158
DOC. 30 FOUNDATION OF GENERAL RELATIVITY
we
also call
a
contravariant four-vector. From
(5a)
it
follows
at
once
that
the
sums
Ao
+
Bo
are
also
components
of
a
four-vector,
if
Ao
and
Bo
are
such.
Corresponding
rela-
tions hold for all
"tensors" subsequently
to
be
introduced.
(Rule
for
the addition and subtraction
of
tensors.)
Covariant Four-vectors.-We
call
four
quantities
Av
the
components
of
a
covariant
four-vector,
if for
any arbitrary
choice
of
the contravariant
four-vector
Bv
ZAvBv
=
Invariant
. .
.
(6)
The
law of
transformation
of
a
covariant
four-vector
follows
from
this
definition.
For
if
we replace
Bv
on
the
right-hand
side of
the
equation
ZA'oB,o
=
ZAvBv
by
the
expression
resulting
from
the inversion
of
(5a),
ZdxvB'o,
we
obtain
Av
=
ZB'OA'O.
o
V
DX
O
O
Since
this
equation
is true for
arbitrary
values of
the
B,O,
it
follows
that the
law of transformation
is
A'O
=
AV
. . . .
(7)
v
dxv
o
Note
on a
Simplified Way
of
Writing
the Expressions.-
A
glance
at
the
equations
of
this
paragraph
shows
that there
is
always
a
summation with
respect
to the indices which
occur
twice under
a
sign
of
summation
(e.g.
the index
v
in
(5)),
and
only
with
respect
to
indices which
occur
twice.
It
is
therefore
possible,
without
loss
of clearness,
to
omit the
sign
of
summation. In its
place
we
introduce the convention:-
If
an
index
occurs
twice
in
one
term of
an
expression,
it
is
always
to
be
summed unless the
contrary
is
expressly
stated.
The
difference
between covariant and contravariant
four-
vectors lies
in the
law
of
transformation
((7) or (5)
respectively).
Both
forms
are
tensors in the
sense
of
the
general
remark
above.
Therein
lies
their
importance. Following
Ricci
and
[12]
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