Pseudo tensors and Levi-Civita tensors

Pseudo tensors and Lavi-Civita tensors

We have so for looked at tensors which transform faithfully not only under rotations and boosts, but also under coordinate inversions.

Recall that under a coordinate inversion,

 $\overrightarrow r \rightarrow -\overrightarrow r;\quad t\rightarrow t$ -----------(*)


This transformation is also called parity transformation. Whose transformation properties

Tensors are consistent with the above expression are called true tensors. They are the analogs of polar vectors which we study in three dimensions.

Let us look at a few examples.

Example 1: The four velocity is given by

$u^\mu=\frac{dx^\mu}{d\tau}$. Hence $u^0 \rightarrow u^0$ and $u^i \rightarrow -u^i$.

This $u^\mu\,$ is a true rector.

Exercise: Verify that the four acceleration is also a true vector.

Example 2: Consider the tensor $X^{\mu v}= x^\mu \, x^\nu$. Clearly, under parity, $X^{00} \rightarrow X^{00}; \quad X^{0i} \rightarrow -X^{i0}$ and $X^{ij} \rightarrow X^{ij}$.

We can verify that the tensors $u^\mu \, u^\nu; \quad u^\mu a^\nu\,$ and $a^\mu \, a^\nu$ have the same property.

We shall come across more nontrivial examples when we discuss relativistic dynamics and electrodynamics.

Example of a pseudo vector

Consider the magnetic field $\overrightarrow B$. The Lorentz force is given by

 $\overrightarrow F = Q \left(\frac{\overrightarrow V}{c}\times \overrightarrow B\right)$


Under coordinate inversions, $\overrightarrow F \rightarrow -\overrightarrow F,\quad \overrightarrow v \rightarrow -\overrightarrow v$. Hence, since $Q \rightarrow Q$ [we will justify this later], $\overrightarrow B \rightarrow \overrightarrow B$ which has the wrong sign. Such quantities are called pseudo vector, and their generalizations are called pseudo tensors.

In three dimensions, such vectors are also called axial vectors.

A similar generalization can be made for 4-vectors as well.

We first construct a nontrivial pseudo tensor, namely, the Levi Civita tensor.

The Levi Civita Symbol:

Example: Consider a matrix $A^i_{oj}$ of dimension 2.

Let us define a symbol

 $e^{lm}=\begin{Bmatrix} 0\, & if\, & l=m \\ 1\, & if\, & l=1,\, & m=2 \\ -1\, & if\, & l=2,\, & m=-1 \\ \end{Bmatrix}$


Recall: In $A^i_j$, $i\,$ represents row and $j\,$ represents columns.

Exercise: Verify the following identity:

 $\epsilon^{lm}\, A^i_l\,A^j_m = \epsilon^{ij} (det\, A)$.


This result can be extended to 4-dimensions To do so we need to remember some simple notations from permutations.

Definitions: Transposition:

Let $(a_1 \, a_2 ...\, a_n)$ be an ordered set of symbols. Suppose we interchange the symbols in the $i^{th}\,$ and the $j^{th}\,$ positions. Thus,

 $(a_1\,a_2...\,a_i...\,a_j...a_n)\rightarrow (a_1\,a_2...\,a_j...\,a_i...\,a_n)$


This operation is called a transposition. The following result holds (without proof here). We state a definition first.

Definition: An arbitrary rearrangement of the symbols $(a_1...\,a_n)$ is called a permutation.

Result: Any permutation can be built out of a sequence of transpositions.

Example: Let $(a_1\,a_2\,a_3\,a_4)\rightarrow (a_1\,a_3\,a_4\,a_2)$ be a permutation.

This can be built as the following sequence.

 $(a_1\,a_2\,a_3\,a_4) \rightarrow (a_1\,a_3\,a_2\,a_4) \rightarrow (a_1\,a_3\,a_4\,a_2)$.


We are now set to give the following definition:

Definition: A permutation which is built of an even number of transpositions is called an even permutation. A permutation which is built of an odd number of transpositions is called an odd permutation.

The previous example is an instance of even permutation since we made two transpositions.

Levi Civita tensor: The definition will be made is two steps. First we define the Levi Civita symbols.

Let

 $\epsilon^{\mu \nu \rho \sigma}= \begin{cases} 0 & \mbox{if any two of indices are identical} \\ 1 & \mbox{if }\mu \nu \rho\, \sigma\mbox{ is an even permutation of}\,(0123) \\ -1 & \mbox{if }\mu \nu \rho\, \sigma\mbox{ is an odd permutation of}\,(0123) \end{cases}$.


Exercise: Let $A^\alpha_\beta$ be $4\times 4$ matrices with $\alpha,\, \beta = 0,\,1,\,2,\,3$.

Then show that

 $\epsilon^{\mu \nu \rho \sigma}\, A^\alpha_\beta\, A^\beta_\nu\, A^\gamma_\rho\, A^\delta_\sigma = \epsilon^{\alpha \beta \gamma \delta}\,(det\, A)$.


We now apply this result when the matrices are Lorentz transformation matrices. Then,

 $\epsilon^{\mu \nu \rho \sigma}\, L^\alpha_\mu\, L^\beta_\nu\, L^r_\rho\, L^\delta_\sigma\, \epsilon = \epsilon^{\alpha \beta \gamma \delta}\, (det\, L)$.


Recall that $det\, L = \pm 1$. Consider the rotations and boosts which satisfy $det\, L = +1$. Then,

 $L^\alpha_\mu\, L^\beta_\nu\, L^\gamma_\rho\, L^\delta_\sigma\,\, \epsilon^{\mu \nu \rho \sigma}\, = \epsilon^{\alpha \beta \gamma \delta}$ ------------(*)


The right hand side (*) has exactly the character of a ranks four tensor, provided, the tensor has the same components in energy inertial frame ! Hence we have the definition:

Definition: The permutation symbol $\epsilon^{\mu \nu \rho \sigma}\,$ is an invariant rank 4 tensor, called the Levi Civita tensor, under proper Lorentz transformation.

 $\bar{\epsilon}^{\,\mu \nu \rho \sigma} = L^\mu_\alpha \, L^\nu_\beta \, L^\rho_\gamma \, L^\sigma_\delta \, \epsilon^{\alpha \beta \gamma \delta}$


where the components have the same value, defined in (I).

However, this property fails if we admit improper transformations. For then it picks up an extra factor $(det\,L)$. Hence it is a pseudo tensor.

Properties of Levi Civita tensor:

(i) By definition, the Levi-Civita tensor is completely antisymmetric. Hence, it has only one independent component: $\epsilon^{0123} \equiv 1$.

(ii) Contraction of a symmetric tensor with the Levi Civita gives zero.

Proof: Exercise.

Thus contraction of Levi-Civita makes sense only with antisymmetric tensors. This leads to the motion of a dual tensors, which we define below: