Write Torsion Tensor: Definition, Metric Tensor & Equation

In summary, the conversation discusses the possibility of writing the torsion tensor in terms of the metric. It is mentioned that a symmetric Christoffel Symbol can be written in terms of the partial derivatives of the metric, but this definition does not apply if the symbols are not symmetric. The question is raised if there is a way to define the non-symmetric Christoffel symbols in terms of a metric and use that to find an equation for the torsion tensor, or if it is independent of the metric tensor. It is also noted that the torsion free condition is necessary to uniquely define a metric compatible connection, and without it, there are multiple possible metrics.
  • #1
Physicist97
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Would it be possible to write the torsion tensor in terms of the metric? I know that a symmetric Christoffel Symbol can be written in terms of the partial derivatives of the metric. This definition of the christoffel symbols does not apply if they are not symmetric. Is it possible to write a definition of the non-symmetric christoffel symbols in terms of a metric, and from that find an equation for the torsion tensor, or is it independent of the metric tensor?
 
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  • #2
The torsion free condition is required to uniquely define a metric compatible connection. If you let it go, there are several possible metrics (one of which is the torsion free one).
 

FAQ: Write Torsion Tensor: Definition, Metric Tensor & Equation

1. What is the torsion tensor?

The torsion tensor is a mathematical object used to describe the twisting or rotational deformation of a coordinate system in a curved space. It is a measure of the failure of a coordinate system to remain rigidly aligned as it moves along a curved path.

2. How is the torsion tensor defined?

The torsion tensor is defined as the antisymmetric part of the connection coefficients, which describe how vectors change as they are transported along a curved path. It is denoted by T and its components are given by Tabc = Γa bc - Γa cb, where Γa bc are the connection coefficients.

3. What is the relationship between the torsion tensor and the metric tensor?

The torsion tensor is related to the metric tensor, which describes the local geometry of a curved space, through the Ricci rotation coefficients. These coefficients are given by Kabc = Tabc - Tacb, and they relate the torsion tensor to the metric tensor through the equation Kabc = gad Tdbc, where gad is the inverse of the metric tensor.

4. How is the torsion tensor used in physics?

The torsion tensor is used in physics, particularly in the field of general relativity, to describe the curvature of spacetime. It plays a crucial role in the theory of gravity, as it is related to the rotation of reference frames in the presence of matter and energy. It is also used in other areas of physics, such as in the theory of elasticity and in the study of crystal lattices.

5. What are some other important equations involving the torsion tensor?

Some other important equations involving the torsion tensor include the Cartan structural equations, which relate the curvature and torsion tensors to the Riemann tensor, and the Bianchi identities, which are differential equations that must be satisfied by the curvature and torsion tensors. The torsion tensor also appears in the field equations of general relativity, known as the Einstein-Cartan equations, which describe the gravitational field in terms of the curvature and torsion tensors.

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