# Torsion tensor in Relativity

While reading Sean Carroll's book on general relativity, I came across something called as a 'Torsion Tensor' which is defined as, $$\Gamma{^\lambda}{_{\mu\nu}} - \Gamma{^\lambda}{_{\nu\mu}} = T{^\lambda}{_{\mu\nu}}$$ When the christoffel symbol is symmetric in it's lower indices, it's known to be torsion-free is what is specified in the book. There's nothing much given about the tensor beyond this other than it's taken as a given in Rienmannian geometry. Is there anyway to imagine this torsion or understand it intuitively? Like what do you mean by the term torsion in space time? Squishing of space-time?

• Some of the answers in the mathoverflow question mathoverflow.net/q/20493 might be helpful to you. – Bence Racskó Jul 3 at 11:10
• I don't have the book at hand, but in the lecture notes on which the book is based (arxiv.org/abs/gr-qc/9712019), he discusses torsion a bit with respect to the commutator of covariant derivatives on pages 75 and following. (See also the note on p 120/121 why theories with torsion do not receive much attention.) – Toffomat Jul 3 at 11:18
• Torsion here is a property of your coordinates and basis, not f the underlying spacetime. By definition $\Gamma^{k}_{ij} = \frac{\partial \mathbf{e}_i}{\partial x^j} \cdot \mathbf{e}^k$. You can always choose your basis / coordinates to make that symmetric (and all "natural" choice do so). If you don't make such a choice, you get extra terms in your covariant derivatives and a general mess that you don't want. – Brick Jul 3 at 15:02
• There is an analogy between torsion and defects in crystals. Google for torsion and Kleinert will turn up abundant references. A new paper here arxiv.org/abs/1907.00023 explores relations between torsion and dislocations/defects in two-dimensional Dirac materials. – MadMax Jul 3 at 15:43
• Maybe useful: Physical Aspects of the Space-Time Torsion - arxiv.org/abs/hep-th/0103093 – Avantgarde Jul 3 at 17:14

Imagining torsion geometrically is not as easy as imagining curvature.

More or less, torsion measures how a curve in the tangent space of a point $$x\in M$$, obtained by parallel transporting each tangent vector of a close curve in $$M$$ back to the point $$x$$, is far from being close as well.

Torsion really plays a role when treating spinors. In fact, it couples with spinors and there you can understand better its physical meaning. For instance you can take a first generalization of GR, called Einstein-Cartan-Sciama-Kibble theory, where we let the Ricci curvature "contain" torsion and we can see that a new field equation arises:

$$Q{^\mu}_{\nu\sigma}=-16\pi\Sigma{^\mu}_{\nu\sigma},$$

where $$Q$$ is the torsion tensor and $$\Sigma$$ the spin tensor.

The torsion tensor $$T^\lambda_{\mu\nu}$$ is defined the antisymmetric part of the affine connection coefficients $$\Gamma^\lambda_{\mu\nu}$$ $$T^\lambda_{\mu\nu}\equiv\Gamma^\lambda_{\mu\nu}-\Gamma^\lambda_{\nu\mu}$$ In General Relativity, it is postulated that $$T^\lambda_{\mu\nu}=0$$. The presence of torsion in affine connection simply implies that the covariant derivative of a scalar field $$\phi$$ doesn't commute, that is $$\nabla_{[\mu}\nabla_{\nu]}\phi=-T^\lambda_{\mu\nu}\nabla_\lambda\phi$$. For a vector $$v^a$$ and a covector $$w_a$$, the following relations are valid: $$\nabla_{[\mu}\nabla_{\nu]}v^\sigma=R_{\mu\nu\lambda}^\sigma v^\lambda-2T^\lambda_{\mu\nu}\nabla_\lambda v^\sigma$$ and $$\nabla_{[\mu}\nabla_{\nu]}w_\sigma=R_{\mu\nu\lambda}^\sigma w_\sigma-2T^\sigma_{\mu\nu}\nabla_\sigma w_\lambda$$ where $$R^\sigma_{\mu\nu\lambda}$$ is the Riemann tensor.

From these definitions, it follows that torsion measures the amount by which the boundary of a loop fails to close after being parallel transported. Thus, non-zero torsion signifies that a loop made of parallel transported vectors is not closed,i.e., geodesics as extremal lines don't coincide with autoparallels.