DOC.
10
RESEARCH
NOTES
229
Drehungsfeld
in
erster Annäherung
[59]
g11dx2
•..•
+
g44dt2
=
ds2
Lagrange'sche
Funktion O-L
=
H
=
ds
dt
2L=
(x-
wrsin(p)2
+
(y
+
wrcoscp)2
+
z
2L
=
x2 +
y2
+
z2
-2corsincoti
+
2corcoscory
+ (02r2
ds
CO
2r2 x2 +
y2
+
z
(D)'i
-co
xy
[eq.
83]
dt
D-L
=
A-
+ (2)corsincpi
-
(2)corcos(py
[60]
dt
berechnet bis und mit
w2&x2
-
A2 +
co2r2sin2cor)
+
co2y2x2
+
w2x2y2
-JlAco2
T-
dt
/.
co2r2
(A-
)
(xL
•2
+
.
f
-2
+1)
.
-2
+
2A(üyx
-
2A(üxy [eq.
84]
[58]See
[p.
21]
for
an
earlier
attempt to
antisymmetrize
a
transformation
to
uniformly
rotat-
ing
coordinates.
[59]Einstein
begins
a
computation
(completed
on
[p.
24])
of
the
components
of the metric
in
Cartesian coordinates
x,
y,
z, t,
which
are
rotating
at constant
angular speed
w
about the
z-axis. If the results that follow hold
locally and
not
just
at
a
point,
the metric
is
Minkowski-
an.
[60]From
the
expression
[eq. 83]
for
ds/dt,
(ds/dt)2
is computed
as [eq.
84],
in
which
2
only constants
and
terms
in
w,
w
,
x, x,
x2
,
etc.,
are
retained
to
recover
the
first
approxima-
tion.
[eq.
85]
2"2\
[61]
*11
+
CO
y
g\2
~
0
*13 =
0
gl4
=
2
A
(ay
V (
w
\
*21 =
0
1
+
(fi?X
823
=
0
g24 =
-2A(ßx
*31
=
0
*32
=
0
*33 =
-
*34 =
0
*41 =
2
A
coy *42 =
~2AC0X
*43 =
0
g44 =
A2-A(02r2
[p. 24]