256 DOC. 10 RESEARCH NOTES
[p. 46]
[117]From
the
quantity
[eq.
169],
which
is
a
tensor
under unimodular
transformations, Ein-
stein
generates
another
tensor,
[eq.
183],
under
an even more
restricted
group
of
transforma-
tions.
It has
the form of condition
(1)
of
the
editorial
note,
"Einstein's Research Notes
on a
Generalized
Theory
of
Relativity,"
without
the
need
to
invoke
the
condition
y
=
0.
In its
place, he
further restricts transformations
by
the
condition that
«
transforms
as a
tensor.
This
quantity
also
plays
an
important
role
on [pp.
7-9].
[118][Eq.
178]
is
[eq.
169]
with the
Christoffel
symbols expressed
in
terms
of
by
means
of
the
identities
[eq.
175]
and
[eq.
177].
[Eq.
178] expands to
[eq. 180]
where
a^Kß^/Ka^/A.ß"
Since both
terms
on
the
right
are
tensors
under unimodular
transformations for which
^
is
a
tensor, the
left-hand
side is
a
tensor
as
well.
[119][Eq.
181],
the covariant
divergence 7™$
is
also
a
tensor
under the restricted
group,
as
is
[eq.
182],
which
is
an
expansion
of
[eq.
181]
with the
discarding
of
terms
in
tensorial under
the
restricted
group.
[Eq. 182]
can
be
further reduced
by
discarding
the final
term,
which
is
equal to
ypßd./p3logG/3.Xß
and
therefore
a
tensor
under
the
restricted
group.
Substraction of
[eq. 182]
from
[eq. 180]
gives the final
result
[eq. 183].
In
expanding
[eq.
178]
to
[eq. 180]
Einstein should have included
y
within
the
scope
of the
operator
in
the
first
two terms
of
[eq. 180],
as
is suggested
by
Einstein's
marking
of these
terms.
This
correction
requires
a
similar correction
to
[eq. 183],
which then does
turn
out to
have tensori-
al
properties
under
the
restricted
group. (An
additional
term
y
ft
can
be discarded since
it
proves to
be
tensorial under
the
restricted
group.)
Y,k
fdgj
K
dgKf
+ +
V
dxi
dXi dXK
/
It
dgu
iK
a*K
I
dxK
^
dx
*gi
kY a
[120]
v
dx,
)
Kixdxx
(VkKSiJ
[eq. 184]
Zurück transformiert
PipPkcPix
PipPkoP^x^ [eq.
185]
Ausführlich
^+PlJ^g"+pj£?gfK
[eq. 186]
x V"X V"X
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