DOC. 14 EINSTEIN AND BESSO MANUSCRIPT
439
[p.
39]
(Einstein)
[176]Page number
in
Besso's hand
(see
note
166).
[177][P.
39]
deals
with
the
perihelion
motion
predicted
by a
special
relativistic
theory
in
which
the
gravitational
field
of the
sun
is
represented
by
a
fourvector,
the
gradient
of the Newtonian
potential
0
=
KM/r.
[Eq.
240]
gives
the
x
and
tcomponents
of the
equation
of motion for
a
test
particle
in this field.
The derivation
on [p.
39] proceeds
in
exactly
the
same way as
the
derivations
in
the
context
of
the
"Entwurf"
theory
on [pp.
815]
and
[pp.
2024] (see the
editorial
note,
"The EinsteinBesso
Manuscript
on
the Motion of
the
Perihelion of
Mercury,"
secs.
II.1b
and
II.2b).
The
perihelion
motion
is
derived from
energy
conservation
([eqs.
242243];
W
is
used
to
abbreviate
yj1

q2/c2
rather than ds/dt
=
yjc2

q2 as one
would
expect
from
its
usage
elsewhere
in
the
manuscript)
and
the
area
law
([eq. 241]
and
[eq. 244];
the righthand side
of
[eq. 241]
should
be
zero).
The result
is
a
perihelion
advance
2/5
the
size
of
the
effect
predicted
by
the "Entwurf"
theory (see
note
179).
For
Mercury,
this
amounts
to
7.2"
per century.
This
value
agrees
with values found
in
other calculations of
Mercury's perihelion
advance
on
the
basis of other
special
relativistic theories of
gravitation.
Poincare arrived
at
a
value
of
7"
per
century (Poincare 1908,
p.
400);
De
Sitter
at
7.15"
(De
Sitter
1911,
p.
405). For
a
discussion of
these
results,
see
Roseveare
1982,
pp.
159162.
[178][Eq.
250], giving
the relation between
dej)
and dr
for
an
orbit
in
the
field
of the
sun,
is
obtained
as
follows. The
starting point
is
the
area
law
squared
([eq.
246]). Writing
q2
in
polar
coordinates,
one
can
rewrite this
equation
as
the
first line
of
[eq. 247]. [Eq.
245],
itself obtained
from
[eqs.
243244],
is
then used
to
eliminate dt. After
some
rearrangement
of
terms (see
[eqs.
248249]),
one
arrives
at
[eq.
250] (in
the
last
term
under
the
square
root
sign,
a
factor
r2 is
omitted).
In
[eqs.
249250], k
should
be
K.
[179]The polynomial
under the
square
root
sign
in
[eq.
250]
can
be
written
in
the
form
r2(r2+
ar

b)
=
r2(r

r1)(r2

r),
where
r1
and
r2 are
the values of
r
at
perihelion and
aphelion,
respectively.
With
the
help
of
[eq.
251],
derived
on [p.
10]
(see note 48),
[eq.
250] can
then
be
integrated to
find
the
angle
between
perihelion
and
aphelion
(see
[eq.
252]). It
follows that the
special
relativistic
theory
considered
on [p. 39]
predicts
a
perihelion
advance
in
fractions of
it
per
half
a
revolution of
1/2K2M2/c2C2.
Comparison with
[eq.
253]
for the
perihelion
advance
predicted
by
the "Entwurf"
theory
(see
[p.
40],
[eq.
263];
[p.
26],
[eq.
177]
and
[eq. 179];
C
=
F)
shows that the effect
predicted
by
this
special
relativistic
theory
is
a
factor
2/5
smaller
than
the
effect
predicted
by
the "Entwurf"
theory.