Principle of covariance

Mathematically, the physical quantities must transform covariantly, that is, under a certain representation of the group of coordinate transformations between admissible frames of reference of the physical theory. This group is referred to as the covariance group.

The principle of covariance does not require the physical laws to be invariant under the group of admissible transformations, although in most cases the equations are indeed invariant. However, in the theory of weak interactions, the equations are not invariant under reflections (but are, of course, still covariant).

Covariance in Newtonian mechanics

In Newtonian mechanics the admissible frames of reference are inertial frames with relative velocities much smaller than the speed of light. Time is then absolute, and the transformations between admissible frames of references are Galilean transformations, which (together with rotations, translations, and reflections) form the Galilean group. The covariant physical quantities are Euclidean scalars, vectors, and tensors. An example of a covariant equation is Newton's second law, $${\displaystyle m\frac{d\overrightarrow{v}}{dt}=\overrightarrow{F},}$$ where the covariant quantities are the mass ${\displaystyle m}$ of a moving body (scalar), the velocity ${\displaystyle \overrightarrow{v}}$ of the body (vector), the force ${\displaystyle \overrightarrow{F}}$ acting on the body, and the invariant time ${\displaystyle t}$.