312
DOC.
411
JUNE
1912
But
the
following
question
obviously
remains
interesting:
Are there nontrivial
cases
in which
the
equivalence
remains
valid
for
finite regions
of
space
as
well?
[Macro-equivalence]
I think
you
have not
treated
this
question
systematically.
Also,
your
procedure
of
working
with
truncated
series
expansions might
be
unwieldy
for this
problem.
My
formulation of the
problem
tries
to
attack the
above
"problem
of
macro-
equivalence"
from the
optical
side.
But
now
you
write:
"It
appears
from
my
latest
paper
that
the
equivalence
principle
can
be valid
only
for
infinitely
small
fields,
and
that,
therefore,[8]
Born's accelerated
finite
system
cannot be
considered
a
static
gravitational field, i.e.,
cannot be
generated
by
masses
at rest."[9]
If
your
third
paper
(which
I
received in
proof
and
made
excerpts from)
is
the latest
one,[10]
then
I would like
to
add
at
once:
Your third
paper
shows
that
generally
speaking
no
macro-equivalence
can
exist. But
I do
not
see
that
your
third
paper
would show
that
macro-equivalence
does
not exist also
in
the
special
case
of Born's
hyperbolic
motion.
For
the time
being
I
would
therefore
like to
maintain the
following
thesis:
It
has
not
been
proved
that
a
finite
field of
hyperbolic
motion
cannot
possibly
be
viewed
as a
static
gravitational
field.
In
any case,
I
do know
one
thing:
Both
in the
x, y,
t
space
and in
the
x, y,
z,
t
space,
Born's
hyperbolic
field satisfies
the
requirements
A
and B
as
well
as
the
requirement
C
(the ascertainability
of
a
wall-clock time
Đ)
and
D
(temporal
constancy
of the
velocity
of
light
from
a
wall
lamp
for
every
fixed
point
of the
laboratory),
which will be
formulated
more
precisely
below.
These
are
of
course
all
purely
optical
requirements.
But the
equivalence requirement
also
comprises dynamical requirements,
and
it is
exactly on
this
point
that
you
have met
with
special
difficulties.
But
I must
say
in this connection: the
dynamical
requirements
are
rather
plastic,
and
if it
is
necessary,
first
break
a
few
more
bones
in
the
body
of
dynamics
on
top
of
those
you
have
been
breaking
during
the
last
7
years (N. B.,
thanks
to
your
training, dynamics
has become
a
real
"lady
contortionist").
For
this
reason
it
did not
seem
uninteresting
to
me
to ask:
What
are
the limits
on
the
validity range
of the
equivalence principle
that
are already
set
by
its
optical requirements alone,
without
regard
to its
dynamical
requirements?
This
is
the
essential
core
of
my
question.
Requirement
A is
unquestionably
contained in the
macro-equivalence requirement.
Naturally,
requirement
B
can
be also rejected.-
But
this
requirement is
"natural"
enough
for the combination of
requirements A
and
B
to
be
tested
at
the
outset.-[I have also
carried
through
to its conclusion
a
careful
investigation
that
did
not
use
the
reversibility
requirement
(B),
but,
instead, combined