224 DOC. 10
RESEARCH
NOTES
Ist
für die
px
ein
System
von
4
Bedingungen,
wenn
dies
stets verschwinden
soll.
Ferner soll Determinante
stets gleich
1
sein.
IX«
=
0.
[eq.
55]
[49]
Ist beides
möglich?
1
+
e
j
l+(e,
+
e2)=l
^ '
dX; dX;
dY
dx
=
-dx+
-
dy
=
2
[eq.
56] 1
+ e.
dx
dy
,,
dY dY
dy
=
^-dx+ ^-dy
dx
dy
djc"Ty
93,
¥
(W
+
X
(•*)
-
2
beide
konstant.
drei Dimensionen[50]
[eq.
59]
=
V*(xy)
[eq.
61]
[eq. 57]
[eq. 58]
[eq.
60]
[48]Einstein considers
yuv,v
under the
transformation
pua
=
dx'../dxa
_
=
8
+
p*a,
where
JXV.V
^IVA. (X
^ |X(X
p^a
are
infinitesimally
small.
nao
=
dxa/dx
o
is
the inverse.
y^v
v
transforms
as a
tensor
provided
[eq.
54]
and
[eq.
55]
are
satisfied,
since
in
the
transformed
expression
^v.v
=
P^ao,
a
+ WVa,
a
"
a
the
second
term
is
zero
because of
[eq. 54]
and the
third
term is
zero as
well,
since
nov
0
=
G
=
-p*v
G
=
-p£c
v,
which
equals
zero
due
to
[eq.
55].
The latter
equation
is
a
consequence
of
det
(p^a)
=
1.
[49]Einstein checks the
compatibility
of
[eq.
54]
and
[eq.
55]
by
trying to construct trans-
formations that
satisfy
them.
He
considers
an
infinitesimal
diagonal
transformation
x'
=
X(x,
y),
y'
=
Y(x,
y),
with
p11 =
1+C1,
p22 =
l+e2,
P12 = P21
=
0, and metric
y^v
=
diag
(-1,
1)
in
a
two-dimensional
space. To first
order
[eq. 55]
is
equivalent to
[eq.
56];
[eq. 54]
is equivalent to
[eq. 57].
The latter
equations yield
[eq. 58]
and
[eq.
59],
for
otherwise undetermined
functions
Y
and
X.
[50]Einstein
considers the
analogous
construction for
a
three-dimensional
space.
[Eq.
61]
and
[eq.
60]
are
compatible
if
Yx1
=
Y1 -
1
etc.
See
also
[p.
5]
for similar calculations.