54
DOC. 9
FORMAL
FOUNDATION OF RELATIVITY
[p.
1055]
Pxx Pxy Pxz
iix
Pyx Pyy Pyz
iiy
Pzx Pzy Pzz
iiz
iSx iSy iSz
-n
is
a
symmetric
tensor
(Tav)
of
rank
two
(energy
tensor);
fx,
fy,
fz,
iw
is
a
four-vector
(Ka).
Of
course,
this is
true
for both
only
under linear
orthogonal
substitutions,
which
are
the
only
ones
admissible in the
original theory
of
relativity.
From
a
formal
point
of
view,
(42) says
that
(Ka)
is the
divergence
of the
energy
tensor
Tav.
Physically,
the
meanings
are
pxx
etc.,
the "stress
components"
i
the
vector
of
momentum
density
§ the
vector
of
energy
current
rj
the
energy density
f
the force
vector
per
unit volume
externally impressed upon
the
system
w
the
energy per
unit volume and unit time
supplied
to the
system.
The
right-hand
sides of
equations (42)
vanish if the
system
is
a
"complete" one.
Our task is
now
to find the
generally-covariant equations
that
correspond
to
equations
(42).
It is clear
that the
generalized equations,
too,
are formally
character-
ized
by
the fact that
the
divergence
of
a
tensor
of rank
two
is
equated
to
a
four-
vector. However,
with each such
generalization
one
faces the
difficulty
that in
generalized relativity
theory-contrary
to
the
original
one-there
are
tensors
of
different character
(covariant, contravariant, mixed,
and furthermore the class
of
V–
tensors);
one
therefore
always
has
to
make
a
choice. Yet this choice does
not
entail
physical
arbitrariness;
it
merely
affects which variables
are
favored for
representation.4
The choice has to
be
made such that the
equations
are
most
comprehensive,
and that the
quantities
used in them have the best
descriptive
physical
[p.
1056] meaning.
It turns out that this
aspect
is best
met
when the tensor
Tav
is
represented
by
the
mixed tensor
Xva,
and the four-vector
Ka
is
represented by
the covariant
V-
four-vector
S*.
The
divergence
is then to be formed
according
to
(41b), whereupon
one
obtains
as
the
generalization
of
(42)
the
generally-covariant equations
4This is connected to the fact that
any
tensor
can
be
changed
into
one
of
another
character
by
multiplying
it with the fundamental
tensor
or
with
\[~g,
respectively.