📂Mathematical Physics

What is a Scalar in Physics?

Definition

In physics, a scalar is a physical quantity defined as follows.

1. A physical quantity that has only magnitude and no direction.¹
2. A physical quantity that remains unchanged under a change of coordinate system, i.e., an invariant.²

Explanation

Time, mass, and temperature are representative scalar physical quantities.

1. In contrast, a quantity that has both direction and magnitude is called a vector. A vector has multiple values called components, whereas a scalar can be thought of informally as a one-component vector. From a rigorous physics viewpoint, the emphasis on ‘invariance under coordinate transformations’ when defining a scalar is because not every quantity represented by a single number is a scalar. Unlike vectors or tensors, whose components change according to specific rules (transformation matrices) when the coordinate system changes, a scalar is the most basic physical quantity whose value remains physically identical for any observer.

2. The mass of an object does not change regardless of which coordinates are chosen. An object whose mass is $10$ $\mathrm{kg}$ has the same value in both the Cartesian coordinate system and the spherical coordinate system. By contrast, vectors are affected by the coordinate system: even for the same physical quantity, the component values depend on the coordinate system. For example, the position and velocity of an object are, in Cartesian coordinates, given as follows: see below. $$ \begin{align*} \mathbf{r} &= x \hat{\mathbf{x}} + y \hat{\mathbf{y}} + z \hat{\mathbf{z}} \\ \mathbf{v} &= \dot{\mathbf{r}} = \dot{x} \hat{\mathbf{x}} + \dot{y} \hat{\mathbf{y}} + \dot{z} \hat{\mathbf{z}} \\ \end{align*} $$ In spherical coordinates, they are expressed as: $$ \begin{align*} \mathbf{r} &= r\hat{\mathbf{r}} \\ \mathbf{v} &=\dot{r} \hat {\mathbf{r}} +r \dot{\theta} \hat{ \boldsymbol{\theta}}+ r \dot{\phi} \sin{\theta} \hat{ \boldsymbol{\phi}} \end{align*} $$

Mathematically, given a $\mathbb{F}–$ vector space, a scalar means an element of $\mathbb{F}$, the field of the vector space.