Galilean Transformation - Translation

Translation

In essence, the Galilean transformations embody the intuitive notion of addition and subtraction of velocities. The assumption that time can be treated as absolute is at the heart of the Galilean transformations, so they presume absolute time and space.

This assumption is abandoned in the Lorentz transformations. These relativistic transformations are applicable to all velocities, whilst the Galilean transformation can be regarded as a low-velocity approximation to the Lorentz transformation.

The notation below describes the relationship under the Galilean transformation between the coordinates (x,y,z,t) and (x′,y′,z′,t′) of a single arbitrary event, as measured in two coordinate systems S and S', in uniform relative motion (velocity v) in their common x and x’ directions, with their spatial origins coinciding at time t=t'=0:

Note that the last equation expresses the assumption of a universal time independent of the relative motion of different observers.

In the language of linear algebra, this transformation is considered a shear mapping, and is described with a matrix acting on a vector. With motion parallel to the x-axis, the transformation acts on only two components:

Though matrix representations are not strictly necessary for Galilean transformation, they provide the means for direct comparison to transformation methods in special relativity.