DOC.
420
DECEMBER
1917
427
_
dg^
f
va
\
ga
\
xßV,
T
-
Qxa
9^'
l
a
J
L
r
9olv
*
]
i
l
Í
a
J
(:Expansion
of
the
fundamental
tensor:)
where
the attached
s
indicates
that the
symbols
for
the
symmetric
tensor
suv
must be formed
(:They
are
defined
below
for the
asymmetric
ones:)
Auo =
-Auv,o

Auv,o
can
be
expressed using
auv
and
{ik/l}
as:
l
A
-
^
va per
^ßis,
(J
-
dx*
üßa
Un
a a
The
combination
appearing
in
the
integral
ƒƒƒ
Auvadxudxvdxa:[5]
da
da,.
daaß
f
=
A
A-A
-)-A
-
i
va
i
iilva
-
Slfiv,r
1-
SiVVíli
-I-
^
f
dx11
f
dxv
is
the
antisymmetric
expansion
of
the
six
tensor
auv.
From
Auv,o
another
tensor is
derived
through multiplication by
guagBv,
which
is
likewise
antisymmetric
in
u,
v:
*~k?+r
dqPv
aa
aa
v
+

P
The reduction
of
this
tensor
according
to
the
first
(or last) upper
index
yields
1

}
+9ßa
\
f

*
d) Asymmetric
triple
index
symbols:
uvo = 1/2
(dgfiv
+
dgt
ßa
a 2V
dxa dxv dxu
gv vg
a
a
=
f
fuva.
A
generalization
of
Riemann’s tensor
(:not
the
only
one:).
Bfiv,
rt
«
d_
VT
d_
va
+
s'

va
¡IT
VT
ga
da
P
dr
P
a
ß
a
ß
Bl/fl,
TT
Bpiy
T(J
=
+B,vß,
=
+B,
vv,tt
B"TfU/.
This tensor
thus
has
36
ra
components.
The 2nd-order terms read:
1/2
(gut/va
+
gva/uT
-gvT/ua -
gua/vT, where
the
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