152
DOC. 30 FOUNDATION OF GENERAL RELATIVITY
of
Z,
permanently
coincide.
We shall
show
that
for
a space-
time
measurement
in the
system
K' the
above definition
of
the
physical meaning
of
lengths
and
times
cannot be
main-
tained.
For
reasons
of
symmetry
it
is clear
that
a
circle
around the
origin
in the
X,
Y
plane
of K
may
at
the
same
time
be
regarded
as a
circle
in the
X',
Y'
plane
of
K'.
We
suppose
that the circumference and diameter
of
this
circle
have
been
measured with
a
unit
measure
infinitely
small
compared
with the
radius,
and
that
we
have the
quotient
of
the
two results.
If this
experiment
were
performed
with
a
measuring-rod
at rest
relatively
to
the
Galilean
system
K,
the
quotient
would be
r.
With
a
measuring-rod
at rest
relatively
to
K',
the
quotient
would be
greater
than
r.
This
is
readily
understood
if
we
envisage
the
whole
process
of
measuring
from
the
"stationary" system
K,
and take into consideration
that the
measuring-rod
applied
to
the
periphery
undergoes
a
Lorentzian
contraction,
while
the
one applied
along
the
radius
does not.
Hence Euclidean
geometry
does not
apply
to K'.
The notion
of
co-ordinates
defined
above,
which
pre-
supposes
the
validity
of
Euclidean
geometry,
therefore breaks
[10]
down
in relation
to
the
system
K'.
So,
too,
we are
unable
to
introduce
a
time
corresponding
to
physical
requirements
in
K',
indicated
by
clocks at rest
relatively
to
K'. To
convince ourselves
of
this
impossibility,
let
us
imagine
two
clocks of
identical constitution
placed,
one
at
the
origin
of
co-ordinates,
and the other
at
the circumference
of
the
circle,
and both
envisaged
from
the
"stationary" system
K.
By
a
familiar result
of
the
special
theory
of
relativity,
the
clock at
the
circumference-judged
from K-goes
more
slowly
than
the
other,
because the former is
in motion
and
the
latter
at rest. An observer at
the
common
origin
of
co-ordinates,
capable
of
observing
the
clock at
the
circum-
ference
by
means
of
light,
would
therefore
see
it
lagging
be-
hind the
clock beside him. As
he
will not
make
up
his mind
to
let the
velocity
of light
along
the
path
in
question
depend
explicitly on
the
time,
he
will
interpret
his observations
as
showing
that the
clock at
the
circumference
"really"
goes
more slowly
than the
clock
at the
origin.
So
he
will be
obliged
to
define
time in such
a
way
that the rate
of
a
clock
depends
upon
where the
clock
may
be.
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