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.
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.
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.
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.
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.