58 DOC.
9 FORMAL FOUNDATION OF
RELATIVITY
[28]
[29]
mean
its
mass per
(co-moved)
naturally
measured unit volume. When it
is
permissible
to
ignore
surface
forces,
then the scalar
of
density together
with the
components
of
dx
velocity
dxu/ds
characterizes matter
completely
in the
hydrodynamic sense.
ds
The
energy
tensor of
mass flow.
Equations
of
motion.
One
can
form
a
mixed
V–
tensor
dx
dx
a:
=
Pov/=I-^£s, "
ds Y"0" ds
(48)
/
from the scalar
p0
and the contravariant four-vector
(dxu/ds)
One
can
anticipate
that
(Iva)
is the
energy
tensor
of
ponderable mass
flow,
and
that
equations (42a),
in combination with
(48), correspond
to the Eulerian flow
equations
of
incoherent
masses, i.e.,
when surface forces
can
be
ignored.
We
prove
this
by deriving
from these
equations
those which
we
have
previously given
for
the
motion of
a
material
point.
Let the extension of the
masses
under consideration in
x1, x2,
x3
be infinitesi-
mally
small. We
integrate
(42a)
in these variables
over
the entire "thread
of
flow"
and
use
the abbreviation
dx1dx2dx3 =
dv. We then
get
d
dx
{/*}
E
{rj"/s}
-
/«.*-•
TV
(50)
Substituting
for
Iva
the
expression
from
(48)
and
considering
that
according
to
(47a)
dv
=
dv0
ds
sFg
dc*
(47b)
and furthermore that
m
-
J
p0dv0,
(49)
[p. 1060]
one
arrives at the
equation:
d
dx
mL8,afi
ds
=
£
vr
f dx
pT
dxA
\a
"
dx
T/i.
ds
or, using as
an
abbreviation the covariant four-vector
i-»E
s*'
ds
f
®"dv.
(50a)
(51)