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 - c^2t'2 = x2 + y2 + z2 - c^2t2 (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.
APPENDIX II
MINKOWSKI'S FOUR-DIMENSIONAL SPACE ("WORLD")
(SUPPLEMENTARY TO SECTION 17)
We can characterise the Lorentz transformation still more simply if we
introduce the imaginary eq. 25 in place of t, as time-variable. If, in
accordance with this, we insert
x[1] = x
x[2] = y
x[3] = z
x[4] = eq.
Pages:
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143