Visualising Torsion Tensor: Is There a Picture?

In summary: The paper by Hehl mentioned in your summary is a great resource for understanding torsion. It talks about how torsion is related to various geometric properties of a manifold, and has diagrams to help illustrate the concepts. It's a bit dense reading, but definitely worth reading if you're interested in the topic.
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chartery
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As I understand it, parallel transport of a vector around a closed loop on a manifold can lead (in the tangent space) to 1) an angular change, given by the Riemann curvature tensor or, 2) a translational defect given by the Torsion tensor.

I can see how the looping on the curvature of a 2D sphere leads to such a changed angle. Is there any equivalent representation of the features of a 2D manifold that would help me visualise pictorially how torsion leads to a translational defect?

Also, is there an easy way to understand how an intrinsic translational defect can be thought of as a generalisation of an extrinsic rotation in flat space?
 
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Isn't the (non)existence of torsion a property of the connection, rather than the manifold? There exists a unique metric-compatible connection that leads to zero torsion. So you can't do anything like the illustration of curved manifolds by embedding a 2-sphere because you can always pick a connection that leads to zero torsion.

But presumably you can also pick a connection that leads to non-zero torsion even in quite trivial cases. Thus (I think!) you could have torsion on a flat 2d plane by picking a non-trivial connection. As far as I understand it, the connection would (via the covariant derivative and hence parallel transport) affect what it means to "travel in a straight line". So you could pick a connection such that the un-accelerated path of a particle on the plane was a parabola (or something) while the shortest distance between two points remained what Euclid would call a straight line.
 
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You are right, there's a unique metric compatible torsion-free connection on a Riemannian manifold, and it's given by the usual Christoffel symbols wrt. a coordinate basis, and by definition a Riemannian manifold is the one, for which the affine connection is defined as this unique torsion-free connection.

However, there are also metric-compatbile connections that are not torsion-free. Then you get a more general type of manifold, an Einstein-Cartan manifold. If you want to define fields with spin, in general you need these more general manifolds. For a review, see

https://doi.org/10.1103/RevModPhys.48.393
 
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vanhees71 said:
However, there are also metric-compatbile connections that are not torsion-free.
Yes, metric compatibility and torsion free-ness are separate choices. I didn't state that clearly.
 
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Thanks all. I guess I should have been clearer about not presuming the manifold was Riemannian. I had seen the mathoverflow link, but find algebra only helps once I have some visual handle on a concept, however simplistic.

On rereading Hehl's paper (https://arxiv.org/abs/0711.1535), the following seems to me to fit the bill:

1673707541910.png

Figure 2: Schematic view on a two-dimensional Cosserat continuum: Undeformed
initial state.

1673707621610.png


Figure 11: Deformation of a cubic crystal by edge dislocations of type α121:
The relative orientations of the lattice plains in 2-direction change. A vector
in x2-direction will rotate, if parallelly displaced along the x1-direction. As
a consequence a contortion κ112 emerges and the closure failure occur of the
“infinitesimal” parallelogram.Although I'm not smart/knowledgeable enough to decide which was the more appropriate:

1673708264923.png
Figure 4: Homogeneous contortion
1673708335241.png

Figure 6: Conventional rotation ∂[1u2] of the “particles” of a Cosserat continuum
caused by an inhomogeneous strain.
 

FAQ: Visualising Torsion Tensor: Is There a Picture?

What is the torsion tensor in general relativity?

The torsion tensor is a mathematical object used in differential geometry and general relativity to describe the intrinsic twisting of a space. Unlike curvature, which measures how space is bent, torsion represents how the geometric structure can be twisted. In the context of general relativity, the torsion tensor is often associated with theories that extend general relativity, such as Einstein-Cartan theory, where it plays a role in describing the spin of matter and its influence on the geometry of spacetime.

Why is visualizing the torsion tensor challenging?

Visualizing the torsion tensor is challenging because it exists in higher-dimensional mathematical spaces that are not easily represented in our three-dimensional intuition. The torsion tensor involves complex relationships between vectors and forms, and these relationships can be difficult to depict graphically. Additionally, torsion is often considered in the context of abstract mathematical structures rather than physical representations, making it harder to create intuitive visualizations.

Are there any existing visualizations of the torsion tensor?

While there are no universally accepted visualizations of the torsion tensor, some researchers have attempted to create graphical representations using computer simulations and geometric models. These visualizations often involve representing the torsion tensor as a field in a simplified context, such as in a two-dimensional space, to illustrate its effects. However, these representations may not capture the full complexity of the torsion tensor in higher dimensions or in the context of real physical theories.

What are the implications of torsion in physical theories?

Torsion has significant implications in various physical theories, particularly those that incorporate spin and intrinsic angular momentum. In Einstein-Cartan theory, for example, torsion is linked to the distribution of spin in matter, which can lead to modifications of the gravitational field equations. This means that torsion can influence the behavior of matter and energy in spacetime, potentially leading to observable effects in high-energy physics and cosmology.

How can one learn more about the torsion tensor and its visualization?

To learn more about the torsion tensor and its visualization, one can explore advanced textbooks on differential geometry, general relativity, and mathematical physics. Online courses and lecture notes from universities can also provide valuable insights. Additionally, research papers focusing on the geometrical aspects of torsion and its applications in theoretical physics can offer deeper understanding and potential visualizations. Engaging with computational tools and software that model geometric structures may also help in visualizing the torsion tensor in specific contexts.

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