DOC. 14 EINSTEIN AND
BESSO
MANUSCRIPT 469
[p. 52]
(Besso)
[241][p.
52]
is
the
verso
of
[p. 51].
[242]On
[p.
52],
deleted
in its
entirety,
three different
attempts
are
made
to
recover
the relation
k
=
(8n/c2)K
(see
[p.
1],
[eq. 3])
between the
gravitational constant
K in
Newton's
theory
and
its
counterpart
k
in
the "Entwurf"
theory through
the consideration of
(slow
motion
in)
the
weak
field of the
sun
in both theories. Besso
may
have done these calculations in connection
with
[eqs.
361-362]
on [p.
53],
where the relation between
K and
the
gravitational constant
k
in
the Nordström
theory
is
derived.
Only
the third of the
attempts
on [p. 52]
succeeds.
In
the
first
attempt (see
[eqs.
346-347]),
the Newtonian
potential
for
the gravitational field
of the
sun is
set
equal
to
minus
(1)Y44.
[Eqs.
352-353]
give
the Newtonian
potentials
0i
and
(pa
for the interior
and exterior
field
of the
sun
conceived of
as a
sphere
of
uniform
density
p,
radius
R,
and
total
mass
M.
A factor pK
is
omitted
on
the
right-hand
side of
[eq. 352].
In
checking
that
[eq.
352]
is
a
solution of the Poisson
equation,
the
relation
A0
=
d2(j)/dr2
+ (2/r)d(j)/dr
is
used
(see
note
12).
An
expression
for
(1)Y44
can
be
read off from
[eq. 354],
the solution of the "Entwurf"
field
equation
[eq. 346]
for this
case (see [eqs. 2, 4, 6] on [p.
1]).
The result of
comparing
(pa
and
(1)Y44
is
k
=
4jtc2K
([eq.
347]).
In
the second
attempt (see
[eqs.
348-349]),
the Newtonian
force
on a
test particle
of
mass m
in the
gravitational field
of
the
sun
is set
equal
to
the force
kx
=
–1/2m(dg44/dx)/
g44 given
by
the "Entwurf"
theory
(g44 can
be
read off
from
[eqs.
5-6]
on [p. 1]).
This
yields
k
=
(8n/c)K. In
the third
attempt (see
[eqs.
350-351]),
the
equation
of motion for
a
test
particle
in
the
field
of the
sun
in the "Entwurf"
theory
is
compared to
its
Newtonian
counterpart
(the same
calculation
can
be
found
on
[p. 13],
[eqs.
94-95]).
This
finally
gives
the
correct
result.