Question

Write the tensor transformation equations for \(\epsilon_{i j k} \epsilon_{m n p}\) to show that this is a (rank 6) tensor ( not a pseudotensor). Hint: Write (6.1) for each \(\epsilon\) and multiply them, being careful not to re-use a pair of summation indices.

Short answer

\( \epsilon_{ijk} \epsilon_{mnp} \) transforms as a tensor of rank 6.

Step-by-step solution

Write the Transformation Equation for \( \epsilon_{ijk} \)

Recall the Levi-Civita symbol transformation under a rotation matrix \( R \). The transformation for the Levi-Civita symbol \( \epsilon_{ijk} \) is given by: \[ \epsilon_{ijk}' = R_{ia} R_{jb} R_{kc} \epsilon_{abc} \]

Write the Transformation Equation for \( \epsilon_{mnp} \)

Similarly, the Levi-Civita symbol \( \epsilon_{mnp} \) transforms under the rotation matrix \( R \) as: \[ \epsilon_{mnp}' = R_{md} R_{ne} R_{pf} \epsilon_{def} \]

Multiply the Two Transformation Equations Together

To find the transformation equation for the product \( \epsilon_{ijk} \epsilon_{mnp} \), multiply the two transformed expressions: \[ \epsilon_{ijk}' \epsilon_{mnp}' = (R_{ia} R_{jb} R_{kc} \epsilon_{abc})(R_{md} R_{ne} R_{pf} \epsilon_{def}) \]

Combine the Rotations and Levi-Civita Symbols

Combining the rotation matrices and Levi-Civita symbols, we have: \[ \epsilon_{ijk}' \epsilon_{mnp}' = R_{ia} R_{jb} R_{kc} R_{md} R_{ne} R_{pf} (\epsilon_{abc} \epsilon_{def}) \]

Analyze the Result

The expression \( \epsilon_{ijk} \epsilon_{mnp} \) transforms by multiplying it with nine rotation matrices, one for each index. Therefore, the product \( \epsilon_{ijk} \epsilon_{mnp} \) transforms as a tensor of rank 6, confirming that it is a tensor, not a pseudotensor.

Key concepts

Levi-Civita Symbol

The Levi-Civita symbol, also known as the permutation symbol, is a mathematical object used in vector and tensor analysis. It is defined as \(\epsilon_{ijk}\) for three dimensions and helps determine the sign of a permutation of coordinates.
It equals +1 for even permutations of \(123\), -1 for odd permutations, and 0 if any indices are repeated. This symbol is pivotal in expressing cross products, determinants, and in the context of the problem, tensor transformations.

When transforming the Levi-Civita symbol under a rotation matrix \(R\), we have \(\epsilon_{ijk}' = R_{ia} R_{jb} R_{kc} \epsilon_{abc}\). This equation tells us how the symbol changes under rotations, maintaining the properties of orientation and the multi-dimensional relationships it encapsulates.

Rotation Matrix

A rotation matrix \(R\) is a matrix used to rotate vectors in a coordinate space. It plays a crucial role in transforming tensors and other mathematical objects under rotations.

In three dimensions, the rotation matrix is a 3x3 matrix satisfying \( R^T R = I \), meaning the transpose of \( R \) times \( R \) equals the identity matrix. This ensures that the length of vectors is preserved under rotation and that angles between vectors remain unchanged.

The transformation laws for the Levi-Civita symbols use rotation matrices to relate the symbols before and after rotation. For example, \( \epsilon_{mnp}' = R_{md} R_{ne} R_{pf} \epsilon_{def} \). This rotation transformation allows us to compute how complex multi-indexed structures like tensors behave when coordinates are rotated.

Rank 6 Tensor

A Rank 6 tensor has six indices and transforms in a specific way under coordinate changes. In the context of our problem, we examine \(\epsilon_{ijk} \epsilon_{mnp}\), which becomes a Rank 6 tensor.

To confirm it as a tensor and not a pseudotensor, we derive its transformation equations under rotation. Using the transformation laws for each Levi-Civita symbol separately, we multiply them:

\(\epsilon_{ijk}' \epsilon_{mnp}' = (R_{ia} R_{jb} R_{kc} \epsilon_{abc})(R_{md} R_{ne} R_{pf} \epsilon_{def})\)

By combining these, we get:
\(\epsilon_{ijk}' \epsilon_{mnp}' = R_{ia} R_{jb} R_{kc} R_{md} R_{ne} R_{pf} (\epsilon_{abc} \epsilon_{def})\)

Thus, the product transforms with six rotation matrices, showing it behaves as a proper Rank 6 tensor under rotations, preserving its tensor properties.

Pseudotensor

A pseudotensor is a tensor that changes sign under improper transformations like reflections, unlike true tensors that remain unchanged or follow specific transformation rules. An important aspect is that pseudotensors incorporate the orientation of the coordinate system.

In the exercise, by transforming \( \epsilon_{ijk} \epsilon_{mnp}\) and showing the result behaves as a proper tensor under rotations without any sign reversal, we confirm it is not a pseudotensor. This distinction is crucial since pseudotensors are used in different contexts, such as describing physical quantities involving handedness or orientation but don't follow the same transformation rules as true tensors.

Tensor Analysis

Tensor analysis is a field of mathematics focusing on the study of tensors and their applications. Tensors extend the concept of scalars, vectors, and matrices to higher dimensions and are essential in physics, engineering, and computer science.

Understanding tensor transformation rules, like those of the Levi-Civita symbol or rotation matrices, is vital in this field. It involves operations such as tensor contraction, addition, and multiplication.

For instance, in our exercise, by analyzing how \(\epsilon_{ijk} \epsilon_{mnp} \) transforms under rotations, we delve into the properties and behavior of a Rank 6 tensor. This analysis provides insight into higher-dimensional spaces and the fundamental framework necessary for advanced studies in fields like general relativity, quantum mechanics, and continuum mechanics.