1 2 4 D O C . 3 1 I D E A S A N D M E T H O D S
to as well as relative to A spherical wave propagating from the origin of the
coordinate system satisfies (under suitable choice) the equation
,
where we put according to the Pythagorean theorem
.
Squaring the equation above we can also write
. (4)
Furthermore, since according to the principle of relativity the propagation of light
must be the same relative to as it is relative to , the same process of propaga-
tion relative to must also be described by a spherical wave of propagation ve-
locity . Therefore, the transformation we are looking for must be such that equa-
tion (4) and also the equation
(4a)
〈are valid and〉 mutually require each other. In essence, this condition determines
the transformation of the space-time coordinates. For the case of coordinate sys-
tems oriented as sketched in fig. 2, one arrives—in a manner not to be explicated
here—at the so-called Lorentz transformation
. (5)
K K′.
r ct =
r2 x2 y2 z2 + + =
x2 y2 z2 c2t2 – + + 0 =
K′ K
K′
c
x′2 y′2 z′2 c2t′2 – + + 0 =
x′
x vt –
1
v2
c2
---- - –
------------------ =
y′= y
z′= z
t′=
t
v-x
c2
----
–
1
v2
c2
---- - –
------------------
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