166
DOC.
30 FOUNDATION OF GENERAL RELATIVITY
If
\/

9 were
to
vanish at
a
point
of
the
fourdimensional
continuum,
it
would
mean
that at this
point
an
infinitely
small
"natural"
volume would
correspond
to
a
finite volume
in
the
coordinates.
Let
us
assume
that this
is
never
the
case.
Then
g
cannot
change
sign.
We
will
assume
that,
in the
sense
of
the
special
theory
of
relativity,
g always
has
a
finite
negative
value.
This is
a
hypothesis
as
to the
physical
nature
of
the continuum under
consideration, and
at the
same
time
a
convention
as
to
the
choice of coordinates.
But
if

g
is
always
finite and
positive,
it
is
natural
to
settle
the
choice of coordinates
a
posteriori
in
such
a
way
that this
quantity
is
always
equal
to
unity.
We shall
see
later that
by
such
a
restriction
of
the
choice of
coordinates it
is
possible
to
achieve
an
important
simplification
of
the
laws
of
nature.
In
place
of
(18), we
then have
simply
dr'
=
dr, from
which,
in view of
Jacobi's
theorem,
it
follows
that
dxu
=
1
....(19)
Thus,
with this
choice of coordinates,
only
substitutions
for
which the determinant
is
unity
are
permissible.
But it
would be
erroneous
to believe
that this
step
indicates
a
partial
abandonment
of
the
general
postulate
of
relativity.
We
do
not
ask
"What
are
the
laws of
nature
which
are co
variant
in
face of
all
substitutions
for
which the determinant
is unity?"
but
our
question
is
"What
are
the
generally
co
variant
laws of
nature?" It is
not
until
we
have
formulated
these
that
we simplify
their
expression by a
particular
choice
of
the
system
of reference.
The
Formation
of
New Tensors
by
Means
of
the
Funda
mental Tensor.Inner,
outer,
and
mixed
multiplication
of
a
tensor
by
the fundamental
tensor
give
tensors
of
different
character and
rank.
For
example,
Au
=
guoAo
A
= guvAuv.
The
following
forms
may
be
specially
noted:
Auv
=
guagvbAab,
Auv
=
gnagyßkaß