36 DOC. 9 FORMAL FOUNDATION OF RELATIVITY
covariant
(symmetric) tensor
of
rank two. We shall call it the "covariant fundamental
tensor."
Note. We could have defined the covariant tensor
alternatively as
the total
of
sixteen
quantities Auv
which transform
exactly
like the sixteen
products
AuBv
of
two
covariant vectors
(Au)
and
(Bv).
If
one
sets
At* =
(6)
one
immediately
obtains from
(3a)
dx0
_
"
dx"
dx
A'
-
A' B'
-
V
g ßA B'
=
V
a
- '
-
&
%
a*;
a*;
and with
regard
to
(5a)
it follows that
Auv
is
a
covariant tensor.
Analogous reasoning
applies
to tensors
of
higher
ranks.
However,
not
every
covariant tensor
can
be
represented
in this
form,
because
(Auv)
has sixteen
components,
but
Au
and
Bv
[p. 1037]
together
have
only eight components;
therefore,
there
are-based
upon
(6)-algebraic
relations
between the
Auv
which tensor
components
in
general
do not
satisfy.
Yet
one
gets
to
any
tensor
by
adding1
several
of
them
of
type
(6),
simply setting
A^
= AßBv
+
CiPD
+
...
(6a)
The
same
is
true,
in
analogy,
for covariant
tensors
of
higher
ranks. This
representa-
tion
of
tensors
formed from four-vectors
proves
useful for
many
theorems. An
analogous
note
applies
to
tensors
of
higher
ranks.
Contravariant
tensors. Just
as
covariant tensors
can
be built from covariant four-
vectors
according
to
(6)
and
(6a),
resp.,
one
can
also form contravariant
tensors
from
contravariant four-vectors
with the
equations
A
MV
=
A
»B v
(7)
and
A^
=
a^Bv
+
CMDV
+
...
(7a)
respectively.
From this
definition,
the transformation law follows
immediately,
according
to
(4), as
A mv'
=
£
*
(8)
aß
°xa
OXß
Contravariant
tensors
of
higher
ranks
are
defined
in
analogy
to those
of
rank
two.
Just
as
above,
the
special case
of
symmetric
tensors
deserves attention.
1It
is obvious that the addition
of
corresponding components
of
a
tensor
again
leads to
components
of
a
tensor,
as
has been shown for tensors of rank
one (four-vectors). (Addition
and subtraction
of
tensors.)