# A Torsion tensor derivation

1. May 22, 2016

### mertcan

hi, I looked up torsion tensor derivation on 2 different books, and encountered 2 different situations, so my mind has been confused. For the first image, I could totally understand how torsion tensor was derived, but for the second image although there are similar things, I can not make a connection between 2 images. For instance, in first image torsion tensor is between r1 and r2 points, but this torsion tensor does not correspond to torsion tensor in second image, because instead there is a [u,v] vector. Besides, as you can see "sr1" is the parallel transported vector according to first image and resembles the vector "uR" ( appearance of these 2 vectors is same I think), but is parallel transported vector according to second image ( it is really parallel to vector, but for the first image the parallel transported vector ("sr1" ) is not really parallel to " pq " vector. To sum up, Could you please help me understand these 2 different situation, and express or explain the truth???

FIRST IMAGE

SECOND IMAGE

2. May 22, 2016

### mertcan

Why anyone can help me?? Is it a though question or long question ??

3. May 22, 2016

### mertcan

Why can anyone help me?? Is it a though question or long ??

4. May 23, 2016

### Markus Hanke

I don't really understand your question, I'm afraid; the two diagrams depict the exact same thing, the only difference is that the various vectors have different labels. In both cases, torsion is demonstrated as being the failure of the parallelogram to be closed.

5. May 23, 2016

### Ibix

I think all this is correct, but do be a bit wary. Still learning this myself.

The second image is comparing and contrasting two similar definitions: "You draw a parallelogram and the failure to close is the torsion" and "You draw a parallelogram and the failure to close is the commutator". The inner parallelogram is the same as your first diagram. The outer parallelogram is something different but confusingly similar.

The inner parallelogram's top and right sides are formed by parallel transporting the bottom and left sides along the left and bottom sides respectively - the failure to close is the torsion. The outer parallelogram's top and right sides are the values of the vector field at R and Q respectively - the failure to close is the commutator.

The difference between the two top sides (i.e. the difference between the parallel transported vector and the "local" value) is the covariant derivative in the direction of the left side. Similarly for the two right sides.

6. May 23, 2016

### mertcan

Markus Hanke they do not depict the same thing, look at the parallel transported vectors. The parallel transported vector "sr1" in first image should correspond to vector not vector in the second image. By the way Ibix, how are these 2 different derivations correct ?? I mean torsion definition must be unique, but there are 2 different meanings for torsion in these images. How can we have 2 different torsion definition ?????? If we say that torsion definition is the the failure of the closure of the parallelogram made up of the small displacement vectors and their parallel transports, we can not use that definition for second image, because in second image we should not have or vectors as a parallel transported vectors

7. May 23, 2016

### mertcan

Guys, I have been waiting for a long time, and I have not received an answer. I really want to make you sure that I can not make any progress in this topic. I got stuck, and I want to proceed to other tremendous, astonishing topics in relativity. I will appreciate, and be so pleased if you return me...........

8. May 23, 2016

### Orodruin

Staff Emeritus
No, this is incorrect, sr1 corresponds to $u_R^\parallel$. The vectors $u_R$/$v_Q$ are not parallel transported along the parallelogram sides.

9. May 23, 2016

### mertcan

Thank you for your return "Orodruin" ,but if or are not parallel transported vectors, for what do they stand??? also What is the meaning of and vector ???

10. May 23, 2016

### Orodruin

Staff Emeritus
11. May 23, 2016

### haushofer

The first one seems to come from Nakahara.

12. May 23, 2016

### haushofer

13. May 23, 2016

### mertcan

Thank you for your informing "Orodruin". As " haushofer" guessed, first image belongs to "Nakahara", whereas second image belongs to "F.W. Hehl and Yu.N. Obukhov". I consider the problem is: There may be a parallel transportation problem of vectors. For example, parallel transported vector "sr1" in first image do not parallel to "pq" vector, but if we pass to the second image we can examine that parallel transported vector is totally parallel to vector, and this situation makes a confusion. Nevertheless, "Orodruin" expressed that sr1 corresponds to . The vectors , and are not parallel transported along the parallelogram sides. After this explanation I have one more question: if or are not parallel transported vectors, for what do they stand??? also What is the meaning of and vector ???

14. May 23, 2016

### Orodruin

Staff Emeritus

15. May 23, 2016

### haushofer

I have a hard time guessing what the problem is, since we have answered your questions already. Why don't you agree with post #8?

16. May 23, 2016

### mertcan

hi, I have been digging up torsion tensor, and I would like to share 2 different images below. At the first image, I suppose that is the difference of directional derivatives, because at the second image this type of representation corresponds to difference of directional derivatives. If we come to the first image we can say that = + -( + ), but we are aware that as I said before is the difference of directional derivatives according to the second image. To sum up, in accordance with the = + -( + ) formula, + should be a directional derivative, but in line with the second image directional derivative is not necessarily tangent to manifold, as you can see first image totally lays on the tangent place, + is tangent to manifold instead of being not tangent according to second image.

First image is quoted from F.W. Hehl and Yu.N. Obukhov
second image quoted from David R. Wilkins

FIRST IMAGE

SECOND IMAGE

17. May 23, 2016

### Orodruin

Staff Emeritus
No, it should not. These are two completely different vectors which a priori has nothing to do with each other.

Also, please use the LaTeX features of the forum rather than editing in images, it will make your posts significantly more readable.

18. May 23, 2016

### mertcan

Well thanks for your return, but why do we put this [u,v] vector (at the second image ) on the image?? How we can derive this vector?? The only derivation I know of this vector is the difference of directional derivatives, but I can not see any directional derivatives ( after " Orodruin" 's explanation ) at the first image. What kind of logic exist for that vector??

19. May 23, 2016

### Orodruin

Staff Emeritus
$[u,v]$ is just the failure of the flows of the vector fields $u$ and $v$ to close. In general, I suggest that you do not try to see properties of a manifold in terms of an embedding as is done in your second image in #16. The properties under discussion are perfectly well defined only in terms of the properties of the manifold and affine connection itself (even without the affine connection when we just discuss the Lie bracket) without reference to an embedding in a higher-dimensional space.

20. May 23, 2016

### mertcan

Do you mean that [u,v] is not the difference of directional derivatives?? If not, how do we calculate the value of it??