# Exploring the Physical Meaning of the Torsion Tensor in General Relativity

• tetraedro
In summary, the three indices of the torsion tensor can represent directions in space. CH is very knowledgeable about GR and may be able to help you further with this question.
tetraedro
Hi, I've been studying extensions of general relativity with the torsion tensor and I have been wondering about the following fact: what is the physical meaning of the three indices of this tensor? That is, do these three indices represent some directions in space? (For example, the translation axis and the plane of rotation). Can you please help me?
Thank you

Thank you for the answer. I've read on the paper "General Relativity with torsion: Extending Wald's Chapter on Curvature", the following sentence:
"...If $T^{z} _{xy}>0$ (where $T^{z} _{xy}$ is the torsion tensor), parallel transport along the x direction will cause v (parallely transported vector) to rotate about the x-axis in a left-handed manner".
Is it correct? This is the reason why I thought that the indices meant spatial directions.

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tetraedro said:

This is a forum right?

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some one must have removed the "answer" because I saw it here not to long ago.

What's the matter with the "answer"? Anyway, does someone know anything more about the issue I have exposed?

If anybody does know more, they don't seem to be talking. I can add that I know very little about torsion (Wald's textbook is probably the most advanced I own, and as you've noticed it talks very little about the subject - MTW doesn't have much either IIRC) and I can also say in general that CH is very knowledgeable about GR.

This is very confusing folks, may I ask what is going on?
I seem to not be able to read certain postings that others apparently can see.

Yes, there was a reply (that was #2) that was apparently deleted or withdrawn from the thread.

In any case,

here is the article mentioned above:
http://theory.uchicago.edu/~sjensen/teaching/tutorials/GRtorsion.pdf
.. on my list of things to read carefully.

FYI: Here are some recent discussions in PhysicsToday:

This may be useful (hopefully this obscure link to Nakahara's text works):

notes by Visser
http://www.mcs.vuw.ac.nz/courses/MATH464/2006T1/Lecture-Notes/notes.pdf

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torsion again

I have another question about torsion, maybe someone can help me.
I have been studying the Einstein-Cartan extension of general relativity and I have seen that apparently (as far as I know) no one reports the mathetical procedure one has to follow in order to derive the field equation satisfied by torsion. Since I know how this is done for the standard Einstein field equation, I wonder if some of you knows the corresponding procedure for the Einstein-Cartan field equation or can suggest me a book or a paper where I can find it. Thanks

Please read Richard Hammond's article "Torsion Gravity",Reports on Progress in Physics,from,I think,2002.It is an excelent article dealing with the evolution of what possible meaning the quantity "torsion" could have in physical theory,from Einstein-Cartan to the String Theory rumpus.Ideas of what torsion "means",geometrically,usually fall victim or prey to the person doing the imagining not distinguishing between a FINITE picture of torsion,e.g.,parallelograms failing to close,and the INFINITESIMAL DEFINITION of the torsion tensor itself,in terms of locally defined quantities in the tangent-plane to a point on a geodesic or curve.The fact that the mathematical form of the torsion tensor is similar to the form of the Maxwell field strength tensor tantalized Einstein for a good three years,1928-1931,in one of his schemes to connect gravitation and electricity geometrically.For that theory,visit Living Reviews In Relativity,Hubert Goenner's "On The History of Unified Fied Theories".Ciao.

## 1. What is the torsion tensor?

The torsion tensor is a mathematical object that describes the twisting or rotation of a vector field in a curved space. It is closely related to the concept of curvature, but while curvature measures how much a surface deviates from being flat, torsion measures how much a curve deviates from being straight.

## 2. How is the torsion tensor defined?

The torsion tensor is defined as the antisymmetric part of the connection, which is a mathematical object that describes how vectors change as they are transported along a curve in a curved space. It can also be thought of as the failure of the commutativity of covariant derivatives.

## 3. What is the physical significance of the torsion tensor?

The torsion tensor has physical significance in the theory of general relativity, where it is related to the presence of matter and energy in the universe. It also plays a role in other areas of physics, such as in the study of crystal defects and in the theory of elasticity.

## 4. How is the torsion tensor measured or calculated?

The torsion tensor can be calculated using mathematical equations based on the curvature of the space and the metric tensor, which describes the geometry of the space. It can also be measured experimentally through physical tests, such as torsion pendulum experiments.

## 5. What are some real-world applications of the torsion tensor?

The torsion tensor has numerous applications in physics, including in the theory of general relativity, crystallography, and continuum mechanics. It is also used in engineering, such as in the design and analysis of structures with complex geometries, such as bridges and aircraft wings.

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