25
and similarly for the accented system K1, then the condition which is
identically satisfied by the transformation can be expressed thus :
x[1]'2 + x[2]'2 + x[3]'2 + x[4]'2 = x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2
(12).
That is, by the afore-mentioned choice of " coordinates," (11a) [see
the end of Appendix II] is transformed into this equation.
We see from (12) that the imaginary time co-ordinate x[4], enters into
the condition of transformation in exactly the same way as the space
co-ordinates x[1], x[2], x[3]. It is due to this fact that, according
to the theory of relativity, the " time "x[4], enters into natural
laws in the same form as the space co ordinates x[1], x[2], x[3].
A four-dimensional continuum described by the "co-ordinates" x[1],
x[2], x[3], x[4], was called "world" by Minkowski, who also termed a
point-event a " world-point." From a "happening" in three-dimensional
space, physics becomes, as it were, an " existence " in the
four-dimensional " world."
This four-dimensional " world " bears a close similarity to the
three-dimensional " space " of (Euclidean) analytical geometry. If we
introduce into the latter a new Cartesian co-ordinate system (x'[1],
x'[2], x'[3]) with the same origin, then x'[1], x'[2], x'[3], are
linear homogeneous functions of x[1], x[2], x[3] which identically
satisfy the equation
x'[1]^2 + x'[2]^2 + x'[3]^2 = x[1]^2 + x[2]^2 + x[3]^2
The analogy with (12) is a complete one.
Pages:
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144