EINSTEIN-BESSO
ON THE MERCURY PERIHELION
345
the basic effect
(i.e.,
the
perihelion
motion
produced
by
the
sun
conceived of
as a
static
mass
distribution)
is
correct,
but
there
is
a
mistake of
a
factor
10
in the
number that
is
inserted for
the
mass
of
the
sun,
which
yields
a
mistake of
a
factor
100 in the final result.
Einstein writes
on [p.
28]:
"1821"
=
30'
independently
checked."[12]
The mistake
is
discovered
in the
following pages
(on [p.
30]
by
Einstein,
on [p. 35] by Besso),
but
there
is
no
clear
and
unambiguous
statement
of
the correct
result
in the
manuscript.
The so-called Scratch
Notebook[13]
contains
an
expression
for
the
perihelion
advance
of
Mercury
which
to
a
good approximation is equivalent to
the
expressions
given
in
this
manuscript.
In the
"Scratch Notebook"
the correct
numbers
are
inserted and
the
end result
is
given as
17".
More
important
than
the
actual numbers
is
Einstein
and
Besso's derivation of
the
expression
for
the
perihelion
motion
predicted
by
the
"Entwurf"
theory.
It turns out
that
the
method used in
1913
is virtually
identical
to the
method Einstein
used in his
November
1915
paper
on Mercury.[14]
This
may
help
to
explain
why
Einstein
was
able
to
write
this
paper
in such
a
short
time.[15]
The remainder of
this
editorial
note
is organized
as
follows.
In
sec.
II
a
brief outline
will be
given
of
the
three main derivations
in
the
manuscript,
with
further details
pro-
vided
in
footnotes
to the
transcription.
These three
derivations, all in the
context
of
the
"Entwurf"
theory, concern
the
motion of
perihelia
in the
metric
field
of
both
a
static
and
a rotating
sun
(see
[pp.
1-30] and
[pp.
32-35],
plus
some
material
on [p.
40]) and the
motion of
nodes
in
the field
of
a rotating
sun
(see
[pp.
45-49],
plus some
material
on [p.
31]
and
[pp.
41-42]).
These three
topics occupy
39
of
the 53
pages
of
the
manuscript.
The
remaining
14
pages
deal
with the
following
topics:
a
plan
for
various corrections
to the
analysis
in Newcomb 1895
on
the basis
of
the
"Entwurf"
theory
([p.
31]);
the
metric
field
inside
a
rotating
shell and the
relativity
of inertia
([pp.
36-38]);[16]
the
perihelion
motion
in
a
special
relativistic
gravitational theory
([p.
39]); an expression
for
the
period
of
a
Newtonian orbit
in terms
of
the
orbiting particle's
total
energy
([p.
40]);
the
"Entwurf"
field
equations
and
Minkowski
space-time
in
a rotating
coordinate
system
([pp.
41-42]);
the
"Entwurf"
field
equations
for
what
is
called
the
"Eulerian
case"
("Eulerscher Fall")
([pp.
43-44]); the
metric
field inside
a
rotating
ring
and the
[12]"...
unabhängig geprüft."
[13]Vol. 3, Appendix A,
[p.
61].
[14]See
Earman and Janssen
1993,
pp.
142-143,
pp.
156-157.
[15]See
David Hilbert
to
Einstein,
19
November
1915:
"...
congratulations
on
conquering
the
perihelion
motion. If
I
could calculate
as
fast
as you,
the
electron
would be
forced
to
surrender
in
the face of
my
equations
and the
hydrogen atom
would
have
to
present
an
excuse
for
the
fact
that
it
does
not
radiate"
("...
herzliche Gratulation
zu
der
Überwältigung
der
Perihelbewegung.
Wenn ich
so
rasch rechnen
könnte, wie
Sie,
müsste
bei
meinen
Gleichungen entsprechend
das
Elektron
kapituliren
und
zugleich
das
Wasserstoffatom seinen
Entschuldigungszettel aufzeigen,
warum es
nicht
strahlt").
[16]On
these
pages
Einstein calculates
the
Machian effects
he
described
in
letters
to
Mach
and
Lorentz of this
period:
the
gravitational
effects
at
the
center
of
a
spherical
mass
shell
when the
shell
either
rotates
uniformly
or
is
accelerated
uniformly
and
rectilinearly
(see
Einstein
to
Ernst
Mach, 25
June
1913
[Vol. 5,
Doc. 448], and
Einstein
to H. A. Lorentz,
14 August 1913
[Vol. 5,
Doc.
467]).