Chapter 16
Two Axiomatic Systems for Special
Relativity
We will now turn to the axiomatic construction of the Special Theory of Relativity.
Here, we will consider our physical spacetime with its distinguished critical velocity, the speed of light c L . The starting point of our considerations in any case is the
reference systems in which we experiment and describe the results of these experiments. Starting out from the oldest physical discipline, mechanics, we arrive at the
distinction of inertial systems as seen in Chap. 4.
In mathematical terms, the problem can be described as follows. The equations of
Newtonian mechanics are identical for all inertial systems, if the Galilei transformation (152) is valid for the coordinates (x, t) and (x
, t
) of an event in two arbitrary
inertial systems o and
, respectively,
x
= x − vt ,
t
= t ,
←→
x = x
+ vt
,
t = t
.
(152)
If a scalar solution f = f (x − c L t) of the wave equation
∂
2 f (x, t)
∂x 2
−
1
c
2
L
∂
2 f (x, t)
∂t 2
= 0
( 1 )
is observed in o , then this describes the propagation of a plane wave signal in
the x-direction with the speed of light c L (see for example Fig. 16.1). Accepting
the Galilei transformation (152) results in f = f (x − c L t) = f (x
+ vt
− c L t
) =
f [x
− (c L − v)t
] for the same signal when observed from
and thus
∂
2 f
∂x 2 −
1
(c L − v) 2
∂
2 f
∂t 2 = 0 .
(164)
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license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_16
153