1 7 0 D O C U M E N T 2 6 6 O C T O B E R 1 9 2 1
Let there be a system of points K in which the axioms I–V are
satisfied.[3]
At the
origin A, instead of uniform time t, a function is introduced and a system
of points is constructed according to Def.
3.[4]
In synchronism and the
lengths from A being equal are defined just as in K. If in axiom III is supposed
to be
valid,[5]
it is evidently sufficient that for a signal between two arbitrary points
in , the time be independent of . It suffices if one imagines the
points from speeding radially away from A, because a system winding in
upon itself cannot lead to the goal. The azimuth hence transforms identically.
The transformation formulas from K to then read for the radius (= distance from
A) and the time (velocity of light set = 1)
(1) In K: t, r
(2) In :
One can now require that for 2 points in , , the sig-
nal time be independent of and then obtain the following condition for
f(t). If , the inverse function of , then must have the follow-
ing characteristic: 3 arbitrary finite quantities are given; hence from
, 3 quantities: are determined by
argument function
If one now varies entirely
arbitrarily for constant , the thus emerging should satisfy the following
relation:
(3)
This is satisfied not just by a straight line but also by such that, depending
on the choice of f(t), is left over. This is the only possibility. If one plugs
f t =
K K
K
B C K
2 1
–
K K
K
1
2
-- - f t r + f t r – + =
1
2
-- - f t r + f t r – – = K ,
K
1 1
and
2 2 1 2
2 1
–
t = f t =
1 2 3
1 2 3
t 1
2 3
1
2
3
1
+
2
+
3
+
t =
t
1
+
t
2
+
t
3
+
1 2 3
1 2 2 3
2
1 3
– +
2 3 1
–
------------------------------------------------ - const
1 2 3
= =
1 – 1. +
t e =
logt =