For the relative orientation of the co-ordinate systems indicated in
Fig. 2, the x-axes of both systems pernumently coincide. In the
present case we can divide the problem into parts by considering first
only events which are localised on the -axis. Any such event is
represented with respect to the co-ordinate system
by the abscissa
and the time
, and with respect to the system
by the abscissa
and the time
. We require to find
and
when
and
are given.
A light-signal, which is proceeding along the positive axis of , is
transmitted according to the equation
Since the same light-signal has to be transmitted relative to with
the velocity
, the propagation relative to the system
will be
represented by the analogous formula
Those space-time points (events) which satisfy (1) must also satisfy (2). Obviously this will be the case when the relation
is fulfilled in general, where indicates a constant; for, according
to (3), the disappearance of
involves the disappearance of
.
If we apply quite similar considerations to light rays which are being transmitted along the negative x-axis, we obtain the condition
By adding (or subtracting) equations (3) and (4), and introducing for
convenience the constants and
in place of the constants
and
,
where
and
we obtain the equations
We should thus have the solution of our problem, if the constants
and
were known. These result from the following discussion.
For the origin of we have permanently
, and hence according
to the first of the equations (5)
If we call the velocity with which the origin of
is moving
relative to
, we then have
The same value can be obtained from equations (5), if we calculate
the velocity of another point of
relative to
, or the velocity
(directed towards the negative
-axis) of a point of
with respect to
. In short, we can designate
as the relative velocity of the two
systems.
Furthermore, the principle of relativity teaches us that, as judged
from , the length of a unit measuring-rod which is at rest with
reference to
must be exactly the same as the length, as judged from
, of a unit measuring-rod which is at rest relative to
. In order
to see how the points of the
-axis appear as viewed from
, we only
require to take a “snapshot” of
from
; this means that we have
to insert a particular value of
(time of
), e.g.
. For this
value of
we then obtain from the first of the equations (5)
Two points of the -axis which are separated by the distance
when measured in the
system are thus separated in our instantaneous
photograph by the distance
But if the snapshot be taken from
, and if we eliminate
from the equations (5), taking into account the expression (6), we
obtain
From this we conclude that two points on the -axis separated by the
distance
(relative to
) will be represented on our snapshot by the
distance
But from what has been said, the two snapshots must be identical;
hence in (7) must be equal to
in (7a), so that we obtain
The equations (6) and (7b) determine the constants and
. By
inserting the values of these constants in (5), we obtain the first
and the fourth of the equations given in Section 11.
Thus we have obtained The Lorentz transformation
for events on the
-axis. It satisfies the condition
The extension of this result, to include events which take place
outside the -axis, is obtained by retaining equations (8) and
supplementing them by the relations
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 and for the system
. This may be shown in the following
manner.
We suppose a light-signal sent out from the origin of at the time
. It will be propagated according to the equation
or, if we square this equation, according to the equation
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 --in accordance with
the corresponding formula
or,
In order that equation (10a) may be a consequence of equation (10), we must have
Since equation (8a) must hold for points on the -axis, we thus have
. It is easily seen that the Lorentz transformation really
satisfies equation (11) for
; 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
be chosen so that they are spatially parallel to those of
. It is
also not essential that the velocity of translation of
with respect
to
should be in the direction of the
-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
, in terms of linear homogeneous functions
of
, of such a kind that the relation
is satisficd identically. That is to say: If we substitute their
expressions in
, in place of
, on the
left-hand side, then the left-hand side of (11a) agrees with the
right-hand side.
As of this snapshot date, this entry was owned by bloftin.