DOC. 14 EINSTEIN AND
BESSO
MANUSCRIPT 433
[p.
36]
(Einstein)
[166]The page
numbers,
in
Besso's
hand,
on [pp.
36-39]
indicate
that,
in
Besso's
ordering,
these
pages
belonged
between
[p.
50]
and
[p.
51].
[167]The
calculations
on
[pp.
36-38]
are
related
to
Einstein's
ideas
about the
relativity
of
inertia. There
are
calculations for
two
related
effects,
the effect of rotation
([pp.
36-37]) and
the
effect of linear acceleration
([p. 38])
of
a
thin
spherical
mass
shell
on a
test particle at
the
center.
Both effects
are
mentioned
in
Einstein 1913c
(Doc.
17),
sec.
9,
pp.
1260-1262, and in
letters
to
Mach and Lorentz
(Einstein
to
Ernst
Mach,
25
June
1913
[Vol. 5,
Doc.
448];
Einstein
to
H. A.
Lorentz,
14 August 1913
[Vol.
5,
Doc.
467]).
[168][Eq.
215] gives some
components
of
the solution of
the
"Entwurf"
field equations in
first–
order
approximation
for
some
stationary
mass
distribution
(see
[p.
18], [eqs.
119-122];
see
also
the editorial
note,
"The Einstein-Besso
Manuscript on
the
Motion of
the
Perihelion of
Mercury,"
sec.
II.2a,
eqs.
(12)-(13)).
[Eq.
216] is
identical
to
[eq. 3] on [p.
1].
[169]From
the horizontal
line
on
[p.
36] to
the second horizontal line
on
[p.
37], a special case
of the metric
field in
[eq.
215]
is considered,
namely
the field
near
the
center
of
a
spherical
shell
with
negligible
thickness,
radius
R,
uniform
density
p0,
and total
mass
M,
rotating
with
a
constant angular velocity
o
around
the z-axis
of
a
Cartesian coordinate
system
whose
origin
coincides with the
center
of
the shell.
Denoting
the
coordinates of
points
near
the
position
of
the
shell, at
large
distance from
the
origin,
by
X,
and
the coordinates of
points
near
the
origin,
where the
field is
to
be evaluated,
by
x,
one can
write
[eq. 215]
somewhat
more
explicitly
as:
u,(1) K
f
gi4(x)
=
-
d3-,J1vPo(X)Xi
47r
J/
\X-x\
where
|X
-
x|
=
R'. Note that
(1)g44
=
0
in this
case,
which
corresponds to
the
fact that,
according
to
Newtonian
theory,
there
will be
no
gravitational
field
inside
a
rotating
shell. In
[eq.
219],
the
integral
in
the above
expression
for
(1)g14
is expressed in spherical
coordinates
(R, 0,
w).
The
relation between
(X,
Y,
Z) and (R,
0, w)
is
given
in
[eq.
221]. Differentiating
this
equation
with
respect to
time and
setting
w
=
o,
one
arrives
at
[eq.
218] expressing
Xi
in spherical
coordinates.
Since
p0
=
0
except
when
|X|
=
R,
the
integral
reduces
to
an
integral
over
0 and
co.
Note that
the factor
R2 in
the
Jacobian
R2sin0
is
canceled
by
the
factor 1/R2
in p0(R)
=
M/4tR2
(see
[eq.
217]). Note
also that
an
equation
for
(1)g24
similar
to
the
one
for
(1)g14
is
indicated
by
the
expression
for
Y
that
is
written
directly
above
the
expression
for
X
in
[eq.
219].
Since Z
=
0,
it is
immediately
clear that
(1)g34
=
0
(see
[eq. 226] on [p. 37]).