DOC.
14
EINSTEIN
AND
BESSO
MANUSCRIPT
465
[p. 50]
(Besso)
[233]On
[p. 50], an
expression
for the metric
field inside
a
rotating ring
is
derived for the
purpose
of
estimating
the effect of
Jupiter
on
the
precession
of
the
nodes of
Mercury and
other
planets
inside
Jupiter's
orbit.
[234]The
first-order "Entwurf"
field
equations
are
solved
to
find the
deviations from
a
flat
Minkowski metric
representing
the
gravitational
field
inside
a
thin
ring
of diameter
R
lying
in
the
xy-plane, rotating slowly
and
uniformly
around the
z-axis.
The
energy-momentum tensor
for
this
problem
will be
of
the
form
given
in
[eq.
332],
and
the
metric
field
y/lv
will
be
of the form
given
in
[eq.
333],
both for
the
special
case z
=
0.
The reference
to
page
"101"
suggests
that
[eq.
332]
and
the
integral
for the
14-component
in
[eq.
333] were copied
from
[p. 18]
(see
[eq.
119]
and
[eq.
121]).
From the
final
results that
are given
for the
components
of
yuv
in
[eq. 333] one
can
infer that Besso made
a
serious
error
in evaluating the integral
for
y14
for the
case
at
hand.
This also affects the
analogous expression
for
y24.
Probably
because of
the
cryptic
notation
used
in
the
manuscript
for calculations
like this
(see
[pp.
18-19]
and
[pp.
36-37]),
Besso confused
the coordinates of
source
points
and
field points in this
case.
When the coordinates of
points
where the
field
is
to be
evaluated
are
called
X,
and
the coordinates of
points occupied
by
the
rotating ring x,
y14
can
be
written
more
explicitly
as
(see
the editorial
note,
"The Einstein-Besso
Manuscript
on
the Motion of
the
Perihelion of
Mercury,"
sec.
II.2a,
eq.
(13))
*
f
Po(x)x
yl4(X)
=
-
/
d-x-^-
J
An
\X
-
x\
The
velocity
x
of
the
rotating ring
can
be
written
as o x
x,
where
o
is
the
angular velocity
vector.
Since
ox
=
oy
=
z
=
0 in this
case,
one
has
x
=
-(2n/T)y,
where T
is
the
period
of revolution of
the
rotating ring.
The
expression
for
y14
given
in
[eq. 333]
strongly
suggests–
using
the notation introduced
above-that
-(2n/T)Y rather
than
-(2n/T)y
was
substituted
for
x
in
the
integral,
and then
put
in
front of
it. Moreover,
a
factor
2
and
a
minus
sign
got
lost
in evaluating
the
integral. Finally,
it
is
somewhat
puzzling
that R'
(or
\X
-
x\
in the notation
introduced
above) is set equal
to R.
After
all,
the field
of
Jupiter
has
to
be
evaluated
near
the
orbit of
Mercury,
not
near
the
sun.
[235]Inserting
guv
from
[eq. 334],
the covariant form of
the metric
field
in
[eq.
333], into
b
=
rot
g (see
note 97),
one
arrives
at
[eq. 335].
[236]The purpose
of
the
calculations
at
the foot of
[p. 50]
seems
to be to find
the ratio between
the b-field of the
sun
and the
b-field
of
Jupiter
near
the orbit of
Mercury
(see
note 220).
When
one
compares
[eq. 320]
for the
b-field
of
the
sun
to
[eq. 335]
for the
b-field
of
Jupiter,
making
the
simplifying assumption
that
the
orbit of
Jupiter,
the orbit of
Mercury,
and the
plane
of
rotation
of
the
sun
all
coincide and
can
be
chosen
as
the
xy-plane,
one sees
that
the b-fields
of
Jupiter
and
the
sun
in
the
plane
of the orbit of
Mercury point
in
opposite
directions
along
the
z-axis.
Both
fields
furthermore contain
a
factor
k,
which will
drop out
when
taking
their
ratio. Hence, in
order
to
find
the ratio of
the two fields
one
only
has
to
evaluate
S0/a3
for the sun's
field,
and M/2T
R
for
Jupiter's
(the
b-field
of
Jupiter
inside
its
orbit
is
tacitly
assumed
to
be
homogeneous). Using
S0
=
4.4
•
1047
g
cm2/s
and
a
=
5.8
•
1012 cm,
one
obtains 2.2
•
109 g/cm
s
for the sun's
field.
For
Jupiter
one
needs
its
mass
(M
-
2
•
1030
g),
its
period
of revolution
(T
~
4
•
108
s),
and
its
distance from the
sun
(R
~
8
•
1013 cm).
In
the
manuscript
the number
4
•
1013
cm
is
used for
R.
It follows that
bz(rotation
of the
sun)
=
-20bz(Jupiter),
a
result stated
explicitly
on [p. 47]
(see note 220).
For
a
discussion of
how,
with the
help
of
this result,
one
arrives
at
the
estimates
at
the foot of
[p.
50]
for the effect of
Jupiter on
the nodes of
Mercury
and
Mars, see
notes 221
and 222.