DOC. 9 FORMAL FOUNDATION OF RELATIVITY
61
With
our special
choice of
coordinates,
the
tensor
we
are looking
for is
given by
[p. 1062]
the
components
-p
0 0 0
0
-p
0
0
0 0
-p
0
0 0 0
p0(1 +
P).
With
an
arbitrarily
chosen
system
of
reference,
this
tensor
obviously
transforms
into
aj
-
-pK'R
-
P^a
*
-
o^E(« )
[31]
p0
ds
^
* ds
After
all,
p0,
p,
and
P
are
scalars
by
definition.
If
we use
the abbreviation
Po^(l
+
-
+ P)
= P*,
Po
equations
(42a)
yield
jxv v
-
+
rl
p*g
=
+8o.
(53)'
V
6
^
^ K
6W
Ac
Ac
1
r ^
Ac
Ac
°
K
ds ds
dg"v dx"
dx
2
rrf
d* ds ds
M-v
dx
These four
equations
determine the five unknown functions
p
and since the
ds
latter share the relation
Yr
=
1
^
ds ds
and
p
is
a
known function
of
p,
with
a
known adiabatic
equation
of
state
of
our
fluid.
The
g^
and
Ra are
to
be considered
as
known.
Equations (53) replace
the Eulerian
equations,
inclusive of the
continuity equation.
This
can easily
be
shown
by
specializing
to the
case
of the
original theory
of
relativity, provided
one
introduces
the
negligible
terms
resulting
from
this,
such
as
velocities small
compared
to
the
speed
of
light
and
pressures so
small that
they
have
no
perceptible
influence
upon
inertia.
§11.
The
Electromagnetic Equations
[32]
The deliberations that will lead
us
to the
generally-covariant equations
of
the
electromagnetic processes are completely analogous
to those
we
had to
carry
out