16 Two Axiomatic Systems for Special Relativity
157
Here, x = x 2 − x 1 , ,t = t 2 − t 1 and for the same points in the primed coordinates
x
= x
2 − x
1 , ,t
= t
2 − t
1 . If one once again takes all three space coordinates,
then the demand for the invariance of the line element runs as follows:
x
2
+ y
2
+ z
2
− c
2
L t
2
= x
2 + y
2 z
2 − c
2
L t
2 .
(170b)
The spacetime continuum for which a distance s is defined, according to (167), is
called Minkowski space. For a detailed discussion of Minkowski geometry, we refer
the mathematical interested reader to G. L. Naber [67]; see also A. P. French [23]
and for an elementary geometrical representation, Liebscher [57].
Let us make it clear for ourselves that the Lorentz transformations (151) are
filtered out from the transformations (165) by the demand made by (170).
We remind ourselves of elementary mathematics:
Let us write
y := c L t .
If we have a plus instead of a minus sign in Eq. (170), hence
x
2
+ y
2
= x
2 + y
2 ,
(171)
then the primed coordinate axes in the x-y-plane, described by Cartesian coordinates,
are created from the unprimed by an angular rotation of ϕ„
x
= x cos ϕ + y sin ϕ ,
y
= −x sin ϕ + y cos ϕ .
(172)
We can now reduce Eq. (170) to (171) by introducing a purely imaginary coordinate
T into the x-t-plane according to
T := i c L t , where i
2
= −1 .
(173)
Thus, (170) becomes x
2
+ T
2
= x
2
+ T
2 and from the solution of this equation
we receive with the purely imaginary angle iϕ
x
= x cos(i ϕ) + T sin(i ϕ) ,
y
= −x sin(i ϕ) + T cos(i ϕ) .
We now take into consideration the relationship between the hyperbolic and the trigonomic functions, namely cos(iϕ) = cosh ϕ, sin(iϕ) = i sinh ϕ , replace T with
(173) and receive
157
Here, x = x 2 − x 1 , ,t = t 2 − t 1 and for the same points in the primed coordinates
x
= x
2 − x
1 , ,t
= t
2 − t
1 . If one once again takes all three space coordinates,
then the demand for the invariance of the line element runs as follows:
x
2
+ y
2
+ z
2
− c
2
L t
2
= x
2 + y
2 z
2 − c
2
L t
2 .
(170b)
The spacetime continuum for which a distance s is defined, according to (167), is
called Minkowski space. For a detailed discussion of Minkowski geometry, we refer
the mathematical interested reader to G. L. Naber [67]; see also A. P. French [23]
and for an elementary geometrical representation, Liebscher [57].
Let us make it clear for ourselves that the Lorentz transformations (151) are
filtered out from the transformations (165) by the demand made by (170).
We remind ourselves of elementary mathematics:
Let us write
y := c L t .
If we have a plus instead of a minus sign in Eq. (170), hence
x
2
+ y
2
= x
2 + y
2 ,
(171)
then the primed coordinate axes in the x-y-plane, described by Cartesian coordinates,
are created from the unprimed by an angular rotation of ϕ„
x
= x cos ϕ + y sin ϕ ,
y
= −x sin ϕ + y cos ϕ .
(172)
We can now reduce Eq. (170) to (171) by introducing a purely imaginary coordinate
T into the x-t-plane according to
T := i c L t , where i
2
= −1 .
(173)
Thus, (170) becomes x
2
+ T
2
= x
2
+ T
2 and from the solution of this equation
we receive with the purely imaginary angle iϕ
x
= x cos(i ϕ) + T sin(i ϕ) ,
y
= −x sin(i ϕ) + T cos(i ϕ) .
We now take into consideration the relationship between the hyperbolic and the trigonomic functions, namely cos(iϕ) = cosh ϕ, sin(iϕ) = i sinh ϕ , replace T with
(173) and receive