DOC.
47
305
two
events
are
simultaneous with respect
to
S',
and
thus also with respect
to
E,
if
,
v
,
v
n
~c*
xi
~
2
~cZ
x2
'
where
the subscripts refer
to
the
one or
to
the other point
event,
respec
tively.
We
shall first confine ourselves
to
the consideration of times that
are
so
short1 that all
terms
containing
the
second
or
higher
power
of
r or
v can
be omitted; taking
(1)
and
(29)
into
account,
we
then
have to
put
x2

xt
=
4

x\
=
£2

h
 &1
h
~
a2
V
=
jt
=
JT
,
so
that
we
obtain
from
the
above
equation
a2
~
= ^7
(£2
"
£l)
•
If
we move
the first
point
event
to
the coordinate
origin,
so
that
rt
=
r
and
E1 =
0,
we
obtain,
omitting
the
subscript
for the
second point
event,
a=t[1+yE/c2].
(30)
This
equation
holds first of all if
r
and
£
lie
below
certain
limits. It is obvious that it holds for arbitrarily
large
r
if the
acceler
ation
7
is
constant
with respect
to £,
because the
relation
between
a
and
t
must
then
be
linear.
Equation (30)
does not
hold for
arbitrarily
large
E.
From
the fact that the choice of the coordinate
origin
must not
affect the
relation,
one
must
conclude that,
strictly
speaking, equation (30)
should
be
replaced
by
the
equation
5=
Nevertheless,
we
shall maintain
formula
(30).
1In
accordance with (1),
we
thereby
also
assume a
certain restriction with
respect
to
the values of
£ =
x'.
[98]
[99]