In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations (in Newtonian mechanics), or by Lorentz transformations or space-time translations (in relativity). (Contrast to vector.)
For example, the distance between two points in space is a scalar, as are the mass, charge, and kinetic energy of an object, or the temperature and electric potential at a point inside a medium. On the other hand, the electric field at a point is not a scalar in this sense, since to specify it one must give three real numbers that depend on the coordinate system chosen. The speed of an object is a scalar (e.g. 180 km/h), while its velocity is not (i.e. 180 km/h north). The gravitational force acting on a particle is not a scalar, but its magnitude is.
Examples of scalar quantities in Newtonian mechanics:
- electric charge and charge density
- mass and mass density
- speed, but not velocity or momentum
- energy and energy density
A physical quantity is expressed as the product of a numerical value and a physical unit, not just a number. It does not depend on the unit distance (1 km is the same as 1000 m), although the number depends on the unit. Thus distance does not depend on the length of the base vectors of the coordinate system. Also, other changes of the coordinate system may affect the formula for computing the scalar (for example, the Euclidean formula for distance in terms of coordinates relies on the basis being orthonormal), but not the scalar itself. In this sense, physical distance deviates from the definition of metric in not being just a real number; however it satisfies all other properties. The same applies for other physical quantities which are not dimensionless.
Scalars in relativity theory
In the theory of relativity, one considers changes of coordinate systems that trade space for time. As a consequence, several physical quantities that are scalars in "classical" (non-relativistic) physics need to be combined with other quantities and treated as four-dimensional vectors or tensors. For example, the charge density at a point in a medium, which is a scalar in classical physics, must be combined with the local current density (a 3-vector) to comprise a relativistic 4-vector. Similarly, energy density must be combined with momentum density and pressure into the stress-energy tensor.
Examples of scalar quantities in relativity:
A related concept is a pseudoscalar, which is invariant under proper rotations but (like a pseudovector) flips sign under improper rotations. One example is the scalar triple product (see vector), and thus the signed volume. Another example is magnetic charge (as it is mathematically defined, regardless of whether it actually exists physically).