Relativity: The Special and General Theory

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. 40: file eq40.gif

or, if we square this equation, according to the equation

          x2 + y2 + z2 = c^2t2 = 0    .     .     .    (10).

It is required by the law of propagation of light, in conjunction with
the postulate of relativity, that the transmission of the signal in
question should take place -- as judged from K1 -- in accordance with
the corresponding formula

                               r' = ct'

or,

       x'2 + y'2 + z'2 - c^2t'2 = 0    .     .     .    (10a).

In order that equation (10a) may be a consequence of equation (10), we
must have

   x'2 + y'2 + z'2 - c^2t'2 = s (x2 + y2 + z2 - c^2t2)       (11).

Since equation (8a) must hold for points on the x-axis, we thus have s
= I. It is easily seen that the Lorentz transformation really
satisfies equation (11) for s = 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 - 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. 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