DOC.
71
PRINCETON LECTURES 331
THE GENERAL THEORY
which
is
an
invariant,
this
cannot
change
in
a
parallel
displacement.
We
therefore have
0
=
5(£"24M’)
=
A*A'dxa
+
g^A^A’
+
g^A'àA“
or,
by
(67),
(dguv/dxa
-
-
AMA
`dXa
=
0.
Owing to
the
symmetry
of the
expression
in the brackets
with
respect to
the
indices
u
and
v,
this
equation
can
be
valid
for
an
arbitrary
choice
of the
vectors
(Au)
and
dxv
only
when the
expression
in
the brackets
vanishes for all
combinations
of
the
indices.
By
a cyclic
interchange
of
the
indices
u,
v,
a,
we
obtain thus
altogether
three
equations,
from
which
we
obtain,
on
taking
into
account
the
sym-
metrical
property
of
the
Tauv,
(68)
[uva]
=
gabTBuv
in
which,
following Christoffel,
the abbreviation
has been
used,
(69)
-
_L
(~!f
~L
-
LaJ -
I
If
we
multiply
(68) by
gao
and
sum over
the
a,
we
obtain
(70)
=
4~g.7a (~
ag,a
~9g~.\
-
+~;-
-.~:) -
in
which
{uva}
is
the
Christoffel
symbol
of
the
second
kind.
Thus the
quantities T
are
deduced
from
the
guv.
Equa-
tions
(67)
and
(70)
are
the foundation for
the
following
discussion.
[71]
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