DOC. 14 EINSTEIN AND
BESSO MANUSCRIPT
451
[p.
43] (Einstein)
[197]The heading
"Eulerscher Fall"
and
the
page
numbers
on [pp.
43-44]
are
in
Besso's hand.
[198]On
[pp.
43-44],
the
leading terms
of
a
metric
of
the
form of
the
Minkowski metric in
a slowly
rotating
coordinate
system
(see [eq.
299])
are
inserted
into
one
of
the
components
of the
vacuum
"Entwurf"
field
equations
(see
notes 199-202),
and the
resulting equation,
[eq.
300], is
studied.
[Eq. 300] is an
equation
for
the
angular velocity
vector
o
of
the
rotation of
the
coordinate
system
as a
function
of
time.
Classically,
the
equations governing
the
time evolution
of
this
vector
are
known
as
Euler's
equations. They
are
discussed
in
Einstein's "Lecture Notes
for
Introductory
Course
on
Mechanics"
(Vol. 3,
Doc.
1,
[pp.
89-98],
where the
components
of
the
angular velocity
vector-or
"Rotationsvektor"
as
it
is
called
in
Einstein's
notes-are
written
as
p, q,
and
r).
It is
unclear what
connection,
if
any,
there
is
between Euler's
equations
and the
heading
"Eulerscher Fall"
on [pp.
43-44].
The vector
o
occurs
earlier
in
the
manuscript
as
the
angular velocity
vector
("Drehungsvektor")
of
the
sun (see
[p.
18], [eq. 123]).
[199][Eq.
293]
is
obtained
by
substituting
a
metric
field
that
has
g11
=
g22 = g33 =
-1,
g44
=
c2,
gi4
=
g4i
1,
and
all
other
components
zero
into
the
14-component
of
(the
left-hand
side
of)
the
vacuum
"Entwurf"
field equations
Auv(y)
-
kUuv =
0 (Einstein
and Grossmann
1913
[Doc. 13], p. 17)
and neglecting all
terms
smaller than of second order
in
gi4.
This metric
field has
the
form
of
the
metric of Minkowski
space-time
in
a slowly
rotating
coordinate
system
with
angular velocity
vector
o,
when the
|o x
x|2
term in
g44
is neglected
(see [eq.
299]).
[Eq.
293]
is
written
over
three
lines,
the
first
containing
five terms
(the
last
one
indicated
by a
dot),
the
second
containing
three,
and the third
containing one.
The
last
three
terms
on
the first line
and
the
terms
on
the second
and
third
lines
are
written
in more compact
form
in
[eq.
294],
which
is
also written
over
three
lines.
The
first two lines
of
[eq.
293]
contain contributions
from
A14(y)
(see
Einstein and Grossmann
1913
[Doc. 13],
p.
16,
eq.
15):
Gg14
and
(dyaß/dxa)(dy14/dxß)
for (aß)
=
(14, 24, 34,
41, 42, 43).
Farther
down
on [p.
43],
it is
discovered that there
is
another
contribution
from
A14(y)
([eq. 301];
see
note 203).
The
third line
gives
the
contribution from
-ktU14 computed in
[eqs.
296-298]
(see note
201).
[200]At
first
sight,
one
would
expect
[eq.
293]
to have
more
contributions
from
A14(y)
than
the
ones
listed
in note 199.
The
term
with
d/dxa
y/-g,
however, only
gives
contributions of
third
order in
gi4.
The
same
is
true
for the
term Eyaßgxp(^Y\T/^xa)(dy4p/dxß)
considered
in
[eq.
295]:
for
r
=
p
=
(1, 2,
3), dy1T/dxa
=
0;
for
r
=
p
=
4, dy4p/dxß
=
0.
[201]Of
the
two
terms in
-k$14
(see
Einstein and Grossmann
1913
[Doc. 13],
p. 15, eq.
13),
only
the
first
one,
considered in
[eqs.
296-298],
gives a
contribution
to
[eq.
293];
the second
one
gives
only
a
contribution of third order
in yi4. Replacing
the summation
over p in
[eq.
296]
by
twice
the
summation
over
i
while
setting
r
=
4 and
using
that
y44
gi4
=
gi4/c2
= yi4
and
Y11
=
-1,
one can
rewrite
[eq. 296]
as
[eq. 298]. (The
intermediate
step, [eq.
297],
contains
a
spurious
factor
2,
and
gi4
is
changed
into
yi4
without
canceling
y44.) [Eq.
298] is
inserted
into
[eq.
293]
(see note
199).
[202][Eq.
300]
is
obtained
by
inserting
[eq. 299]
for
yi4
(for the
most
part
without the factor
c2
that
seems to
have been
added
later)
into
[eq.
293]. Dots indicate
time derivatives. Since
dyi4/dxi
=
0 (no
summation
over
i),
the
part
of
[eq.
293]
summarized
in
[eq.
294] only gives
four
terms
that
are
non-zero.
[203][Eq.
301] gives
another contribution from
A14(y) to
[eq.
293].
[Eq.
299]
for
yi4
(without
the
factor
c2) is
substituted into
this extra
term
(see
[eq.
302]) and the
result
is
added
to
[eq.
300], giving [eq. 303]
("= 0"
is
omitted from
this
equation).
[204]Besso began
writing,
then
deleted, the comment he
added
to
[p. 44] (see
note 211).