DOC. 24 RESPONSE TO
QUESTION
BY REIßNER 279
that the
gravitational
mass
of
a
closed static
system
be determined
solely by
the total
energy,
in such
a
way
that the
gravitational
field contributes
to
the total
mass
in
a
manner
analogous
to
the
matter.
Instead,
the
same
must
also be
true
of
the inertial
mass
of
the
system.
One
can
prove
in the
following way
that this
is
clearly
the
case.
[8]
First of
all,
we
note
that
1/-g
(2ov
+ tov)
is
a
mixed
tensor
with
respect
to
linear
transformations
(covariant
with
respect
to
the index
o, contravariant
with
respect
to
the index
v).
Let
21ov
be
an
arbitrary
mixed
tensor
of this kind. Then Ev
Uov/dXv
is
a
covariant
vector
or,
what
amounts to
the
same,
fr/-
^
^
*
But under linear transformation
\-g
changes only by
a
constant
factor.
Hence,
dlgv-g/dXv
is
a
covariant
four-vector,
and the
same
holds for the second
term
of
the
above
expression.
From this
it
follows that the four
quantities
=
E 92lav
sf^g
\f~~g v
also
form
a
covariant four-vector.
On the other
hand,
the
product
of the four-dimensional
space
element dx and
yf-g is
a
scalar. From this
it
follows that
5A-v
is also
a
covariant
vector.
The
same
holds also for the
integral
of this
quantity over
an
arbitrary part
of four-dimensional
space.
Now
we assume
the
following
about the 2lav:
1.
All
2lov
may
vanish
for
(positive
and
negative) infinitely large
x1,
x2, x3.
2.
The
sum
Ev
duov/dXv
should differ from
zero
only
in
a
finite
region
of
x4,
but
v
CL*v
should
vanish for
larger
and smaller values of
x4.
Given these
assumptions,
the
integral
/*E
321
av
extended
over
all of
the values of
x1, x2,
x3
and between
two
values
t1
and
t2
of
x4