DOC.
556
JUNE
1918 577
That this
actually is
so can
be
seen
as
follows.
If
a
closed
system
is incom-
pletely
described,
then the
energy
law for
the
described part reads
E
dT^y
dx"
=
Pn
where
pu
is
the
four vector of
the
force
density acting
on
the
system.[8]
If
this
equation is integrated
over a
segment
of
a
four-dimensional world strand,
then
one
obtains
on
the
right-hand
side
the
time
integral
J
dt
ƒ
PftdV,
this
is
the total
momentum and
energy
increase
of
the
system, on
the left-hand
side,
the
expressions
/
T^dV
=
A
J".
Since
the
result of
this
integration
has
a
vector
character
just
as
the
integrand,
the
same
is
valid for
AJu, that
is,
for
the
increase
that the
momentum and
energy
experience
on
the
segment
of
the
world
strand
under consideration. Given this,
it
cannot
be
doubted that the
Ju's themselves have this characteristic.
The
proof
can
be
furnished, incidentally,
also for the
Ju's
themselves in
that
the
Ax4Ja
=
Ao4’s are
components
of
a
tensor
Aar
(whose
components
A11
...
A33
vanish
(as can
easily
be
proven)[9].
I
do
not
want to
go
into
this
more
formal
proof
here,
though. So
that
out
of
the
tensor
character the
possibility
of
describing
it in
the
form Ja
=
E0
dxa/ds
follows,
it
is still
necessary,
however,
that the
momentum
of
a
system
at rest
vanishes,
which
is not
self-evident
per se.[10] Physically,
this
means
that
a
system
at rest
can
be
brought
into
another
position
through
rotation
without
a
finite exertion of
force.-[11]
Your first
letter[12]
was
very
instructive
for
me;
the
consideration
was new
to
me.
From
the
physical
point of
view,
I believe
I
can
assert
very definitely
that
this,
because
it
is four-dimensionally uniform, mathematically
more
elegant
conception
of
the
world,
does not
correspond
to
reality.
For
the
universe
seems
to
be
built
in such
a way
that
its
finely
distributed
matter could remain at
rest,
with
a
suitable
choice
of
the
coordinate
system.[13]
This
requires
that
g44
=
const.
A
physical
interpretation
of de Sitter’s solution is
easily
obtained
on
the
basis
of
de
Sitter’s
own
considerations.
Namely,
he
finds
that,
through
a
choice of
variables
such
that the
guv's
become
independent
of
t,
ds2
can
be
brought
into
the
form[14]
ds2
=
-dr2
-
R2 sin2
-j-
(dip2
-I-
sin2
ipdi)2)
+
cos2
-j-
c2dt2.
R
v
'
R