DOC. 9 FORMAL FOUNDATION OF RELATIVITY
41
Note. That the
dXa
correspond
to the
customary
coordinates in the
original
theory
of
relativity
is
seen
from
(12).
Three of these coordinates
are
real-valued,
and
one
is
imaginary (e.g.,
dX4). Consequently,
dT0
is
imaginary.
On the other
hand,
the
determinant
g
with real-valued time coordinates in the
original theory
of
relativity
is
negative,
since the
guv
(under
suitable choice
of
the time
unit)
get
the
values
-1
0
0 0
0
-1
0
0
0
0 -1
0
0 0
0
1
(18)
/g
is therefore also
imaginary.
This is
quite generally
the
case,
as we
will show in
§17.
In order
to
avoid
imaginaries, we put
[p. 1042]
dr0
=
-
i
JfdX1dX2dX3dX4
and instead
of
(17) we
write
\f~gdr
=
dr0.
(17a)
The
anti-symmetric fundamental
tensor
of
RICCI
and
Levi-Civita.
We claim that
^iklm = ^S^iklm
(19)
is
a
covariant tensor.
8iklm
denotes
+1
or
-1
depending upon
1234 results from
iklm
by an even or
odd
permutation
of
the
indices.
In order to
prove
this,
we
note
first that the determinant
£
S[klmdx^dx^dxPdx^
=
V
(20)
iklm
is-aside
from
an
irrelevant factor-equal
to
the
volume
of
an
elementary
pentahedron
whose
corners
are
represented by one point
of the continuum and the
four
end
points
of
the
arbitrary
line elements
(dxi(1)),
(dxk(2)),
(dxl(3)),
(dxm(4))
extending
from this
point. According
to
(19)
and
(20),
one
has
£
G^dx^dx^dx^dx^
=
fgV.
Since from
(17)
the
right-hand
side is
a
scalar,
(Giklm)
is
a
covariant
tensor,
precisely
because
Siklm
is defined
as an
anti-symmetric
covariant
tensor.
From the
tensor
Giklm
one easily
forms
a
contravariant
tensor
by
mixed
multiplication according
to
the scheme
£
"
=
GikIm.
(21)
aß\f±
The contravariant
tensorial character
now
follows
directly
from
§4.
Due
to
(19),
the left-hand side takes the form
[10]