DOC. 30 FOUNDATION
OF GENERAL
RELATIVITY
165
mation
of
the determinant
g
=

guv
.In
accordance
with
(11)
TtXn
^x
9
7X'
Hence,
by
a
double
application
of
the rule
for
the
multipli
cation
of
determinants,
it
follows
that
2
f
()X 'ÖX
9w
I
=
7)x
9
= Verl
'
7)X.
V
or
J9
=
s/g•
dx
a
On
the other
hand,
the
law of
transformation
of
the element
of
volume
dr
=
^dxldx2dx3dx4
is,
in
accordance
with the theorem
of
Jacobi,
dr'
=
dr.
bXu
By multiplication
of
the last two
equations,
we
obtain
Jg'dr'
=
J
gdr
...(18).
Instead
of
g,
we
introduce in what
follows
the
quantity
/

g,
which is
always
real
on
account
of
the
hyperbolic
character
of
the
spacetime
continuum.
The invariant

gdr
is
equal
to
the
magnitude
of
the
fourdimensional
element
of volume
in the
"local" system
of reference,
as
measured
with
rigid
rods and clocks
in the
sense
of
the
special
theory
of
relativity.
Note
on
the
Character
of
the
Spacetime
Continuum.Our
assumption
that the
special
theory
of relativity
can
always
be
applied
to
an
infinitely
small
region,
implies
that
ds2
can
always
be
expressed
in
accordance
with
(1) by
means
of
real
quantities
dX1
...
dx4.
If
we
denote
by dr0
the
"natural"
element
of
volume
dX1,
dx2,
dx3,
dx4,
then
dr0
=
J

gdr
...(18a)