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  Can the Torsion tensor have a non-zero value?

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  1. The symmetry of basis vectors leads to the property that the Christoffel symbols are symmetric with respect to the bottom indices (Ref: Mathematical methods for Physicists by Arfken, Weber and Harris). (Γij = Γji). The symmetry of basis vectors is a property based on geometry of space.

    (A flat space supports this geometrical property.)

    This property is used in the derivation of the Christoffel symbols formula. The formula therefore becomes dependent upon geometry of space.

  2. Therefore, is it possible to get a non-zero value for the Torsion tensor, (Tij = Γji - Γij)?

  3. Is it possible to derive a Christoffel symbol formula which is not symmetric, so that the Torsion tensor is non-zero? What will be the associated coordinate system and geometrical picture?

Question:

What I wish to know is the "Geometrical Picture" behind a non-zero torsion tensor. What will be the non-symmtric Christoffel symbol formula to obtain it? How this formula can be derived from geometry of space?

Ref: https://physicsoverflow.org/44303/a-classical-scrutiny-of-the-schwarzschild-solution

asked Jul 7 in Theoretical Physics by NSRG [ no revision ]

1 Answer

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One of the assumptions of General Relativity is a vanishing torsion tensor. One can, for example, modify the formalism to accommodate torsion by introducing a "torsion potential" to the vielbein, or equivalently gauging the translation group.
Nonzero torsion in physics is seen in teleparallel theories of gravity on Weitzenböck spacetime and in Einstein-Cartan Theory which generalizes GR. I'm not sure what you mean by "Geometrical Picture," but a parallel transport over a closed path in a space with torsion produces a displacement from the original position, like how curvature changes a vector's orientation. On a discrete lattice, this means that torsion is equivalent to dislocation defects.

The Christoffel symbols can't be derived purely from the metric tensor in this case. See https://en.wikipedia.org/wiki/Contorsion_tensor

answered Jul 18 by QuantumGhost (75 points) [ no revision ]

QuantumGhost

Thanks for the very informative reply. We have two situations:

1.  No-torsion condition:   The path independence property (Section A, in the image) of the incremental displacement vector leads to symmetry of basis vectors.  The Christoffel symbol formula (Section B in the image) is not merely a mathematical definition but its derivation is based on symmetry of the incremental displacement vector, which is the property of geometry of space.

A metric gives us the incremental displacement vector and the corresponding unit vectors. The unit vectors are essential for suggesting a coordinate system corresponding to the space under study. This is required to write any vector in the given space.

By a geometrical picture, I mean a coordinate system, corresponding unit vectors and a metric describing curvature (if any) of the space.

2. Non-zero torsion: Kindly refer to the reply:

A non-zero torsion situation is described as: “A parallel transport over a closed path in a space with torsion produces a displacement from the original position, like how curvature changes a vector's orientation. On a discrete lattice, this means that torsion is equivalent to dislocation defects”.

If it is difficult to describe a metric for this non-zero torsion situation, then it will also be difficult to write the unit vectors. It will be difficult to write a coordinate system or write any vector in such space. Therefore, I desired to know the geometrical aspects such as a metric, corresponding unit vectors and a coordinate system to describe such a space. It is appropriate to derive a revised Christoffel symbols formula corresponding to such space, based on these geometrical characteristics.

3. I also wonder, if it is absolutely necessary to define a curved space (non-zero torsion) to describe a non-closing shape such as a parallelepiped?

A Cartesian coordinate system is suitable to describe a straight line or a three dimensional cube. We shift to the spherical coordinate system to study a circle or a sphere. But the space still remains flat. Similarly a non-closing parallelogram (or parallelepiped) may also be mathematically described in a flat space. May be, each side will have to be studied independently by dividing it into small increments. The analysis may be more complicated but the space need not be called curved.

Thanks once again for your reply and I am raising these comments for remarks by the experts.

Ref: Physics Overflow submission: 

A classical scrutiny of the Schwarzschild solution

@NSRG 
I have to admit that I am not very familiar with a pure mathematical formalism of torsion past what is relevant for gravity theories, although I don't think the answer would vary much. Maybe someone more experienced in the mathematical side of things can add more, but to my understanding:

For points 1 and 2:
The coordinate picture of torsion can be understood by analogy of crystal defects. See for reference Kleinert's book Multivalued Fields – particularly chapter 12 and beyond. Kleinert uses the vielbein formalism, which can be converted in terms of a metric tensor as $g_{\mu\nu}=\eta_{ab}e^a_\mu e^b_\nu$. I think he explains that approach much better than I currently can :).

If one wants to avoid the multivalued maps suggested by Kleinert, then as far as I know one would have to espouse alternative interpretations such as treating gravity as a gauge field of the translation or Poincaré group, or augmenting the metric formalism or doing away with it entirely. The "revised Christoffel symbols formula" is given on the Wikipedia page above; alternatively, in some conventions the torsion tensor is the antisymmetric part of the Christoffel symbols themselves.
A more radical modification motivated by string theory in non-geometric backgrounds would be to modify the metric tensor to an asymmetric tensor with the antisymmetric part corresponding to the torsion potential, but that would introduce complications to the geometry, as on one hand it is unclear what an asymmetric metric geometrically means, and the differential Bianchi identity would be violated.

@QuantumGhost 

Thanks for your reply. I think the questions about a curved space are not purely mathematical but require a complete shift in how we understand the nature.

I have raised an important question today: 

Does the curved space-time metric suggest a curvature of the space as a physical change or it is just mathematical shift to a curved coordinate system?

https://physicsoverflow.org/44474/suggest-curvature-physical-change-mathematical-coordinate

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