Consider the now-classic scenario in which observer *K* is at rest and observer *K′* is moving in the positive direction of the *x* axis with constant velocity *v*. The basic problem is that if they both observe a point event *E*, how should one convert the coordinates of *E* from one reference frame to the other?

First assume time is absolute and space is relative with no characteristic speed. Only the spatial coordinates in the positive direction of the *x* axis are affected. The other coordinates do not change. The transformation equation for the positive direction of the *x* axis is

*r _{x}′ = r_{x} − vt_{x}*

where *r _{x} * is the spatial coordinate and

*t*is the temporal coordinate in the positive direction of the

_{x}*x*axis. The inverse transformation is

*r _{x} = r_{x}′ + vt_{x}*

*′*.

Adding them together gives

*r _{x}′ + r_{x} = r_{x} + r_{x}′ − vt_{x}*

*+ vt*

_{x}*′,*

which easily leads to

*t _{x}*

*′*=

*t*.

_{x}This is called the Galilean transformation.

Now consider the case in which space is absolute and time is relative with no characteristic speed. Only the temporal coordinates in the positive direction of the *x* axis are affected. The other coordinates do not change. The transformation equation for the positive direction of the *x* axis is

*t _{x}′ = t_{x} − r_{x}/v*

and the inverse transformation is

*t _{x} = t_{x}′ + r_{x}′/v.*

Adding them together leads to

*r _{x}*

*′*=

*r*.

_{x}This could be called the *dual Galilean transformation* since only temporal coordinates change.