DOC.
13
GENERALIZED THEORY OF RELATIVITY
185
im
kl il km
+
E
y
pa
pa
\
P
.
a P. a
/
By
means
of covariant
algebraic
and differential
operations
we
obtain the
complete system
of
differential
tensors (thus
also the differential
invariants)
of the
manifold from the Riemann differential
tensor
and the discriminant
tensor
(§3,
formula
38).
The
(ik, lm) are
also called the
Christoffel four-index
symbols of
the
first kind.
In addition
to
these,
of
importance are
also the
four-index
symbols
of the
second kind
(44) {ik, lm}
=
•B.
+
[/ml
jp/
dxm dX[
k
pJU
/
which
are
related
to
the former
in
the
following way:
{ip, lm}
=
^2
YP
or,
when
solved,
(45)
(ik, lm)
= E&fcp
0'P,
lm}.
P
In
general
vector
analysis,
the four-index
symbols
of the second kind take
on
the
meaning
of the
components
of
a
mixed
tensor
that
is
covariant of
third rank and
contravariant of first
rank.14
The
extraordinary importance
of these
conceptions
for the
differential
geometry15
of
a
manifold that
is
given by
its
line element makes it
a
priori probable
that these
general
differential
tensors
may
also be of
importance
for the
problem
of the
differential
equations
of
a gravitational
field.
To
begin
with, it
is,
in fact,
possible
to
specify a
covariant differential
tensor
of
second rank and second order
Gim
that
could
enter
into those
equations, namely,
(46)
Gim
=
E
Yu (ik
Im)
=
E
0*,
km}.
kl
k
It turns
out, however,
that
in
the
special case
of the
infinitely
weak,
static
gravitational
field this
tensor
does
not
reduce
to
the
expression
Ap.
We
must
therefore leave
open
the
question
to
what
extent
the
general theory
of the differential
tensors
associated with
a
gravitational
field
is
connected with the
problem
of the
14This
follows from the first of
equations
45.
15The
identical
vanishing
of the tensor
Riklm
constitutes
a necessary
and sufficient
condition for the differential form's
being
transformable
to
the form
dxi2.
i
[68]
[69]
[71]
[72]
[70]