DOC.
21
GENERAL RELATIVITY
101
(ik,
lm)
=
&8im
+
yg«
_
yg,7
_
ygmt
1
dxkdx¡
dx¡dxm
dxkdxm
dxldx¡
+
£gp1
pT
im/kl
p

iol
p
fon
a
(10)
(11)
It is in the nature
of
gravitation
that
we are
most interested in tensors
of
rank
two,
which
can
be formed
by
inner
multiplication
of
this tensor
of
rank four with the
gmv.
Due to the
symmetry properties
of the
Riemannian
tensor,
apparent
from
(10), viz.,
(ik,
lm)
=
(lm,
ik)
(ik, lm)
=
(ki,
lm),
this
multiplication can
be formed
only
in
one way; whereby one
obtains the tensor
Gim
=
£gkl(ik,
lm).
(12)
kl
It is
more advantageous
for
our purposes
to derive this tensor from
a
different form
of
(10)
which
Christoffel
has
given,2
i.e.,
Ja
lim
aU
CT
{ik,
lm}
=
£
8kp(ip,
lm) =
^^
p
dxm
dxl
p
When this tensor is
multiplied
(inner
multiplication)
with the tensor
8‘k
=
Egfaig“'
(im
I
Jp.
(13)
one
obtains
Gim,
viz.,
Gim
= {il,
lm)
=
Rim
+
Sim
Rim =
"
liml
d\l\
dx,
p
:
r
(13)
[p.
782]
{1}
(13a)
[4]
2A
simple
proof
of the tensorial character
of
this
expression can
be
found
on
page
1053
of
my repeatedly
quoted paper.