DOC. 14 EINSTEIN AND BESSO
MANUSCRIPT
471
[p. 53]
(Besso)
[243]On
[p.
53],
the
perihelion
motion of
an
orbit
in
the weak field
of
a
static
sun
is
calculated
on
the
basis of
the
Nordström
theory
(for
a
discussion of
the role this
theory played
in
Einstein's
thinking in
the
years
19131914,
see
the
editorial
note,
"Einstein
on
Gravitation
and
Relativity:
The Collaboration
with
Marcel
Grossmann,"
pp.
xxxx,
and
Norton
1992b).
This
one page
discusses
both how the
gravitational
field
of
the
sun
is
found
in the
Nordström
theory
and how
the
perihelion
motion of
an
orbit
in
this field
is
found.
The
strategy
followed
in
the
latter
part
of these
calculations,
[eqs.
364376],
is
the
same as
the
strategy
followed
on [pp.
811, 1415]
that contain similar calculations
in the
context
of
the
"Entwurf"
theory.
This
part
of
[p.
53]
is
similar
in
character
to
the
summary
of
[pp.
8, 9,
14] on [p. 15]
(see notes
72 and
73).
For
an
outline of
the
strategy, see
the
editorial
note,
"The EinsteinBesso
Manuscript
on
the
Motion of
the
Perihelion of
Mercury," secs.
II.1b and II.2b.
The
first
part
of
the calculation,
[eqs.
355363],
relies
heavily
on
the
discussion of
the
Nordström
theory in
a
lecture delivered
by
Einstein
in
Vienna in
September
1913 (Einstein 1913c
[Doc. 17]).
At the
top
of
[p.
53],
there
is
an
explicit
reference
to the
Vienna lecture
("Wiener
Vortrag"),
and the equation
numbers
given on [p.
53]
all
refer
to equation
numbers
in
this
paper.
[244][Eq.
355] is
the
gravitational field equation
for
the
Nordström
theory.
For the
case
of
an
inhomogeneous
mass
distribution, the
source
term, the
trace
of the
energymomentum tensor, is
given by [eq. 356].
The factor
co
is
defined
in
[eq. 357].
For the
static
case,
[eq. 355]
with the
help
of
[eqs.
356357]
reduces
to
[eq. 358],
which
can
be
rewritten
as [eq. 359]. [Eqs.
355357]
can
be
found
in
Einstein 1913c
(Doc. 17),
pp.
12531254.
[245][Eq.
360]
gives
the field 0
of
the
sun at
distances
that
are large
compared to
the
sun's
radius.
[Eq.
360] is
the
solution of
[eq.
359]
for
the
case
of
a
static
mass
distribution
representing
the
sun.
Using
modern
notation,
one can
write
[eq.
359]
more
explicitly
as
A(p(x)
=
C
k c
J
d3x'8(x

x')/3(x')
po(x').
The solution of
this
equation
is
^
f
/
03(*')Po(*')
0(jc)
=
constant

C
k c
I
dx
.
J
4n
\x

jc'
The
constant
is equal to
the value
of
the field
0
at
infinity,
written
as
0^.
Although
this is
not
done
in
the
manuscript,
0oo
can
be
set
equal
to
1.
When
x is
large
compared to
the
radius of
the
sun,
4(x')
in
the
integral
in
the
second
term
can
be
replaced
by
0,
an average
value of 0 inside
the
sun,
and
\x

x'\
can
be
replaced
by
\x\
=
r.
Setting
/
d3x
po(x)
=
M, M
being
the
mass
of
the
sun,
one
then
arrives
at
[eq. 360].
With the help
of
[eqs.
361362], the
relation between
the
constants
in
[eq. 360]
and
the
Newtonian
gravitational constant
K
is
derived
(see
the upbracket
under
the
second
line
of
[eq.
362];
the
factor
1/47T
should
be
part
of
K).
With the
help
of
this
relation,
[eq.
360] turns
into
[eq.
363].
[246][Eq.
364]
gives
the
x
and
ycomponents
of
the
equations
of motion
in
the
Nordström
theory
for
a mass
point
in
a
gravitational
field.
When
terms
of order
q2 are
neglected, it
follows
from
[eq.
364]
and
[eq.
363]
that
the
area
law
holds
(see
[eq. 365];
F
is
twice
the
area
velocity).
[Eq. 366]
gives
the
Hamiltonian for
a
point
mass
in
a
gravitational
field
in
the
Nordström
theory.
The
equations
referred
to
by
the
numbers
"(2)" and "(2a)," which
are
identical
to
[eq. 364]
and
[eq.
366],
can
be
found
in
Einstein
1913c
(Doc. 17),
p.
1252.
[247]In polar
coordinates
r
and
\j/
(0
is already
taken for
the
gravitational
field), [eqs.
365366]
can
be
rewritten
as
[eqs.
367368].
Eliminating
dt
from
these
two
equations,
one
arrives
at
[eq.
369]
for the
relation between
d\jr
and dr for
an
orbit
in the field
of
the
sun
according
to the
Nordström
theory.
The
term in
square
brackets
in
[eq.
369]
should
be
(p2K2M2/c2E2
+
F2/c2.