Tensor transformation under reflections

[Mesmerized]

I guess it should be an easy question and I'm just missing smth, but I already spent on it much time and didn't get the answer. Here's a little quote from Landau's book "The Classical Theory of Fields" (2nd volume of 'Theoretical Physics' series)

"With respect to rotations of the coordinate system, the quantities e_iklm (iklm in the superscript) behave like the components of a tensor; but if we change the sign of one or three of the coordinates the components of that tensor, being defined as the same in all coordinate systems, do not change, whereas some of the components of a tensor should change sign. Thus e_iklm (iklm again in superscript)" is strictly speaking not a tensor, but rather a pseudotensor. Pseudotensors of any rank, in particular pseudoscalars, behave like tensors under all coordinate transformations except those that cannot be reduced to rotations, i.e. reflections, which are changes in sign of the coordinates that are not reducible to a rotation."

Now when I'm trying to consider the reflection of only one coordinte axis, which means the transformation matrix is A_ik=(1,1,1,-1) on main diagonal and all the rest 0s, trying to see how Levi-Chevita tensor (pseudotensor) and Kroneker delta (real tensor) transform, i.e. A_ip*A_kr*A_ls*A_mt*e_prst (again prst in the superscript, sorry I don't know how to put it there), I come to the exact opposite, some elements of the pseudotensors do change their signs, tensors do not!
Hope someone could help me, I'm close to despair :) Any help will be much appreciated

 

[bcrowell]

Hi, Mesmerized,

Please use LaTeX to write math in your posts. Here's a sample: $e^{iklm}$. If you click on the QUOTE button on my post, you can see how I did that.

By writing down the tensor transformation law, you are assuming that e transforms like a tensor, and therefore you come to the conclusion that it transforms like a tensor. But if e doesn't transform like a tensor, then it doesn't obey the tensor transformation law in the first place.

-Ben

 

[arkajad]

Expanding bcrowell's note: the Levi-Civita pseudotensor is defined to have the same components in all rectangular coordinates, including the reflected ones. What you have shown is that then it cannot be a tensor, because a tensor would have to change its sign.
That is the reason why it is called pseudotensor rather than tensor.

 

[Mesmerized]

OK, thanks, but then another question. The Kroneker delta (which is a tensor I guess) when applied that reflection transformation should change it's sign as I understood. but
$$\delta'_a^b=\delta_c^d$$*A_ac*A_bd (where the A matrix is the reflection transformation matrix with 1,-1,-1,-1 on the main diagonal and 0s in all other places) neither element of that tensor does change it's sign??

 

[arkajad]

$$\delta^b_a$$ is the identity tensor. It is constant under any linear transformation automatically. This follows from the fact that upper and lower index transform one with $$A$$ and the other with $$A^{-1}$$

I am not sure whether what you wrote is what you meant to write because you are new to latex. But I would suggest check carefully in whatever textbook you are using the transformation law for upper (contravariant) and lower (covariant) tensor indices.

 

[Rasalhague]

The coefficient matrix diag(1,1,1,1) represents the identity tensor in any basis, so it shouldn't (and doesn't) change sign:

$$(\delta')^a_b=\frac{\partial (x')^a}{\partial x^m}\frac{\partial x^n}{\partial (x')^b} \, \delta^m_n = \delta^a_b$$

$$A I A^{-1} = AA^{-1}= I$$

$$\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{pmatrix}$$

$$= \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}$$

 

[Mesmerized]

"the transformation law for upper (contravariant) and lower (covariant) tensor indices." yeah, I know, just for this reflection transformation matrixes A and $$A^{-1}$$ are the same.
So is this the case, in general tensors should change their signs under reflections but there are such special cases like the identity tensor (and metric tensor too, cause making the same calculations that Rasalhague did in previous post, it also remains the same), that do not change under those reflection transformation?

 

[arkajad]

So is this the case, in general tensors should change their signs under reflections but there are such special cases like the identity tensor (and metric tensor too, cause making the same calculations that Rasalhague did in previous post, it also remains the same), that do not change under those reflection transformation?

Not exactly. A general rule, following from the transformation law, is that for a given, tensor, some components of this tensor may change the sign. What really happens depends on the tensor. For instance $$\delta_{ab}$$ is constant with respect to all orthogonal transformations, including reflections. But the reason for it is of a somewhat different origin than for $$\delta^a_b$$.

$$\delta_{ab}$$ is constant because $$A^tA=I$$ for orthogonal $$A$$.

$$\delta^a_b$$ is constant because $$A^{-1}A=I$$ for all transformations

$$\epsilon_{abcd}$$ in 4 dimensions changes sign because $$\det(A)$$ is -1 for reflections with respect to an odd number of coordinates.

 

[Mesmerized]

Thanks, so just to be sure that I understood corrctly, does that mean that whether the tensor is a real tensor or a pseudotensor is determined by how we assume it to be in the beginning when we define it? Like that we say, let Levi-Civita tensor be determined as the same in all coordinate systems and not tranform under any orthogonal coordinate transformations, and it makes it a pseudotensor. And at the same time we say let the metric tensor be a real tensor, even though we then see that neither of it's elements changes it's sign under reflections?

 

[Mesmerized]

damn, I'm getting more and more confused, here's a passage from wikipedia

"In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation)." http://en.wikipedia.org/wiki/Pseudotensor

It looks like the exact opposite to what is written for a pseudotensor in Landau's passage

 

[arkajad]

Additional sign flip can well mean (-1)(-1) = +1. Pseudotensors are particular cases of http://en.wikipedia.org/wiki/Tensor_density" .