Question

Show how to decompose the tensor \(T_{i j}\) into three tensors, $$ T_{i j}=U_{i j}+V_{i j}+S_{i j} $$ where \(U_{i j}\) is symmetric and has zero trace, \(V_{i j}\) is isotropic and \(S_{i j}\) has only three independent components.

Short answer

Decompose \( T_{i j} \) as: \( U_{i j} = T_{(i j)} - \frac{1}{3}\text{tr}(T_{i j})\text{Id} \, V_{i j} = \frac{1}{3}\text{tr}(T_{i j})\text{Id} \ and \ S_{i j} = T_{i j} - U_{i j} - V_{i j} \

Step-by-step solution

- Define the symmetric, traceless tensor

A symmetric tensor has the property that it equals its transpose, i.e., \( U_{i j} = U_{j i} \). A traceless tensor means that the sum of the diagonal elements is zero, i.e., \( \text{tr}(U_{i j}) = U_{ii} = 0 \). To extract the symmetric part of \( T_{i j} \) without its trace, calculate \( U_{i j} = T_{(i j)} - \frac{1}{3}\text{tr}(T_{i j})\text{Id} \), where \( T_{(i j)} = \frac{1}{2}(T_{i j} + T_{j i}) \) and \( \text{Id} \) is the identity tensor.

- Define the isotropic tensor

The isotropic part of the tensor \( T_{i j} \) can be characterized as being proportional to the identity tensor. This part is captured by \( V_{i j} = \frac{1}{3}\text{tr}(T_{i j})\text{Id} \). This ensures it is the same in all directions and has the same value along the diagonal.

- Define the tensor with three independent components

The remaining part of the tensor \( T_{i j} \), denoted by \( S_{i j} \), has only three independent components. It is the part of \( T_{i j} \) that is neither symmetric traceless nor isotropic. Thus, \( S_{i j} = T_{i j} - U_{i j} - V_{i j} \).

- Assemble the final decomposition

Combine all parts to confirm the original tensor: \( T_{i j} = U_{i j} + V_{i j} + S_{i j} \). Here, \( U_{i j} \) is symmetric and traceless, \( V_{i j} \) is isotropic, and \( S_{i j} \) has only three independent components.

Key concepts

Symmetric Tensor

A symmetric tensor is fundamental in understanding tensor decomposition. Let’s break it down.

**Symmetry Definition**: A tensor is symmetric if it equals its transpose. In mathematical terms for the tensor\( T_{ij} \), symmetry means \( T_{ij} = T_{ji} \).

**Tracelessness**: A tensor is considered traceless if the sum of its diagonal elements is zero. This is represented mathematically as \( \text{tr}(U_{ij}) = U_{ii} = 0 \).

To get a symmetric and traceless tensor \( U_{ij} \) from \( T_{ij} \), we use the formula:

\( U_{ij} = T_{(ij)} - \frac{1}{3}\text{tr}(T_{ij})\text{Id} \)

where:
* \( T_{(ij)} = \frac{1}{2}(T_{ij} + T_{ji})\) (symmetric part).
* \( \text{tr}(T_{ij}) \) is the trace of \( T_{ij} \).
* \( \text{Id} \) is the identity tensor.
By following this method, the resultant tensor \( U_{ij} \) is both symmetric and traceless.

Isotropic Tensor

An isotropic tensor represents an aspect of a tensor that is uniformly distributed in all directions.

**Uniformity**: An isotropic tensor has properties that are invariant under any rotation. This means it behaves the same regardless of the orientation.

**Mathematical Representation**: For a given tensor \(T_{ij}\), its isotropic part \(V_{ij}\) is given by:

\( V_{ij} = \frac{1}{3}\text{tr}(T_{ij})\text{Id} \)

where:
* \( \text{tr}(T_{ij}) \) is the trace (or sum of the diagonal elements) of \( T_{ij} \).
* \( \text{Id} \) is the identity tensor, ensuring the isotropy by having the same value along its diagonal.
This ensures that the isotropic tensor component remains consistent across all directions, making it truly uniform.

Independent Components

In tensor decomposition, identifying the parts of the tensor with independent components provides critical insights.

**Definition**: Independent components refer to the minimal set of elements needed to describe the tensor's properties without redundancy.

**Calculation**: For a tensor \( T_{ij} \), the component with independent elements \( S_{ij} \) is derived by removing the symmetric traceless part \( U_{ij} \) and the isotropic part \( V_{ij} \):

\( S_{ij} = T_{ij} - U_{ij} - V_{ij} \)

This approach ensures that \( S_{ij} \) retains only the essential characteristics of \( T_{ij} \) that are not covered by the symmetric traceless and isotropic parts.

**Three Independent Components**: For second-order tensors, the additional constraint is that \( S_{ij} \) contains only three independent components. This simplification helps in easier analysis and understanding of the tensor’s nature.

Traceless Tensor

A traceless tensor is an important concept in tensor decomposition and various physical contexts.

**Tracelessness Definition**: A tensor is traceless if the sum of its main diagonal elements equals zero. Represented mathematically for tensor \( U_{ij} \) as:
\( \text{tr}(U_{ij}) = U_{ii} = 0 \)

**Symmetric and Traceless**: In tensor decomposition, we often combine the properties of symmetry and tracelessness:

• \(U_{ij} = \frac{1}{2}(T_{ij} + T_{ji}) - \frac{1}{3}\text{tr}(T_{ij})\text{Id}\)

This combination helps in isolating the specific part of the tensor that fits these properties.

**Applications**: In various fields such as physics and engineering, traceless tensors are used to describe stress, strain, and other physical phenomena without redundant scalar information from the trace.