# Why is There No Factor of 1/2 in the Torsion Tensor Definition?

• B
• Shirish
In summary, the conversation discusses the definition of the torsion tensor in the context of two covariant derivative operators. The torsion tensor is defined as ##T^{\alpha}_{\mu\nu}\equiv 2C^{\alpha}_{[\mu\nu]}##, where ##C^{\alpha}_{\mu\nu}## is a tensor that satisfies ##\nabla_{\mu}\omega_{\nu}=\nabla'_{\mu}\omega_{\nu}-C^{\alpha}_{\mu\nu}\omega_{\alpha}## for all covariant vectors ##\omega_{\nu}##. The conversation also discusses the convention for defining anti-symmetrisation and the importance
Shirish
Let's say we have any two covariant derivative operators ##\nabla## and ##\nabla'##. Then there exists a tensor ##C^{\alpha}_{\mu\nu}## such that for all covariant vectors ##\omega_{\nu}##,$$\nabla_{\mu}\omega_{\nu}=\nabla'_{\mu}\omega_{\nu}-C^{\alpha}_{\mu\nu}\omega_{\alpha}$$
Now I'm quoting the relevant section on torsion tensor definition:
What if the no-torsion requirement is dropped? Set ##\omega_{\nu}=\nabla_{\nu}\phi=\nabla'_{\nu}\phi##: (which gives) ##\nabla_{\mu}\nabla_{\nu}\phi=\nabla'_{\mu}\nabla'_{\nu}\phi-C^{\alpha}_{\mu\nu}\nabla_{\alpha}\phi##. Antisymmetrize over ##\mu## and ##\nu##, and assume ##\nabla'## is torsion free, but ##\nabla## is not. In that case ##\nabla_{[\mu}\nabla_{\nu]}\phi=-C^{\alpha}_{[\mu\nu]}\nabla_{\alpha}\phi##. The torsion tensor is defined as ##T^{\alpha}_{\mu\nu}\equiv 2C^{\alpha}_{[\mu\nu]}##, implying that $$(\nabla_{\mu}\nabla_{\nu}-\nabla_{\nu}\nabla_{\mu})\phi=-T^{\alpha}_{\mu\nu}\nabla_{\alpha}\phi$$
I don't understand why this is so. I mean the LHS is can also be notationally represented as ##\nabla_{[\mu}\nabla_{\nu]}\phi##, so either there should be a factor of ##1/2## on the RHS, or the torsion tensor should be defined as ##T^{\alpha}_{\mu\nu}\equiv C^{\alpha}_{[\mu\nu]}##, or am I missing something?

Looks OK to me.$$\begin{eqnarray*} \nabla_\mu\nabla_\nu\phi-\nabla_\nu\nabla_\mu\phi&=&-C^\alpha{}_{\mu\nu}\nabla_\alpha\phi+C^\alpha{}_{\nu\mu}\nabla_\alpha\phi\\ 2\nabla_{[\mu}\nabla_{\nu]}\phi&=&-2C^\alpha_{[\mu\nu]}\nabla_\alpha\phi\\ &=&-T^\alpha{}_{\mu\nu}\nabla_\alpha\phi \end{eqnarray*}$$You appear to be defining anti-symmetrisation as$$\nabla_{[\mu}\nabla_{\nu]}\phi=\nabla_\mu\nabla_\nu\phi-\nabla_\nu\nabla_\mu\phi$$Carroll, at least, defines it as$$\nabla_{[\mu}\nabla_{\nu]}\phi=\frac 12\left(\nabla_\mu\nabla_\nu\phi-\nabla_\nu\nabla_\mu\phi\right)$$Carroll is consistent with your OP (note: in general the prefactor is ##1/n!## when ##n## indices are being summed over). I seem to recall reading that not all sources put the ##1/n!## prefactor in - are you possibly taking a definition of anti-symmetrisation from a source that doesn't include it and a definition of torsion from a source that does?

Shirish
Ah you're right. I didn't know that the antisymmetrization operator is defined along with a factor of ##1/2##

Shirish said:
Ah you're right. I didn't know that the antisymmetrization operator is defined along with a factor of ##1/2##
It certainly appears to have been defined that way here. The argument for the prefactor comes from the notion that if you have an antisymmetric tensor ##A^{\mu\nu}## then ##A^{[\mu\nu]}=\frac 12\left(A^{\mu\nu}-A^{\nu\mu}\right)=A^{\mu\nu}## and antisymmetrising an already antisymmetrised tensor does nothing. A similar argument can be made for wanting a symmetrised symmetric tensor to be unchanged, and hence placing the same prefactor there.

However, it's purely a convention. Having had a quick glance at Carroll's notes confirms my recollection - he warns that not everyone puts in the prefactor. So you do have to keep an eye on what convention is in use!

Last edited:
Shirish

## 1. What is the definition of torsion tensor?

The torsion tensor is a mathematical quantity that describes the amount of twisting or rotation in a given space. It is a measure of the curvature of a space and is commonly used in fields such as physics, engineering, and mathematics.

## 2. How is the torsion tensor calculated?

The torsion tensor is calculated using the Christoffel symbols, which are derived from the metric tensor. It is a complex mathematical process that involves taking derivatives and solving equations.

## 3. What is the difference between torsion tensor and curvature tensor?

The torsion tensor and curvature tensor are both measures of the curvature of a space, but they describe different aspects of curvature. The torsion tensor describes the amount of twisting or rotation in a space, while the curvature tensor describes the amount of bending or stretching.

## 4. How is the torsion tensor used in physics?

The torsion tensor is used in physics to describe the behavior of objects in curved spaces, such as in Einstein's theory of general relativity. It is also used in fields such as quantum mechanics and string theory to describe the properties of space and time.

## 5. Can the torsion tensor be visualized?

The torsion tensor is a mathematical concept and cannot be directly visualized. However, it can be represented graphically using diagrams and equations to help understand its properties and behavior.

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