the fault is entirely mine - I didn’t make my question clear enough, and the additional clarification I just added came too late for your post, and somehow got tacked onto Lorentz Jr’s post instead of the OP. It’s all gone a bit messy here. Apologies for causing you lots of typesetting work. 

My question was actually about vertical alignment of indices - how does Bμν formally relate to (eg) Bμν , with the emphasis on vertical alignment or lack thereof of tensor indices? Or is there perhaps no difference at all?

Vertical alignment is exactly what I'm talking about. If you have a 2-index tensor, the first index is the column index, and the second one is the row index. Unless I'm missing something, that's what the vertical alignment is all about.

[Markus Hanke]

16 minutes ago, joigus said:

The Schaum series perhaps?

Ok, thank you As it happens I have both Schaum and Synge/Schild in my personal library, but haven’t read either one of them yet. Just leafing through them, they both look pretty good. Still, it will be helpful to see what other recommendations people here might have!

[joigus]

In other words, one should never write the tensors with vertical alignment. That could lead to errors if you have rotations mixing in in you brew. Some people do it, I know.

[Markus Hanke]

2 minutes ago, joigus said:

Vertical alignment is exactly what I'm talking about. If you have a 2-index tensor, the first index is the column index, and the second one is the row index. Unless I'm missing something, that's what the vertical alignment is all about.

Yes sure - but if both indices are written in one vertical line, as in \(B^{\mu}_{\nu}\), how do you know which is which? 

[joigus]

As to index gymnastics, I'm very fond of Anomalies in Quantum Field Theory, by Reinhold A. Bertlmann --the famous mathematician of John Bell's article. The first third of the book has a lot of it, because he deals with gravitation a lot. Not in detail, but you can check the calculations page by page as a good gymnastics.

[Markus Hanke]

2 minutes ago, joigus said:

In other words, one should never write the tensors with vertical alignment. That could lead to errors if you have rotations mixing in in you brew. Some people do it, I know.

Ok, so basically this is just a sloppy way of writing them?

[joigus]

Just now, Markus Hanke said:

Yes sure - but if both indices are written in one vertical line, as in Bμν , how do you know which is which? 

Exactly. Wrong notation!

Ambiguous, as you rightly said.

[Markus Hanke]

1 minute ago, joigus said:

Exactly. Wrong notation!

Ambiguous, as you rightly said.

Ok, that clarifies it! I was simply wondering if that was just sloppiness, or whether there is any significance to the notation.
Thank you

[joigus]

1 minute ago, Markus Hanke said:

Ok, so basically this is just a sloppy way of writing them?

You bet. You can't imagine how many people I've met who don't know this. As well as the reason why one set of coordinates is co-variant, and the other contra-variant.  

4 minutes ago, joigus said:

As to index gymnastics, I'm very fond of Anomalies in Quantum Field Theory, by Reinhold A. Bertlmann --the famous mathematician of John Bell's article. The first third of the book has a lot of it, because he deals with gravitation a lot. Not in detail, but you can check the calculations page by page as a good gymnastics.

It's all in the language of differential forms, but that's not gonna be a problem for the likes of you.

 

[Genady]

They are different, I think. Consider

B^a_b = B^acg_bc = B_d^cg^adg_bc ≠ B_b^a

B^a_b and B_b^a are transpose of each other, like B^ab and B^ba. No difference if the tensor is symmetric.

 

edit: the thread was open on my screen for way too long, so this is x-posted with the entire exchange above. Still, the last remark, about symmetric tensor, is an addition, I think. 

Edited February 25, 2023 by Genady

[Markus Hanke]

13 minutes ago, Genady said:

edit: the thread was open on my screen for way too long, so this is x-posted with the entire exchange above. Still, the last remark, about symmetric tensor, is an addition, I think.

Yes, thank you I’m clear about index symmetries…it was rather about that strange notation where indices are written vertically aligned one atop the other. Turns out it’s just sloppy notation.

[joigus]

49 minutes ago, Genady said:

They are different, I think. Consider

B^a_b = B^acg_bc = B_d^cg^adg_bc ≠ B_b^a

B^a_b and B_b^a are transpose of each other, like B^ab and B^ba. No difference if the tensor is symmetric.

 

edit: the thread was open on my screen for way too long, so this is x-posted with the entire exchange above. Still, the last remark, about symmetric tensor, is an addition, I think. 

Yes, that was exactly my point. Swapping indices cavalierly is dangerous when you're in a non-Euclidean context. If you want to correlate observers and you need boost + rotation to do that, transpose Lorentz transformations do not coincide with given transformation. B^T not equal to B.

@Genady, what's your code for that?

I use,

\left.B_{a}\right.^{b}

which produces,

\[ \left.B_{a}\right.^{b} \]

Never mind. I've just realised while I was making myself a sandwich! LOL

[Genady]

12 minutes ago, joigus said:

Never mind. I've just realised while I was making myself a sandwich! LOL

Yes, I'm a simple guy.  

[joigus]

20 minutes ago, joigus said:

Swapping indices cavalierly is dangerous when you're in a non-Euclidean context.

Now that I think about it, it's also dangerous when you're in a Euclidean context too.  

[studiot]

5 hours ago, Markus Hanke said:

I have some questions here, which I’m hoping someone might be able to help with. I’ve spent the last few years focussing on other things in my life, so I’m afraid I’ve lost touch with the some of the basics - I’ve recently attempted to once again put pen to paper and actually work out some GR tensor calculus practice problems from scratch by hand, and…let’s just say it didn’t go so well 😕

1. Notational question - assume we are working in the context of GR, ie we are on a semi-Riemannian manifold endowed with the Levi-Civita connection and a metric. What is the actual significance of the vertical alignment (or lack thereof) of indices on tensors and spinors? In other words, what is the actual difference between the following three notations (let B be a rank-2 tensor), if any at all?

 

Bμν vs Bνμ vs Bμν

 

2. I need to really revise and - above all - practice my tensor calculus index gymnastics, but I’m having trouble finding a suitable text that actually focuses on the mechanics of index manipulation, rather than abstract definitions and proofs (which is what you most often get in GR texts). Does anyone here have recommendations? What I am specifically after is something not too high-level that goes through the various concepts in tensor index manipulation, provides worked examples, and then gives exercises to work through. The relevant chapters of MTW actually are good in that regard (they’re on a level I can follow easily enough), but I think the material is presented too concisely and quickly - I’m looking for something that introduces it more slowly and in more detail, including worked examples, and gives many more exercises of varying levels of difficulty to do. I understand the concepts involved reasonably well if I see them written down in an equation, I just need much more practice in actually using them in a pen-on-paper kind of way - which is an entirely different skill set. So I’m after something that really drills home the mechanics of index manipulation through worked examples and exercises. Any suggestions, anyone?

TIA.

 

 

OK to answer this I would like to start with a bit of history.

The first person to use the word tensor was Hamilton, I think in 1841,  though he actually meant something different when he introduced dyadics.

The first person to use the word tensor in the modern sense was Voight in 1898 for studies in crystallography.

Inbetween those times the pillars of maths were changed substantially by many workers, Ricci and Levi Cevita being at the forefront of developing tensor maths, although they did not call it that. 

In those days and into the early part of the 20th century it was called 'the absolute differential calculus'.

Then Einstein introduced his summation convention (1916).

Again at first there was no standard but by the 1930s the convention of contravarient upstairs only , covarient downstairs only and mixed both upstairs and downstairs had emerged. But with no regard to index order or spacing.
This convention carried on to the end of the 20th century and beyond. Some still use it today.

But by the end of the 20th century the advantage of proper index ordering had become apparent so today newer authors are beginning to leave spaces where some indexes are not used for some of the terms, as in this extract from Fleisch Vectors and Tensors  - Cambridge.

OOps pressed the go button before I was finished so I am carrying on editing.

 

So the moral of this history is that you need to have your date calendar about your person when reading the literature.

 

Your second question about exercises is much more difficult as you presumably want exercise with answers and perhaps worked examples ?
The idea of drill exercises is a very good one though.

Here is a series of videos, this one in the middle focused on your question.

https://cosmolearning.org/video-lectures/few-tensor-notation-exercises/

 

As to books, Fleisch is already mentioned,

Sokolnikoff -  Tensor Analysis - Wiley  - is a good and clear mid 20th cent text with exercies and some answers

Zafar Ahsan - Tensors - PHI learning Delhi is modern (2018) again with some exercises and some answers.

Bickley and Gibson  - Via Vector to Tensor - EUP  again offers some exercises and answers.

Lawden - Introduction to Tensor Calculus, Relativity and Cosmology  Wiley or Dover - offers exercises but no answers.

Sadly your Synge and Schild offers neither exercises nor answers  it is a treatise.

 

Finally there is the issue of associated mathematics  - The Hodge star, wedge and V products the geometric v the algebraic viewpoint and so on.

 

I hope some of this helps.

Edited February 25, 2023 by studiot

[Genady]

Here is a different approach:

Quantum Field Theory and the Standard Model: Schwartz, Matthew D.: 8601406905047: Amazon.com: Books

[Markus Hanke]

14 hours ago, studiot said:

I hope some of this helps.

It does, thank you
I did notice though that you haven’t mentioned Schaum’s Outline in your literature list - is the omission deliberate, ie is there something about the text I should be weary of? I’m just asking because I happen to have that text in my possession already - I haven’t read it yet, but it looks good at first glance.

[studiot]

5 hours ago, Markus Hanke said:

It does, thank you
I did notice though that you haven’t mentioned Schaum’s Outline in your literature list - is the omission deliberate, ie is there something about the text I should be weary of? I’m just asking because I happen to have that text in my possession already - I haven’t read it yet, but it looks good at first glance.

No I had simply forgotten all about it.

David Kay has done a good job as a specialist in line with the Schaum philosophy.

There is many wrinkles in that book and of course quite a few exercises.

But these are scattered throughout the book, there is no specific section or chapter  explicitly on your desired topic, you will find bits and pieces all over the place.

 

[Markus Hanke]

16 minutes ago, studiot said:

No I had simply forgotten all about it.

David Kay has done a good job as a specialist in line with the Schaum philosophy.

There is many wrinkles in that book and of course quite a few exercises.

But these are scattered throughout the book, there is no specific section or chapter  explicitly on your desired topic, you will find bits and pieces all over the place.

 

Great thanks +1

PS. I accidentally hit the downvote button instead of the upvote one (touchscreen)…I tried to correct it, but it’s now displaying an upvote in red color. Not sure what that’s about, but it’s definitely meant to be an upvote.

[studiot]

15 minutes ago, Markus Hanke said:

Great thanks +1

PS. I accidentally hit the downvote button instead of the upvote one (touchscreen)…I tried to correct it, but it’s now displaying an upvote in red color. Not sure what that’s about, but it’s definitely meant to be an upvote.

Looks green to me, thanks.

 

Yes it is a good workbook so +1 to @joigus for remembering it.