DOC. 14
EINSTEIN AND BESSO MANUSCRIPT
445
[p.
41]
(Einstein) (continued)
in
[eq.
268]
and
[eq.
282],
[eq.
270]
has
-8a2
instead.
Next,
consider
Kt44
(see
[eq.
271],
which
is equivalent
to
Einstein and Grossmann
1913
[Doc. 13], p.
16,
eq. 14). Only
the
second
term
contributes.
It
contributes
for the
four
combinations of
values
for
p, r,
and
a
(= ß)
listed
next to
[eq.
271],
giving
-1/4a2
for
each
combination
(-a2 with the
spurious
factors
2 in
[eq.
268]
and
[eq.
282]). So,
Kt44
gives
-a2
(the
factor
16a2/c2
below
[eq. 271]
should
be
4a2/c2). Adding
the contributions from
D44
and
kt44,
one
arrives
at
-Ag44(2)
-
2a2
-
a2
=
0
(see
[eq. 272];
the
coefficients of
a2
are a
factor
4 too
large
and
the third
term
has the
wrong sign). So,
the
equation
for
g44(2)
becomes
Ag44(2)
=
-3a2
(see
[eq. 273];
4 should
be
3).
With
the
help
of
[eq.
285]
on [p.
42] (see
note
192),
it is
easily
verified that
[eq.
274], giving
the
44-component
of
the
Minkowski
metric in the
rotating frame, is a
solution of
[eq. 273].
This
explains
Einstein's
comment
"is
correct"
("stimmt").
When
[eq. 273]
is corrected, its
solution becomes
g44(2)
=
-
3/4a2p2.
So,
the
extra
factors 2 in
the
expressions
for
yi4
and
gi4
together
with
the
sign error
in
[eq.
272]
combine
to
hide
the fact
that
the
approximation
procedure
does
not
reproduce
the
44-component
found
through
direct transformation
of
the
Minkowski metric
to
a
rotating
frame
(see
note
185).
The
mistake
is
not
discovered
in
the
manuscript,
even
though
on [p. 42] a
result
incompatible
with
the result
on [p.
41]
is
found
(see note
193).
[187]The
calculations in
[eqs.
276-280] may
have been
prompted
by [eqs.
290-291] on
[p. 42].
[Eq. 276]
is
obtained
by
inserting
[eq. 267]
for
the Minkowski metric
in
a
rotating
coordinate
system
into
[eq.
275],
the
x-component
of the
equation
of motion for
a
point particle
in
a
metric
field (see
[p. 8],
[eq.
48];
W
=
ds/dt).
[Eq. 267]
contains
several
errors.
It
should have
g14
=
g41
=
-ay
and
g24
=
g42
=
ax.
[Eqs.
276-280]
inherit the
errors
in
[eq. 267]. [Eq.
276]
is
rewritten
as [eq.
277]. Inserting
[eq. 267]
into
W2
=
(ds/dt)2
=
Eguvxuxv
and
its
time
derivative,
one
arrives
at
[eqs.
278-279].
Inserting
[eqs.
278-279]
into
[eq.
277]
and
setting
x
=
y
=
0
(which amounts
to
the
assumption
that the
rotating
frame
is
the
particle's
instantaneous
rest
frame),
one
arrives at
[eq. 280].
In
the
two
lines
following
[eq.
280],
it is
checked whether the
area
law
holds
in this
case
(see
the editorial
note,
"The Einstein-Besso
Manuscript on
the Motion of the Perihelion of
Mercury,"
sec.
II.1b,
eqs.
(5)-(6)).
[188]The
material written
sideways
at
the bottom of
[p.
41] is
related
to
calculations
on [pp.
45-49]
for the effect
predicted
by
the "Entwurf"
theory
that the rotation of the
sun produces
a
motion of
the
nodes of
planetary
orbits
(see
the editorial
note,
"The Einstein-Besso
Manuscript
on
the Motion of
the
Perihelion of
Mercury,"
sec.
II.3).
The
trigonometric expression
in
the lower
right corner
resembles the
expression
for
S
sin 6 in
[eq.
329]
on [p.
49].
The numbers
-2.3"
and
-0.9"
per century
in
the lower left
corner are
the end results
given on [p.
49]
for the motion
of
the
nodes of
Mercury
and
Venus,
respectively, produced
by
the
rotation of
the
sun.
These
numbers
are a
factor
in
the order of
103 too large
(see
note 231).
Next
to
these numbers
are
values for the
discrepancies
between
(Newtonian)
theory
and
observation for the motion of the
nodes of
Mercury
and
Venus,
based
on
numbers
given
in Newcomb 1895 (there is
a
reference
to
this
book
on [p.
31],
see
note
138).
On
pp.
109-110 of
this book,
Newcomb
gives a
table
of
"discordances,"
as
he calls
them,
between the theoretical and observed secular variations
Dtx
of
four different
quantities
x
for
Mercury,
Venus,
Earth,
and
Mars (this
table
is
reproduced
and discussed
in
Roseveare
1982,
pp.
44-46).
One
of
the quantities he
considers
is sin i Dt0,
where
i
is
the inclination of the orbital
plane
to
the
ecliptic
and
where
0 is
the
longitude
of
the
ascending
node. For the difference between theoretical and
observed
value
of
this
quantity
Newcomb
gives
0.61" ± .52" for
Mercury
and 0.60" ±
.17"
for
Venus
(Newcomb 1895,
p.
109).
For the inclinations of the orbits of
Mercury
and
Venus
he
gives
7°
0'
7.00" and
3°
23'
35.26",
respectively (ibid.,
pp.
181-182). Using
rounded-off values for these
numbers,
one
arrives
at
5.0"
and 10.2"
for the difference
between
theoretical
and
observed value for
Dt6,
the
change
in
the
longitude
of
the
nodes
per century,
for
Mercury
and
Venus, respectively,
which
are
the
numbers
given in
the
manuscript.
Since the rotation of the
sun produces
a
retrogression
of the
nodes rather than
an
advance,
the effect
only
makes the
discrepancy
between theoretical and
observed values
slightly greater.