 |
 |
 |
 |
|
 |
 |
74
Relativity: The Special and General Theory
or,
x'² + y'² + z'² ? c²t'² = 0
.
.
.
(10a).
In order that equation (10a) may be a consequence of equation (10), we must have
x'² + y'² + z'² ? c²t'² = A (x² + y² + z² ? c²t²)
(11).
Since equation (8a) must hold for points on the x?axis, we thus have A = I. It is easily seen that the
Lorentz transformation really satisfies equation (11) for A = I; for (11) is a consequence of (8a) and
(9), and hence also of (8) and (9). We have thus derived the Lorentz transformation.
The Lorentz transformation represented by (8) and (9) still requires to be generalised. Obviously it
is immaterial whether the axes of K¹ be chosen so that they are spatially parallel to those of K. It is
also not essential that the velocity of translation of K¹ with respect to K should be in the direction of
the x?axis. A simple consideration shows that we are able to construct the Lorentz transformation
in this general sense from two kinds of transformations, viz. from Lorentz transformations in the
special sense and from purely spatial transformations. which corresponds to the replacement of the
rectangular co?ordinate system by a new system with its axes pointing in other directions.
Mathematically, we can characterise the generalised Lorentz transformation thus :
It expresses x', y', x', t', in terms of linear homogeneous functions of x, y, x, t, of such a kind that the
relation
x'² + y'² + z'² ? c²t'² = x² + y² + z² ? c²t²
(11a).
is satisficd identically. That is to say: If we substitute their expressions in x, y, x, t, in place of x', y',
x', t', on the left?hand side, then the left?hand side of (11a) agrees with the right?hand side.
Next: Appendix II: Minkowski's Four Dimensional Space
Relativity: The Special and General Theory