DOC. 32
INTEGRATION OF FIELD EQUATIONS
357
Published
in Königlich Preußische
Akademie
der
Wissenschaften (Berlin). Sitzungsberichte
(1916):
688-696. Submitted 22
June
1916, published 29
June 1916.
[1]De
Sitter's
letter
is not
available.
See, however,
Einstein's
reply (Einstein to
Willem
de
Sitter, 22 June
1916),
in
which
he
thanks him for his
comments
and summarizes this
paper
(which
was
submitted the
same
day).
[2]Einstein
1915h
(Doc.
24).
[3]This is
the
first
occurrence
of
the field
equations
in
this
particular
form.
See,
e.g.,
Einstein
1916e
(Doc. 30),
§12,
for
an
earlier
indication;
see
also Einstein 1915i
(Doc. 25),
eq.
(2a).
In
the third
equation
a
minus
sign
is
missing
in
front of the summation
symbol;
"3"
in
the first
term
should
be
"32."
[4]"dy'uv"
in
the
first
term
should be
"d2y'nv."
[5]"z
-
z"
should be
"z
-
z0."
[6]"Yuv" should be
"y'uv."
[7]"3oa"
should
be
"Saa"; a
summation
over
a
is
implied.
[8]As
Einstein later realized,
this
statement is wrong.
In
the
approximation
used
in
the
expression
dgpa/dxu
in
eq. (7a),
g can
be
replaced
by y
but
not
by
y',
as
is
done
here.
As
a con-
sequence,
§§2
and
3
contain
erroneous
results, in
particular
concerning
the existence of
waves
that
do
not
transport
energy
and the radiation
by spherical
symmetric
objects.
Einstein
acknowledged and
corrected his mistake
in 1918 in
a
second
paper
on
gravitational
waves
(Einstein 1918).
[9]The
minus
sign
should be
a
plus sign;
in
eq.
(14)
the second minus
sign
of
the 44-com-
ponent
should
be
a
plus sign
as
well. These
errors are
silently
corrected
in
Einstein
1918.
[10]
In
expressions
b)
and
c) a
factor
2 is
missing
on
the
right-hand side; d), e),
and
f)
give
the
41-component
of
t.
[11]"y2" should
be
"yz."
[12]Tullio
Levi-Civita blamed this
"strange
result"
on
the
use
of the
nontensorial
quantity t
to represent stress-energy
of the
gravitational
field-a feature that
Einstein's
final
version of
general relativity
shared
with all earlier
versions.
See
Levi-Civita 1917 and also the
ensuing
discussion
on
this
topic
in
Einstein
to
Tullio
Levi-Civita, 2 August
1917,
and Tullio Levi-Civ-
ita
to Einstein,
between
2
and
9 August 1917;
see
Cattani
and
De Maria
1993
for
a
historical
discussion, both of the
present paper
and of Einstein's
correspondence
with Levi-Civita.
[13]This point
in
particular
was
emphasized
in
the
letter
to
De Sitter cited
in
note 1.
See also
Einstein
to
Willem
de
Sitter,
15 July
1916,
for
further
comments
by
Einstein.