DOC.
71
PRINCETON LECTURES 327
THE GENERAL THEORY
The Fundamental Tensor.
It
follows from
the invariance
of
ds2
for
an
arbitrary
choice of
the
dxv,
in connexion
with
the condition
of
symmetry
consistent
with
(55),
that
the
guv
are
components
of
a
symmetrical
co-variant
tensor
(Fundamental
Tensor). Let
us
form
the determinant,
g,
of the
guv,
and
also
the
cofactors,
divided
by
g,
corresponding
to
the various
guv.
These
cofactors,
divided
by
g,
will be
denoted
by guv,
and
their co-variant
character
is
not
yet
known.
Then
we
have
(62)
rrrf
=
ôî
=
1
if
a
=
ß
0
if
a
*
ß
If
we
form the
infinitely
small
quantities
(co-variant
vectors)
(63)
=
g
¡tadxa
multiply
by
guß
and
sum over
the
u, we
obtain,
by
the
use
of
(62),
(64)
dx0
=
g^d^.
Since
the ratios
of
the
d£u
are
arbitrary,
and the
dxB
as
well
as
the
dEu
are
components
of
vectors,
it
follows
that
the
guv
are
the
components
of
a
contra-variant tensor*
(contra-variant
fundamental
tensor).
The
tensor
charac-
ter
of
dBa
(mixed
fundamental
tensor) accordingly follows,
*
If
we
multiply
(64)
by
dx'a/dxB',
sum over
the
ß,
and
replace
the
dEu
by
a
transformation
to
the accented
system,
we
obtain
dx'a
dx'f dx'a
aß
dx"
dxß
S
f'
The
statement
made above follows from
this, since, by (64), we must
also
have
dx'a
=
goa'dE'o,
and both
equations must
hold for
every
choice of the
dE'o.
[67]