DOC. 14
EINSTEIN
AND
BESSO MANUSCRIPT 437
[p. 38]
(Einstein)
[174]Page
number
in
Besso's hand
(see
note
166).
[175]On
[p.
38],
the
effect of
a
rectilinear acceleration rather than
a
rotation of
a
spherical
shell
on a
test particle
at its center is
studied. Similar calculations
in the context
of Einstein's
1912
theory
can
be
found
in
Einstein
1912e (Doc.
7).
The result of
the
calculation
on [p. 38]
is-in
accordance
with the
requirements
of
the
relativity
of inertia
(see
Einstein
1912e
[Doc.
7], p. 40;
Einstein 1913c
[Doc. 17],
p.
1261)-that
the
induced acceleration of
the
test particle
is proportional
to the acceleration of
the shell.
The calculation
proceeds
as
follows. Consider
the
equation
of motion dJx/dt
=
sx
(see
[p.
8],
[eqs.
48-49];
Einstein and Grossmann
1913
[Doc. 13], p. 7, eqs.
5-8;
Einstein 1913c
[Doc. 17],
p.
1260,
eq.
1b")
for
a
point
mass m
at
rest at
the
center
of
the shell.
Since
g44
=
0
and dxi/dt
=
0,
sx
=
0
([eq.
235]).
Hence, the
equation
of
motion reduces
to [eq.
236]. From
[eq.
236] it
follows that
x
=
dg14/dt,
the first
part
of
[eq. 238] ([eq. 237]
should
be
differentiated
with
respect to
t).
When
the
expression
for
g14(x)
given
in
note 169
is
used to
evaluate
the field
produced at
the center
of
the shell
when
the shell
undergoes
a
rectilinear
acceleration X,
one
finds
g14(x
=
0)
=
k/4tz(MX/R).
This
is
the
expression
that
is
inserted for
gl4
in
[eq. 238] (x is
changed into X in
[eq.
239]).
It
follows
that the acceleration
x
of
the test
particle
is
proportional to the
acceleration
X
of
the shell.
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