376 DOC.
382
SEPTEMBER
1917
(Here a
brief
parenthetical remark.
An
equation quite
similar to
(7)
is
obtained
from
(18)
Berl.
Ber.[11]
The vector
U
thus
has
the
same
divergence
as
the
vector
of which
the
a-component is
£
90*
dg\
Whether
these two
vectors themselves
are
identical
I
do not
know.)[12]
Now
one
should
assume
that the
gravitational field is stationary.
Then the
four-dimensional
divergence
in
(7)
reduces to
the
three-dimensional
divergence
in
the
spatial part
of
U.
Integrated
over
the
entire
three-dimensional
space,
(7)
gives
furthermore,
on
the
basis
of
Laue’s
theorem,
the total
energy
E and
hence
also
the
mass
of
the
system:[13]
E=
Í
£(£«
+
Odv.
J
a
(8)
This
integral
must be extended
not
only
over
the material
system
but
also
over
the entire
gravitational
field.
E
can
also be described
by
an integral over
the material
system
alone.
First,
we
also have for
E
E
=
J{S*
+
t\)dV
(9)
integrated
over
the
entire three-dimensional
space. However,
because for
a
sta-
tionary field
gua4
equals
zero,
your
last
expression
(20)
in the Berl.
Ber.[14] yields
t\
=
^0*.
(10)
Thus, according
to
(6),
*4
=
^£C
(10a)
and
equation
(9)
multiplied by 2 and,
with
(8)
subtracted
from
it,
gives
the result
E
=
J(2%i-'£ 1°)dV
=
ƒ
(ïJ
-
T¡
-
S|
(11)
Here
the
energy
is
expressed by an
integral
that
only
needs
to
be extended
over
the material
system.
The
volume
integral
in
(11)
can
also be
transformed
into
a
surface
integral
over
a
surface
enclosing
the material
system.
This
is
because
your
equation
(18)