DOC.
382
SEPTEMBER
1917
377
in the
Berl.
Ber.[15]
shows
that
(I44
-
t44)
can
be
expressed
as a
divergence
of
a
three-dimensional
“quasi-vector”:
($4
+
ú
=
div®)
(where
^^
•
(12)
This
equation
(12)
multiplied by 2
and with
(7)
subtracted
from
it,
taking
(10a)
into
account, yields
T4
-
X]
-
T2
-
T3
=
div(2$
-
21) =
div
C,
(13)
and
by applying
this
and Gauss’s
law, (11)
changes
into
E
=
ƒ €ndf,
(14)
integrated
over
surface
ƒ enclosing
the material
system. Unfortunately, however,
the
quasi-vector
E is not
a
four
vector,
not
even
in
the
sense
of
the
special
theory
of
relativity.
I
have verified
equations
(11)
and
(14)
for the
case
of
a
spherically symmetric
body.[16]
When
I
continued to calculate
t44
in
the
gravitational field
outside of
the
body, however,
I
obtained
a
result
that
does not
agree
with
your
formula
(11)
in
“Approximative
Integration
of
the
Field
Equations.”[17] [When
this had
already
been
written, I
discovered
a
possibility
of
explaining
the
inconsistency.
See
page
5.][18]
I
would
like
to
present
this calculation
now.[19]
First
I
calculate
the
vector
U
for
the
field
outside
the
center.
In
this
field
one
has,
e.g.,
according
to
Droste
(dissertat[ion]
equation
(28)),[20] if
the coordinate
system
is
conveniently chosen,
ds2
-
(l
- -
^
dt2
- -
r2(dd2
+
sin2
ddp2).
(15)
To avoid difficulties
with the
integration
boundaries, etc.,
I
use
Cartesian
co-
ordinates,
however,
and
retaining
Droste’s
designations as
much
as
possible,
I
set[21]
9n
=
~P2
T2
-
|(“2
712
_
XiX2
rp"2.
(U2-P2).,
9u
=
724
=
=
734
=
0,
744
=
w2.)
where
p,
u,
w
are
functions
of
r.
Then[22]
nn
1
x\(
1
"12
-
XIX2
Í1 1
9
-
p2
r2
\u2 p2)
’
9
- _ r2
\u2
p2
9U
=
to
II
to
34
=
0,
to
II
1
w2
’
V=~9
=
uwp2.
etc.
etc.
(16)
•
(16a)