232
DOC. 10
RESEARCH NOTES
[64]Einstein's
nomenclature
is
nonstandard. The
terms
"Punktvektor"
(point vector)
and
"Ebenenvektor"
(plane
vector)
are
not
used
in the
vector analysis
of
Föppl 1894,
Abraham
1901, Abraham/Föppl 1904,
Minkowski
1908, Sommerfeld 1910a,
1910b,
or
Laue
1911a. An
interpretation
consistent
with the
examples
listed,
with
the
exception
of
[eq.
89],
is
that
"Punktvektor"
designates
a
contravariant
vector
and "Ebenenvektor"
a
covariant
vector; "."
then
designates
a
contravariant index and
"-" a
covariant
index,
so
that
a
"••- tensor"
has
two
contravariant
and
one
covariant index. The
same
prefixes
are
applied to tensors
as
well:
see, e.g.,
[p.
33],
where
the
fully
covariant fourth-rank Riemann
curvature tensor
is
called
"Ebenentensor" (surface tensor) and its
fully
contravariant second-rank contraction
is
"Punkttensor" (point tensor).
[p. 26]
[65]
I
a
r
gmvYaßaxaaxßy
d2y
'JIV
^
-*x
2
^
dxm
y«ß
a*aaxß
[eq. 90]
aß(iv UA|i
V
Dritte
Ableitungen treten
nicht
auf,
wenn
^
dy
UV
=
0 ist.
[66]
n
[65][Eq.
90]
arises from
the
substitution of
YaßYuv,
aß
for the
stress-energy tensor
Guv
in
the
energy-momentum
conservation
law
as
given
in
Einstein and Grossmann
1913 (Doc.
13),
p.
10, equation
(10)
for
the
case -g
=
1.
This substitution
suggests
that Einstein
was
consid-
ering
the
gravitational
field
equation
YaßY^v
aß
=
K®^V'
from which
it
would follow that
[eq. 90]
equals
zero.
[66]Einstein
expands
[eq.
90]
under the
assumption
y^v
^
=
0,
which
ensures
that
no
third-
derivative
terms
in the
metric arise.
mv
1
dg
(IV
\
Y«ßax"axn
d\v
+
..X
v
s.
dxß
2
dxm
J a~"ß
(ivaß
,
^a^ß
a
(X«XvYuv) =
0
IV
V
Xy
=
0
dx
a
f
äYaß
a
3Yaß^
mv
g
Y
dxa
mv
dxa
{
rJxp
dxß
,
V ß
/
3Yaß
dr
dxa