280 DOC. 24
RESPONSE
TO
QUESTION
BY REIßNER
that include the
value interval indicated in
(2),
has the value
ff
J
21
a\frgdxxdx1dx1
t
=
|
Ia\\
According
to
what has been said
above,
this
integral,
when extended
over a
specified part
of the four-dimensional
space,
is
a
covariant
four-vector.
This
property
is
retained if the domain of
integration
is
extended
to
regions
in which the
integrand
vanishes. From this it follows that the
integral
we
considered last
is
also
a
four–
vector.
The
same
will also hold for the three-dimensional
integral
Ia
=
f
2la4 sTgdV,
as
long
as
for all of the
prospective
reference
systems
the three-dimensional
integration
considered falls within
a
four-dimensional
region,
in which
we
have
everywhere
T
92U
=
0
v
0*v
From this
we
draw the
conclusion,
after
having
substituted the
ten-
sor
1/v-g
(Sov
+ tav)
for
2lov,
that the four
three-dimensional
integrals,
f~~g
Ia
=
f
($04
+
to4) dV,
extended
over a
closed
(complete) system
E,
form
a
covariant four-vector.
It is clear that the three first
integrals,
I1, I2, I3,
represent
the
components
of
the
momentum (with
a
negative sign),
while the last
one
(I4)
represents
the total
energy
of
E.
From
this
it
follows that the inertial
properties
of
a
closed
system (considered
as a
whole)
are
identical with those of
a
material
point
of
arbitrarily
small
mass.
It
only
remains for
us
to
investigate
how the "mass" of
the
system
is related
to
the
indicated
integrals.
If
we
denote the covariant
momentum-energy
vector
of
a
material
point
of
mass
m
with
Io*,
then5
=
___
where
we
set
tic2
=
We have assumed
that the
gravitational
field vanishes
at
infinity, i.e.,
that the
guv are
constant at
infinity.
Hence,
we can
choose the reference
system
in such
a
way that,
5Cf.
the
pamphlet
"Entwurf einer
Verallgemeinerung
der Relativitätstheorie und einer
Theorie
der
Gravitation"["Outline
of
a
Generalized
Theory
of
Relativity
and
of
a
Theory
of
Gravitation,"],
§2
(B. G. Teubner,
1913).