Is a Tensor a Physical Quantity?

Understanding whether a tensor is a physical quantity involves delving into its nature, its applications, and its mathematical representation. While scalars and vectors describe physical quantities, tensors provide a more comprehensive framework to understand complex physical phenomena.

What is a Tensor?

Think of a scalar or a vector: they are not physical quantities but descriptors of how the quantities behave under transformations. For example, charge is a scalar, and current is a vector, with the physical quantities being charge and current. Similarly, a tensor describes how a physical quantity behaves under a transformation. So, is a tensor a physical quantity? The answer is yes, but it's more nuanced than that.

A tensor can be thought of as a description or value of an object that requires multiple properties to be fully characterized. For instance, location and velocity.

Tensor in Solid Mechanics

Consider a cup of coffee. Its width, height, and length can be described as a tensor because these dimensions are invariant and can be determined regardless of the object's orientation. Similarly, the temperature of the coffee can also be described as a tensor if we maintain the idealization that it remains uniform across the coffee.

A More Detailed Example

Let’s imagine a big, non-ideal table with a tablecloth hanging off the edges. Place a cup of coffee on this tablecloth. Now, pull the tablecloth, and the cup of coffee moves. The length, width, height, and temperature of the coffee, which were previously known and assumed to be invariant, would change to maintain their ‘tensor’ quality. This means that the value and properties of these quantities remain consistent relative to the new orientation, even though their absolute orientation has changed. This is the essence of a tensor.

Scalars and Vectors as Tensors

A scalar is a zeroth-order tensor, as it only has a magnitude without direction. For instance, the mass of an object (10 kg) is a scalar. A vector, on the other hand, is a first-order tensor, which includes both magnitude and direction, like the distance between two cities.

Example in Geography

To travel from City A to City B, you would require both the distance (50 km northeast) and the direction to locate the city. This can be represented as a vector, indicating both magnitude and direction, and thus, a first-order tensor.

The Stress Tensor

In solid mechanics, tensors play a crucial role in describing physical quantities like stress. Consider an object under the influence of external forces, like the Earth’s gravity. To describe the stress at a point within an object, we use a tensor known as the stress tensor.

The Concept of Stress Tensor

A stress tensor can be reduced to a simple matrix or a dyad, which represents all states of stress. For example, the stress tensor in a solid can be represented as:

Stress Tensor [σ11 σ12 σ13 σ21 σ22 σ23 σ31 σ32 σ33]

Here, the elements of the matrix represent the stress components acting in all possible directions. For instance, if the only stress acting is along the x-direction (σ11), then a simple matrix or dyad would represent just this scalar aspect, providing a 1st-order tensor. This 2nd-order tensor fully characterizes the stress state within the object.

Adequacy of Tensors in Real-World Systems

In real-world applications, especially in continuum mechanics, tensors go beyond simple scalar or vector descriptions. The stress tensor, in particular, is essential for modeling the behavior of materials under stress in various engineering applications.

Note: Images of tensors and stress matrices were taken from the Wikipedia page on Cauchy stress tensor. Visit _stress_tensor for more details.

Thus, tensors, while not simple physical quantities themselves, are fundamental in describing and understanding complex physical phenomena in a multi-dimensional context.