314 DOC. 71 PRINCETON LECTURES
SPECIAL
RELATIVITY
-
T\\
-
a
dxidxi
*
df~
P*“=a
~
P-
In the absence
of
any
force,
we
have
dT"
du" d(mi,) dp
-"“'dï,
+
u~&r
+d^"
~
°-
[65]
If
we
multiply
this
equation
by
uv
(=dxu/dT)
and
sum
for
the
u’s
we
obtain,
using
(40).
(52)
d(ru,)
,dp_n
dx,
dr
where
we
have
put
dpdxu/dxudT=dp/dT.
This
is
the
equation
of
continuity,
which
differs from
that of
classical
mechanics
by
the
term
sp/dT,
which, practically, is vanishingly
small.
Observing
(52),
the conservation
principles
take the
form
(53)
du»
~dr
+
a,
j-h
dp dp
dr
dx»
=
0.
The
equations
for
the
first
three
indices
evidently
corre-
spond to
the Eulerian
equations.
That the
equations
(52)
and
(53)
correspond, to
a
first
approximation, to
the
hydrodynamical equations
of classical
mechanics,
is
a
[66]
further confirmation of
the
generalized energy principle.
The
density
of
matter
(or
of
energy)
has
tensor
character
(specifically,
it constitutes
a
symmetrical
tensor).
[54]