36: file eq36.gif
But from what has been said, the two snapshots must be identical;
hence Dx in (7) must be equal to Dx' in (7a), so that we obtain
eq. 37: file eq37.gif
The equations (6) and (7b) determine the constants a and b. By
inserting the values of these constants in (5), we obtain the first
and the fourth of the equations given in Section 11.
eq. 38: file eq38.gif
Thus we have obtained the Lorentz transformation for events on the
x-axis. It satisfies the condition
x'2 - c^2t'2 = x2 - c^2t2 . . . (8a).
The extension of this result, to include events which take place
outside the x-axis, is obtained by retaining equations (8) and
supplementing them by the relations
eq. 39: file eq39.gif
In this way we satisfy the postulate of the constancy of the velocity
of light in vacuo for rays of light of arbitrary direction, both for
the system K and for the system K'. This may be shown in the following
manner.
We suppose a light-signal sent out from the origin of K at the time t
= 0. It will be propagated according to the equation
eq.
Pages:
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141