Let us consider the two systems of coordinates K and K' represented figure 14 in which the axis x and x' are common.
First case: The events are on x-axis only.
One of these events is represented in the system of coordinates K by the
abscissa x and the time t and in the system K' by the abscissa x' and the
time t'. A light signal going along the axis x can be represented by:
X = ct or X - ct = 0 (I)
In the system K' the equation becomes:
X'- ct'= 0 (II)
As the events that satisfy (I) must also satisfy (II) we have:
(X'- ct') = (X - ct) (III), is a constant.
In the same way, for light ray that are transmitted along the negative
X axis:
(X'+ ct') = µ(X + ct) (IV)
By adding (III) and (IV) we get: 2X' = X( + µ) - ct( - µ)
By subtracting (III) and (IV): -2ct'= X( -µ) - ct( + µ)
X' = X ( + µ)/2 - ct( - µ)/2
-ct'=X( - µ)/2 - ct( + µ)/2
And If we define: ( + µ)/2 = a
( - µ)/2 = b
We Obtain: X' = aX - bct (V-1)
ct' = act - bX (V-2)
For the origin of K' we always have X' = 0 and (V-1) gives us: x = (bc/a)t
If v is the velocity of the origin of K' relative to K we have: v = bc/a (VI)
If fact v is the relative velocity of the two systems of coordinates.
The principle of relativity tells us that seen from K, the length of a unit measuring-rod at rest relative to K' is the same as the length of the same unit rod seen from K' at rest relative to K.
If in (V-1) we make t=0, then X'=aX (V-3)
Let us now takes two points on X' separated by the distance X'=1 as measured in the K' coordinates. In the K coordinate, (V-3) gives us: X' = a X = 1 or X = 1/a (VII)
Let us consider K'(t'=0) and, eliminate t between the two equations (V-1)
and (V-b):
We get X' = a(1-b²/a²) or, taking (VI) into consideration: X'
= a(1- v²/c²)
From this we conclude that two points on the X axis separated by the unit distance 1 in relation to K will be represented in K' by: X' = a(1 - v²/c²) (VII-a)
But as X = X' and taking (V-3) into consideration: a² = 1/(1 - v²/c²) (VII-b)
A and b are obtain from (VI) and (VII-b) and introducing them in (V-1) and (V-2) we get:
X' = (X - vt)/ sqrt(1 - v²/c²) (VIII-1)
t' = (t - vX/c²)/ sqrt(1 - v²/c²) (VIII-2)
These are the Lorentz transformation for events on the X axis. It satisfies
the condition: X'² - c²t'² = X² - c²t² (VIII-a)
To include events that takes place outside the X axis we keep the (VIII)
equations but add: y' = y (IX-1)
Z' = z (IX-b)
A light signal emitted at the origin of K at t = 0 will travel according to the equation: r = Sqrt(x²+y²+z²) = ct that can also be written x²+y²+z²-c²t² = 0 (X)
As the velocity of the light is constant and following the principle of
relativity, the transmission of the light signal seen from K' we must have:
r' = ct or x'²+y'²+z'²-c²t'² = 0 (X-a)
(X) and (X-a) gives x'²+y'²+z'²-c²t'² = x²+y²+z²-c²t² (XI) as (VIII-a) must apply on the X axis.
If:
The axis K' are not parallel to the K axis
The velocity of translation of K' relative to K is not in the direction
of the X axis
We also have: x'²+y'²+z'²-c²t'² = x²+y²+z²-c²t²
(XI-a)