DOC. 9 FORMAL FOUNDATION OF RELATIVITY 57
nw
=
(46)
[27]
2
"
3*a
i ;
l±
are
of crucial
importance,
and for this
reason we
want to
call them the
"components
of the
gravitational
field."
§10.
Equations
of Motion of
Continuously
Distributed Masses
Naturally
measured
quantities.
It has
already
been
emphasized
that it is
not
possible
in
a generalized theory
of
relativity
to
select
a
coordinate
system
such that
spatial
and
temporal
coordinate differences
are
in
a
direct connection with the
results obtained
from
measuring
rods and
clocks,
as
has been the
case
in the
original theory
of
relativity.
Such
preferred
choice of coordinates is
only possible
in the
infinitesimally
small
by setting
ds2
=
£
g^dxtidxv =
-dl\ -d\\ -d\\
-dl\.
(46)
{11}
/IV
The
d\
are (see
§2)
just
as
measurable
as
the coordinates in the
original theory
of
relativity,
but
they
are
not
complete
differentials. It is
possible
in the
infinitesimally
small to refer all
quantities upon
the coordinate
system
of the
d\.
When this is
done,
we
call them
"naturally
measured"
quantities.
And
we
call the coordinate
system
the
"normal
system."
According
to
(17a) we
have for
infinitesimally
small four-dimensional volumina
\pg
J
dx1dx2dx3dx4
=
J
d'iyd^2d^id^i. (47)
Let the volume under consideration consist
of
an
infinitesimally
short and
infinitesimally
thin four-dimensional
thread.
And let dv
be the
integral
J
dx1
dx2 dx3
extended
over
it.
We
now
choose the
dE,
such that the
dE4-axis
coincides with the axis of the
thread,
which effects
dE4
=
ds and the
integral
JdE1
dE2 dE3
to
be the
naturally
measured volume
dv0
of the thread
at rest.
According
to
(47) we
then find
sf^gdvdx^
=
dv0ds
(47a)
Unit of
mass.
The
comparison
of
the
masses
of
two
mass points can
be done
by
[p.
1059]
the usual methods. We
merely
need
a
unit
mass
to
measure masses.
Let
it
be defined
as
the
quantity
of
water that fits into
a naturally
measured unit volume
1
that is at
rest.
The
mass
of
a
material
point
is
by
definition
an
invariant under all transforma-
tions.
The
scalar of
density. By
scalar
density
of
continuously
distributed matter
we