DOC.
43
JANUARY
1915
51
Can
we
find
a
way
out
by visualizing
the
four-dimensional
space
E
you
focus
on
as
unlimited in all
directions? Not
likely, as
it
appears
to
me.
For,
the coordinate
systs.
adjusted to
the
gravitational field
can
just
be found when
the
field
in
the
examined
region
is
known,
and
we
know
nothing
about the
physical phenomena
for t
=
-oo.
Systems
adapted to
the
field
could
perhaps
best be found if
purely
periodic processes (such
as
in
the
old
theory
where the motion of two
mutually
attractive
mass-points) were
involved. If such
processes
existed also in
the
new
theory
of
gravitation,
t
=
F(x,
y, z)
and t
=
F(x,
y, z)
+
T
(T period)
could be
chosen
as
the
boundary
of
region
E and
the
Axu’s
could be chosen in such
a way
that
for
x,
y,
z,
t
+
T the
Axu’s
would have
the
same
value
as
for
x,
y,
z,
t.
As
concerns
the
linear
substitutions,
these do not
pose
a
problem,
since
they
are
introduced
for all values
of
x,
y,
z,
t. They incidentally
do not
belong
in
the
class
of
transformations
presently
being
examined
by
you,
since
they
are
incompatible
with the
boundary
conditions
adopted for
the
Axu’s.
With
these remarks
I
do
not want to
deny
in
any
way
that the introduction
of
adapted
coordinate
systems
is
very
appealing
and
that
it has
proven
very
useful
in
the
derivation
of
the
differential
equations
for
the
gravitational
field.[5]
If
this
derivation alone
is
involved,
then
we can
limit ourselves to the interior of
a
finite
region
and do not need
to
worry
about
the discontinuities at
the
boundary.
I just
wanted to
say
that
it
will
be difficult to
uphold actually finding
suitable
adapted
systems
and that
therefore in the end
the
need
for
nonlinear
transformations
against
which
the
equation
of
the
physics
should be covariant
is
satisfied
only
to
a
limited
degree.
I
would like to
say
a
few
words
about
this
“need”
as
well. Apparently
it
was
much less keen in
my
case
than in
yours.
I have
nothing against
the
existence
of
“outstanding”
coordinate
systems
for
the
physical
phenomena
to be
“preferred”
in
a
certain
sense over
all others. With this I do
not
mean
that when
basing
the
description
of
the
phenomena
on
such
an
outstanding
system something
“ab-
solute”
is
being
described in
the
processes; only
that the
description
is
formed
more simply or
attractively
than
through
another
coordinate
system
selection,
so
that
it
offers
us more
satisfaction.[6] In
many
cases
it
will
be
easy
to
agree
on
the
question
of what is
more
and
what
is
less simple. Incidentally
the
equal
status of
both
systems
is
already
obstructed
when
upon implementation only
the
description
of
the
one or
the
other looks different.
Take, e.g.,
the
case
of
a
rotating
system,
which Newton and Mach discuss and
which
you
also touch
upon
briefly.[7]
Experience
instructs
that
coordinate
system
I
can
be selected in such
a
way
that the
motion
of
a
body
in
the
proximity
of
the
Earth
can
be
described,
at least to
a
close
approximation,
with
the
equations
d2x/dt2
=
-
ax/2
-
dx2
(1)
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