60 DOC. 9 FORMAL FOUNDATION OF RELATIVITY
t
_
m(\x
~a1
.
\/i
-
q2
(52)
m
^7
if
q represents
the
three-dimensional
vector
of
velocity
and
q
its
magnitude.
This is
in
agreement
with the
theory
if
we
recall that
we
have-with
the definition
(18)-chosen the
"light
second"
as
our
time unit.
[p.
1061]
Assume
that Ra vanishes in
(50b),
that
is,
the outer forces
vanish,
but
not
those
originating
from
the
gravitational
field.
If
then the
equation
is
multiplied
1
dx*
by
-
• --,
one
obtains for the motion
of
a
material
point
in
a gravitational
field,
m
ds
after
a
simple
calculation,
equation
(23a),
which is
equivalent
to
(1).
This confirms
our anticipation
that in
(48)
is indeed the
energy
tensor
of
flowing
matter.
The
energy
tensor of
an
ideal
fluid.
We
now
want
to
complete
(48)
such that
we
obtain the
energy
tensor
of
an
ideal fluid while
allowing
for the surface forces
(pressure)
and the
changes
in
energy
due to
changes
in
density.6
Without
difficulty,
one
can
find the
energy
tensor
at
a point
in the medium for
that
specific
normal
system
whose
d£4-axis
(at
the
point
under
consideration)
coincides
with the element
of the four-dimensional
line
of
flow.
Let
(f)
be the
(naturally measured)
volume
of
such
an
amount
of
substance
that,
when
brought
to
pressure 0,
the volume
$0
has
mass 1.
The
naturally
measured
energy
e
of
this
amount
with volume
f
is
then-considering
only
adiabatic
changes
of
state-
-
fpdcb,
where
p
denotes the
naturally
measured
pressure. According
to
(52),
the
energy
of
the unit
mass
at
rest is
1
when
pressure
vanishes. The
negatively
taken
integral
is
only a
function
of
pressure,
and
we
call it P. The
energy
unit
per
volume follows
after multiplication with
p0
=
Consequently, the energy density is
p0(1
+
P).
6In
saying
this,
we
restrict ourselves to adiabatic
processes
of
flow in
a
fluid with
a
uniformly
adiabatic
equation
of
state.