DOC.
14 EINSTEIN AND
BESSO MANUSCRIPT
443
[p. 41]
(Einstein)
[184]The
top
half of
[p.
41],
with the
exception
of
[eq.
267],
is in
Einstein's
hand; [eq. 267]
and
the bottom half of
[p.
41],
starting
with
[eq.
275],
were
later added
by
Besso
(see
the
editorial
note,
"The EinsteinBesso
Manuscript on
the Motion
of
the
Perihelion of
Mercury," sec.
III).
Besso's contribution consists of
two
parts.
In
[eqs.
276280], he
goes
through
the calculation
Einstein started but did
not
finish in
[eqs.
290291]
on [p. 42].
The material written
sideways is
related
to
Besso's calculations
on
the motion of nodes
on [pp.
4549] (see
also
note
189).
[185]On
[pp.
4142],
Einstein checks whether
he
can
use
the "Entwurf"
field
equations
to
derive
the
44component
of the Minkowski metric
in
rotating
coordinates from
its i4components. In
a
coordinate
system rotating
counterclockwise and with
angular velocity
a
around the zaxis
of
some
Lorentz
frame, the
Minkowski metric
is
given by gij
=
diag(1,
1, 1),
g14
=
g41
=
ay,
g24 = g42
=
ax,
and
g44
=
c2(1

a2
p2
/c2) (with
p2
=
x2 +
y2).
For
small
angular
velocities
(making ap
£
1),
this metric field
has the
same general
form
as
the metric
field
of
the
sun
considered earlier
in
the
manuscript:
guv =
guv(0)
+
guv(1)
+
guv(2).
In
the
case
of
the
Minkowski
metric
in
rotating
coordinates,
guv(1)
contains
the
terms
proportional
to
a,
guv(2)
the
term
proportional
to
a2. It would
seem
that,
starting
with
guv(0)
and
guv(1),
one
could
use
the
approximation procedure
used
to find
the metric
field
of the
sun
to
second order
to find
g44(2)
for the Minkowski metric
in
a rotating
coordinate
system
(see
[eqs.
268274]
and
note
186).
The
question
then becomes
whether the
g44
found
in
this
way
is
the
same
as
the
g44(2)
read off
from
the
expression
for
g44
above
obtained
through
direct transformation of the Minkowski metric
to
the
rotating
frame. On the
last
page
of
his
"Scratch Notebook"
(Vol.
3, Appendix A,
[p.
66]),
Einstein
explicitly posed
this
question.
Next
to
expressions
for
g44, g14,
and
g24
for
the
Minkowski metric
in
a
rotating
frame,
he wrote:
"Is
the first
equation
a
consequence
of
the
other
two
on
the
basis
of
the
theory?" ("Ist
die
erste Gleichung Folge
der beiden letzten auf Grund der
Theorie?").
The
conclusion drawn from
the calculation
on [pp.
4142]
is
that
this
question
be
answered
in
the affirmative
(see
Einstein's
comment
"is
correct"
["stimmt"] next to
[eq.
274]).
However,
the calculation contains several
errors
(see note
186).
When these
are
corrected,
one
finds
that
the
approximation procedure
gives
g44(2)
=
3/4a2p2,
whereas direct transformation
gives
g44(2)
=
a2p2.
In
1915,
Einstein
redid the
calculation and discovered the
discrepancy
between the
two expressions
for
g44(2)
(see
Einstein
to
Erwin
Freundlich, 30
September 1915).
The
reason
the
approximation procedure
gives
the
wrong
result
is that the
"Entwurf"
field
equations, contrary to
what Einstein
thought during
the
period
19131915,
do
not
hold
in
the
rotating
frame.
[186]The
purpose
of the calculation
in
[eqs.
268274] is to find
the secondorder
component
g44(2)
for the Minkowski metric
in
a slowly
rotating
coordinate
system
with
the
help
of
the
vacuum
"Entwurf"
field
equations
([eq.
269]; k
should
be k)
and
the
firstorder
components
gi4(1)
=
g4i(1)
and
Yi4(1)
=
y4i(1).
The
expressions
in
[eq.
268]
for
yi4
contain
two
mistakes:
a
factor
1/c2
is
omitted,
and the factor
2
should be
1.
The former mistake
is
corrected
in
the actual
calculations,
the latter
is not.
Spurious
factors 2 also
occur
in
[eq.
267]
and in
[eq.
282]
on [p.
42]
for
gi4.
In
[eqs.
270271]
these
expressions
for
yi4
and
gi4 are
inserted
into all
terms
of
the
44component
of
[eq.
269]
that
are
quadratic
in
yi4
and
gi4
to
obtain
an
equation
for
g44(2).
The calculation
is
similar
to
the calculation
on [pp.
12] (see
the editorial
note,
"The EinsteinBesso
Manuscript on
the
Motion
of
the Perihelion of
Mercury," sec.
II.1a,
eq. (2)). First,
consider
D44(g)
(see
Einstein
and Grossmann
1913
[Doc. 13], p. 16, eq. 16). D44(g)
consists of
two
terms. The first
term
only
gives
 Ag44(2).
The
second term
gives
a
(,) (,)
E(0)
(O)
dg4x
dg4pa
YaßYrp
«
0
aßrp
dX"
3Xß
The
only
contributions
to
this
sum
come
from the
terms
with
(a
=
ß
= 1,
r
=
p
=
2)
and
(a
=
ß
=
2,
r
=
p
= 1),
giving
a2
each.
So,
D44
gives
 Ag44(2) 
2a2. Due the
extra
factors
2