Chapter 13
The Lorentz Transformation
We consider an observer in the reference system
moving with respect to our
reference system o with the velocity v. The measuring-rods and clocks used in
are described by us from out of o using the solutions q
I
(x, t) and q
III
(x, t) of the
sine-Gordon equation. In order to make measurements, the observer in
distributes
these measuring-rods and clocks and synchronises these clocks according to our
elementary principle of relativity. Because we now know that the observer measures
the ‘universal’ velocity c o (or the equivalent sound velocity c T ) for the propagation of
a ‘sound signal’; he could proceed with this synchronisation using the velocity c o , as
in the traditional description of Special Relativity. The initial points of his space and
time measurement are determined by the initial conditions (134), the coincidence
of the coordinate origins of o and
that we have described as event O with
(x, t) = (x
, t
) = (0, 0).
Our measuring section, X = x L o = x
L
, rests in
. Its left end point
lies in the coordinate origin of
with x
1 = 0 and thus has in o the coordinates
x 1 = v t. The length of the measuring section is arbitrary. We can thus write for
the right end point in
the coordinate x
2 = x
= x
and in o the coordinates
x 2 = x = x + v t, compare to (130), so that with x
= x
and x = x − v t, we
get from Eq. (113) for the Lorentz contraction
x
(x, t) =
x − v t
1 − v 2 /c 2
o
(149)
with arbitrary x and t. The hand setting t
(x, 0) of the clocks of
for an arbitrary
position x and at the uniform time t = 0 of o was calculated using our elementary
principle of relativity in Eq. (142). We know how these clocks run. They go behind
our clocks in o by the factor
1 − v 2 /c 2
o . The clock that was at x at the o time
t = 0 and had the hand setting (142) finds itself at time t at the position x + v t. It
has moved on by t
1 − v 2 /c 2
o and thus has the hand setting
© The Editor(s) (if applicable) and The Author(s), under exclusive
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_13
133
The Lorentz Transformation
We consider an observer in the reference system
moving with respect to our
reference system o with the velocity v. The measuring-rods and clocks used in
are described by us from out of o using the solutions q
I
(x, t) and q
III
(x, t) of the
sine-Gordon equation. In order to make measurements, the observer in
distributes
these measuring-rods and clocks and synchronises these clocks according to our
elementary principle of relativity. Because we now know that the observer measures
the ‘universal’ velocity c o (or the equivalent sound velocity c T ) for the propagation of
a ‘sound signal’; he could proceed with this synchronisation using the velocity c o , as
in the traditional description of Special Relativity. The initial points of his space and
time measurement are determined by the initial conditions (134), the coincidence
of the coordinate origins of o and
that we have described as event O with
(x, t) = (x
, t
) = (0, 0).
Our measuring section, X = x L o = x
L
, rests in
. Its left end point
lies in the coordinate origin of
with x
1 = 0 and thus has in o the coordinates
x 1 = v t. The length of the measuring section is arbitrary. We can thus write for
the right end point in
the coordinate x
2 = x
= x
and in o the coordinates
x 2 = x = x + v t, compare to (130), so that with x
= x
and x = x − v t, we
get from Eq. (113) for the Lorentz contraction
x
(x, t) =
x − v t
1 − v 2 /c 2
o
(149)
with arbitrary x and t. The hand setting t
(x, 0) of the clocks of
for an arbitrary
position x and at the uniform time t = 0 of o was calculated using our elementary
principle of relativity in Eq. (142). We know how these clocks run. They go behind
our clocks in o by the factor
1 − v 2 /c 2
o . The clock that was at x at the o time
t = 0 and had the hand setting (142) finds itself at time t at the position x + v t. It
has moved on by t
1 − v 2 /c 2
o and thus has the hand setting
© The Editor(s) (if applicable) and The Author(s), under exclusive
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_13
133