Page:CunninghamExtension.djvu/11

II.

4. The group of conformal transformations in four dimensions referred to in the introduction can be built up of transformations by inversion in the hyperspheres of the space.

To establish the new theorem of relativity it is sufficient to consider a single member of the group, i.e., a single inversion. Let the centre of inversion be taken as origin.

The analysis is best carried out in spherical polar coordinates, so that the geometrical correlation of the two systems is

$R={\frac {k^{2}r}{r^{2}-c^{2}t^{2}}},\ T={\frac {k^{2}t}{r^{2}-c^{2}t^{2}}},$

the angular coordinates remaining unaltered.

Geometrical relations arising immediately are first given.

If u, U are the velocities of a moving point and the corresponding point in the other system,

$U_{R}={\frac {\left(r^{2}+c^{2}t^{2}\right)u_{r}-2c^{2}rt}{2rtu_{r}-\left(r^{2}+c^{2}t^{2}\right)}},$ $U_{N}=-{\frac {\left(r^{2}-c^{2}t^{2}\right)u_{n}}{2rtu_{r}-r^{2}+c^{2}t^{2}}}.$

In particular, if v is the velocity in either system of a point corresponding to a stationary point in the other,

$v={\frac {2c^{2}rt}{r^{2}+c^{2}t^{2}}},$

and is a radial velocity.

As in the former transformation,

$\beta =\left(1-v^{2}/c^{2}\right)^{-{\frac {1}{2}}}={\frac {r^{2}+c^{2}t^{2}}{r-c^{2}t^{2}}};$

and, using this, the above equations become

$U_{R}=-{\frac {u_{r}-v}{1-u_{r}v/c^{2}}},$ $\beta U_{N}={\frac {u_{n}}{1-u_{r}v/c^{2}}}.$

If we write $k^{2}\Lambda =\left(R^{2}-c^{2}T^{2}\right)$, and consider a small volume at rest in