Tensor Density Transformation Law: Order of Jacobian Matrix?

[binbagsss]

I just have a quick question on which order around the numerator and denominator should be in the jacobian matrix that multiplies the expression.

As in general Lecture Notes on General Relativity by Sean M. Carroll, 1997 he has the law as

$\xi_{\mu'_{1}\mu'_{2}...\mu'_{n}}=|\frac{\partial x^{\mu'}}{\partial x^{\mu}} | \frac{\partial x^{\mu_{1}}}{\partial x^{\mu'_{1}}}\frac{\partial x^{\mu_{2}}}{\partial x^{\mu'_{2}}}...\frac{\partial x^{\mu_{n}}}{\partial x^{\mu'_{n}}} \xi_{\mu_{1}\mu_{2}...\mu_{n}}$

whereas on wiki the law is

$\xi^{\alpha}_{beta}= |[\frac{\partial \bar{x}^{t}}{\partial x^{\gamma}}]|\frac{\partial x^{\alpha}}{\partial \bar{x}^{\delta}}\frac{\partial \bar{x}^{\epsilon}}{\partial x^{\beta}}\bar{\xi}^{\delta}_{\epsilon}$

So both sources seem to have the matrix the other way around relative to the 'orginal' tensor and what is being transformed.

Does the order not matter?
Thanks.

 

[Orodruin]

It looks to me both your definitions are equivalent. How the Jacobian appears will also depend on the weight of the relative tensor.

 

[binbagsss]

It looks to me both your definitions are equivalent.

How? The first has the partial derivative of the transformed coordinates with respect to the partial derivative of the ' origninal' coordinates.
The second has the partial derivative of the orginial coordinates with respect to the partial derivative of the 'transformed' coordinates.

How the Jacobian appears will also depend on the weight of the relative tensor.

Whilst I see the second is transforming a (1,1) tensor density and the first a (n,0) tensor density, the first therefore lies in the same subset of the second so would expect the transformation law , coming from the lower indices, to take the same form.

 

[Orodruin]

How? The first has the partial derivative of the transformed coordinates with respect to the partial derivative of the ' origninal' coordinates.
The second has the partial derivative of the orginial coordinates with respect to the partial derivative of the 'transformed' coordinates.

This has to do with the contravariant/covariant indices in the tensor density that you are describing, not with the Jacobian.

Whilst I see the second is transforming a (1,1) tensor density and the first a (n,0) tensor density, the first therefore lies in the same subset of the second so would expect the transformation law , coming from the lower indices, to take the same form.

If you lower the $\alpha$ of the second transformation law using the metric (or construct a new tensor density with only lower indices by contraction with any rank two covariant tensor), the transformation laws are equivalent. Being a relative tensor does not have to do with how the individual indices transform, but how the entire tensor transforms apart from the transformations imposed by the location of the indices.

 

[binbagsss]

This has to do with the contravariant/covariant indices in the tensor density that you are describing, not with the Jacobian.

.

Sorry?My comment above and the OP is referring to the Jacobian only and not the other matrices, I understand the other matrices ok I think.

 

[Orodruin]

I see, I missed the subtlety in that you have used $x$ as the original coordinates in one of the definitions and as the transformed coordinates in the other.

Still, it is only a difference in how the weight of the relative tensor is defined. One definition is the negative of the other.

 

[Orodruin]

In fact, there is a (quite heated) discussion about this very issue on the Wikipedia talk page. The TLDR of that is basically that it is all conventional.