230
DOC.
10
RESEARCH NOTES
G
=
2r2
co
y)
,
2
{(i
-
-
co2r2
cozx
-}o(ÖA*
-
-
cozr
o?
)
+
4
or*
9i
}
+ 4co2j
C02r2
2
co2r2
-
+ cry
+
-
+
co2jc2
+
r
-4coz;t
+
4cozy
=
-1+2,
5co2r
-
co2r
-
3co2jc +
5co2y
[62]
Substitutionen mit Determinante
1.
Infinitesimal
in
2
Variabeln
[63]
dx
=
dx+
(pxxdx
+
pX2dy)
Pn +
P22
~
0
[eq. 86]
dy'
=
dy+
(p2ldx
+
p22dy)
dX dY
=
0
[eq.
87]
Q71
ax
oy
52\(/ 52\|/
3\|/ 3\|/
dx
=
(1
+
^
)
dx
+
---
dy
X
=
Y
=
-
[eq. 88]
dxdy
dy
dy
3x
dx
=
d\\l
P11
=
52\|/
Pn
=
52y
dy
dxdy
a7
8
y
=
-
3\|/
32\|/ 32\|/
dx
P
21
-
dx2
p
22
=
dxdy
[61]The components
of the metric for
a
uniformly rotating
coordinate
system
follow from
[eq.
84]
on [p.
23].
[62]Einstein computes the
determinant
G
of metric
[eq. 85], retaining
only
constants
and
2
2
terms
in
w,
w, x, y, x,
y,2
etc.
[63]Einstein considers
an
infinitesimal transformation
x'
=
x
+
X(x,
y)
,
y'
=
y
+
Y(x, y)
where
p^a
=
dx'n/dxa
=
Öllfy+/?^a
and
pxaa
are
infinitesimally
small.
[Eq.
86]
and
[eq.
87]
(ia
r
are
equivalent
to
det
(pua) =
1
in first-order
quantities. The
transformation
is
generated
from
an
arbitrarily
chosen
function
y in
[eq. 88] to
ensure
the
fulfillment of
the
determinant
condition.
See
[p. 5]
and
[p.
19]
for similar calculations.