Principle of relativity

Remarks

Example: Newton's second law $m \ddot{\mathbf{x}} = \mathbf{F}$ for forces depending only on relative positions satisfies the principle with $G$ the group of Galilean transformations. Check: under $\mathbf{x}' = \mathbf{x} - \mathbf{v} t$ with constant $\mathbf{v}$, one has $\ddot{\mathbf{x}}' = \ddot{\mathbf{x}}$, and relative positions do not change. Non-example: the wave equation $\partial_t^2 u = c^2 \partial_x^2 u$ does not satisfy it for the Galilean group. The solution $u = f(x - ct)$ becomes $f(x' - (c - v) t')$ under $x' = x - vt$, $t' = t$, which solves the equation only with $c$ replaced by $c - v$. The wave equation does satisfy the principle for the group of Lorentz transformations. The principle of special relativity is the case of the Poincaré group; the general principle of relativity of general relativity asks for invariance under all smooth coordinate changes.