What is the difference between a scalar and a tensor in physics?

In physics, the concepts of scalar and tensor are fundamental to describe quantities and their transformation properties under changes of the coordinate system. Both scalars and tensors are used to describe a myriad of physical phenomena, ranging from motion and mechanics to electromagnetism, relativity, and quantum mechanics.

Scalars

A scalar is the simplest type of physical quantity and is characterized by having only magnitude. Scalars are invariant under coordinate transformations, meaning their value does not change if we switch from one coordinate system to another.

Examples of scalar quantities include mass, temperature, energy, speed, and charge. These quantities can be fully described by a single real number. If we have two scalar quantities, we can add them together directly without worrying about any other attributes.

Consider temperature, for instance. If it’s 20 degrees Celsius outside and the temperature rises by 5 degrees, the resulting temperature is simply 25 degrees. It doesn’t matter which direction you look from, the temperature remains the same.

Tensors

Tensors, on the other hand, are more complex mathematical objects. Like scalars, tensors can describe magnitude. However, unlike scalars, tensors also include information about direction.

At the most basic level, a tensor is a geometric object that maps in a multi-linear manner geometric vectors, scalars, and other tensors to a resulting tensor. Tensors have a rank, or order, which determines the complexity of the tensor.

• A tensor of rank zero, or a tensor with no direction, is a scalar.
• A tensor of rank one is known as a vector. Vectors have both a magnitude and a direction in space.
• A tensor of rank two represents a linear map from vectors to vectors, or it can be thought of as a type of matrix. An example is the stress tensor in continuum mechanics, which takes a direction as input and outputs a force vector.
• Higher rank tensors can also exist, which can be represented as multidimensional arrays.

One of the key properties of tensors is their transformation rules. When you change the coordinate system (say, by rotating your axes), tensors transform in a particular way according to their rank. Scalars (rank-0 tensors) don’t change at all, vectors (rank-1 tensors) rotate with the coordinate system, and higher-rank tensors follow more complex transformation rules.

Conclusion

So, the essential difference between scalars and tensors lies in the type of information they can convey and their transformation properties under coordinate changes. Scalars only have magnitude and are unchanged under coordinate transformations. Tensors, on the other hand, can include both magnitude and direction and follow specific transformation rules depending on their rank.

Understanding these mathematical structures and their properties is vital in physics because it allows us to formulate physical laws in a way that is independent of the choice of coordinate system, thereby revealing the underlying symmetries of the physical world.