and is positive; hence t' increases as t increases, if (x, y, z) are kept constant.
The transformation which corresponds to a reflexion in the four-dimensional space is also of considerable interest. In the particular case when the reflecting space passes through the plane x = 0, s = 0, the reflexion may be replaced by a rotation round the plane x = 0, s = 0, and a reflexion in the space x = 0. The corresponding transformation is thus made up out of a transformation of Lorentz
and a change in the sign of x'. Putting
and changing the sign of x', we get
The quantity u is introduced because the angle of rotation in the four-dimensional space is twice the angle between the reflecting space and the space x = 0. Its geometrical meaning in the case of the spherical wave transformation is indicated by the equation
which implies that a plane moving with the constant velocity u is transformed into itself. Further, when x = ut, we have
hence every point of the plane is transformed into itself.[1]
The formulae of transformation of the electromagnetic vectors are found from the first set of equations for a transformation with negative Jacobian.
- ↑ Geometrically the transformation is equivalent to a reflexion in a moving plane mirror.