EINSTEIN-BESSO
ON THE MERCURY PERIHELION
353
a
Taylor
series in
x
(see
[p.
18],
[eq.
124])
and
using
the
spherical symmetry
of
the
problem, one
finds
that
y4i
can
be
written
as (see
[p.
19],
[eq.
129])
yI (it)
=
4^
(oxx)
,
(14)
Cq
R
V
h
where
R
=
|X|, and
where
S
=
|
/
dr
p0r4
(see
[p.
19], [eq. 128],
and
[p. 34],
[eq.
205]),
which
up
to
a
constant is
just
the moment
of inertia of
the
sun
in
ordinary
Newtonian mechanics.
At this
point
a
change
of notation
occurs
in the
manuscript,
which
is
only
mentioned
explicitly
on [p.
20].
The coordinates of
the
points at
large
-*
distance from
the
sun
at
which
the field is
evaluated
are no
longer
denoted
by
X but
by
x,
the coordinates
that
until this
point
referred
to
points
in the
interior of
the
sun.
A
JJ-
(1
(1)
O
(0)
_
. Adding
y4i,
yi4
and,
y44
to
yMV,
one
finds the metric field
y^v
of
a
rotating
sun
to first
order, which
can
then
be
inverted
to
find
g^ to first
order
(see
[p.
19],
[eqs.
131-132]).
2b.
Precession
of
the
perihelion
of
an
orbit
in the
field
of
a
rotating
sun
to
first
order
([pp.
20-24],
[p. 29], [pp. 32-35]).[43]
To find the
perihelion
motion
in the field
of
a
rotating sun,
one
needs
the
Euler-Lagrange
equations
and the
Hamiltonian for
a
unit
point
mass
moving
slowly
in this field.
For
a
metric
field such
as
that of
a
rotating
sun
to first
order,
the
Euler-Lagrange
equations
can
be
written
as (see [p. 20], [eq. 137]; [p. 37], [eq. 227]; [p. 47],
[eq.
319];
Einstein 1913c
(Doc. 17),
p.
1261,
eq. 1d)

x
=
curl
g x x
-
\
grad
g44,
(15)
where
g
=
(g14,
g24, g34) =
(Stc/r3) (3 x
x).[44]
Now, considering
the
special case
of
an
orbit in
the
plane
of rotation of
the
sun
and
choosing
this
plane
to be the
xy-plane
(see
[p.
21]),
the first term
on
the
right-hand
side
of
eq.
(15)
becomes[45]
a
S
o K
curl
g
x x
=
--
(y, -x,
0).
(16)
r3
Since both
curl
g
x
x
and
grad
g44
lie in the
xy-plane,
it
follows
from
eq. (15)
that


x
will
also lie
in
the
xy-plane.
Hence, the
planet's
orbit
will
never
leave the
plane
of
rotation of
the
sun.
From
eqs. (15)
and
(16) one can
derive
an
expression
for
the
area velocity
f
(see
[p.
21], [eq. 143]):
2/
=
xy
-
yx
=
C +
(17)
r
[43]The argument
is
completely analogous to the argument
outlined
in
sec.
1b
above.
[44]In
the
manuscript,
a
Gothic
g is used
for
this
quantity.
[45]In
the
manuscript,
a
factor
k
is
tacitly
absorbed into
S at this
point
([p. 20], [eq. 138]).
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