50
DOC.
43 JANUARY
1915
the condition
that
at
the transition
from
one
system
to
another
one
that
differs
from
it
to
an
infinitely
small
degree,
the
Axu
and
dAxu/dxa
terms vanish
at
the
boundary.[4]
If
it
is
assumed
that the
second differential
gradients
of Ax
with
respect
to
x
differ from
zero
at
the
boundary,
then,
as
formula
(63a)
shows,
this
also
applies
to
Aguv.
The
latter
do not vanish
on
the
inner side of
the
boundary,
and
since
they
are zero
at
the
outer side where
nothing
has been
changed,
an
inconsistency
thus
arises for these values.
Therefore,
if
to
begin
with
you
relate
a given
gravitational field
to
a
coordinate
system,
and
thereafter
to
a
second
system
adapted to
the
field
and
differing
from the
first
only
in
the
interior
by E,
you
are
then
introducing
inconsistencies at
the
boundary.
A
description
of the
phenomena
involving
this
can
hardly
be called
satisfactory,
however.
The
difficulty
remains
Thus
it
is
not
impossible
that
at
that
place
higher
differential
quotients
of
Ax
with
respect to
x
are
other than
zero.
If this
is
the
case,
then
as can
be
derived from formula
(63),
the variation
A,
with
respect
to
the coordinates
at
the
boundary,
for
certain
differential
quotients
of
the
guv’s
will
also differ from
zero.
Now,
since
nothing
has been
changed
on
the
outer side of
the
boundary,
an
inconsistency
would
necessarily
occur
in
the
values
of
those differential
quo-
tients,
if
you
introduce
a
coordinate
system
that
differs
only
in
the interior
of
the
delimited
area
from
the
one
first used. The
introduction
of such
an
inconsistency
in
the
description
of
the
gravitational
field is
scarcely gratifying, though.
Were
the
second differential
quotients
of
the
Ax’s with
respect
to
x
other
than
zero
at
the
boundary,
then
an
inconsistency
would result
already
in the first
differential
quotients of
the
guv's.
Such
an
inconsistency
can
exist
only
when
a
finite amount
of
the attractive
“agent” (thus
here
the
energy,
etc.)
is
distributed
throughout
a
surface.
It
is
clear, however,
that
when such
a
surface
distribution
does not exist in
one
description
of
the
gravitational
field,
it also cannot exist in
the
new
description.
Similar observations also
necessarily
follow
when
you imagine
that material
processes
take
place
in
the
interior of
E.
Only,
in this
case
attention
must
be
directed
to all
the
equations, therefore,
not
only
to
the
gravitation
eqs.
but
also
to those
that
determine
the material
processes. By
the
way,
it
seems
to
me
that
the
difficulty always
exists
even
when,
for
ex.,
the
only
restriction consists in
that
the
processes
are
also
analyzed
from
a
particular moment
onward,
so
that
region
E
is denoted,
shall
we
say,
by
the
inequality
t
F(x,
y,
z).
As
I
see
it,
it
is
permissible
to
require
that the
description
of
the
phenomena
after the
moment
t0
=
F(x,
y,
z)
conform with
the
description
of
the
processes
preceding
this
time,
and
that
therefore
no
discontinuities
re
introduced
at
the
boundary
t0.