DOCUMENT 487 MARCH 1918
689
ALS.
[14 399]. Significant
variations
between
the
text
of
this document and
a
draft
(GyGöU,
Cod.
Ms. Klein
22B,
87-88
[14 399.1])
are
noted.
[1]Doc.
480.
[2]Klein,
F. 1917,
p.
481.
[3]Klein,
F. 1917,
pp.
473-474.
[4]Two
weeks
earlier,
Klein
had
alerted David Hilbert to the
same
mistake,
explaining
that his
con-
fusion stemmed from the fact that the coordinate condition
7-g
=
1
used in Einstein 1916e
(Vol. 6,
Doc.
30)
obscures the distinction between
tensors
and tensor densities
(see
Felix Klein
to
David
Hilbert, 5
March
1918,
GyGöU,
Cod. Ms. Hilbert
179;
published
in Frei
1985,
p.
142).
[5]In
Doc. 480, Einstein took
exception to
the claim in section 9
of
Klein,
F.
1917 that the
equation,
which
according
to Einstein
expressed energy-momentum
conservation,
is
a
mathematical
identity.
[6]Einstein 1916o (Vol.
6,
Doc.
41).
[7]Einstein used Gothic characters
for
these two scalar
densities, out
of
which he constructed the
action
integral:
B
is the
gravitational,
m
the
matter
part.
Klein focused
on
the
special
case,
in which
the
only
matter
is the
electromagnetic
field. The constant
a
is Einstein’s
gravitational
constant.
[8]The
Lie variation
of
the
electromagnetic
four-vector
potential,
6qp =
q'p(x)
-
qp(x)
(see
Doc.
492, note
12,
for further
discussion
of
the
usage
of
Lie
variations).
[9]In
the second
of
the two
equations
below,
"+"
should be
"-."
[10]After
criticism
by
Einstein,
Klein decided not
to
publish
this lecture
(see
Doc.
540).
[11]This
same proposal
of
interpreting
the left-hand side
of
the field
equations as
the
gravitational
energy-momentum
tensor
was
made in
Lorentz
1916d and in Levi-Civita 1917b. For Einstein’s
earlier
arguments
against
this
proposal, see
Doc. 368 and Einstein 1918a
(Vol.
7,
Doc.
1), p.
167. For histor-
ical
discussion,
see
Cattani and
De
Maria
1993.
[12]Schwarzschild
1916b
(see
Doc. 188
for
an
outline
of
this
paper).
[13]In the
equation below, a
should be
v.
[14]Hilbert
1915.
[15]In
other
words,
Klein did
not
use any
matter
terms,
whereas Lorentz did
(see,
in
particular,
Lorentz
1916b,
sec. 1,
and Lorentz
1916d,
secs.
40 and
43).
[16]In
Schwarzschild 1916b,
the
expression
for
the
energy-momentum
tensor
for
an
ideal fluid
given
in Einstein 1914o
(Vol.
6,
Doc.
9),
p.
1062,
is used.
[17]The
expression
below should be:
(p0c2/16)(7X2a
-
5X2).
[18]In
the
draft,
Klein added
two comments.
Referring
to
Lorentz
1916d, sec. 52,
he wrote:
"Höchst
merkwürdig
finde
ich,
dass sich Lorentz
am
Schluss seiner Vorles.
von
1916
befriedigt erklärt,
indem
er
im Anschlusse
an
die
Feldgleichungen
a)
einen
doppelten
Energiestrom postuliert,
dessen beide
Bestandteile
sich
fortgesetzt kompensieren." Referring
to the "agreement"
("Uebereinstimmung")
mentioned
in the
text, he also
added:
"Sie enthält
gerade
die
Uebereinstimmung
der
grav.
Masse
und
der
Trägheitsmasse,
auf die Sie Gewicht
legen."
[19]In
the
equation
below,
T should be
S,
and
"-"
should be "+." The
sign error
is
an error
in
eq.
(22)
in Einstein 1916o
(Vol. 6,
Doc.
41).
The corrected
equation
is
equivalent
to
the
vanishing
of
the
covariant
divergence
of Tuv.
The covariant
divergence
of
-Kuv/a,
the
gravitational energy-
momentum tensor
proposed by
Klein,
also vanishes
on
account
of
the contracted Bianchi identities.
[20]Carl
David Tolmé
Runge
(1856-1927)
was
Professor of
Applied
Mathematics at the
University
of
Göttingen.
After criticism
by
Einstein
(see
Doc.
492),
his
lecture,
"Über den Satz
von
der
Erhal-
tung
der
Energie
in der Gravitationstheorie"
(Königliche Gesellschaft
der
Wissenschaften zu
Göttin-
gen. Gesellschaftliche Mitteilungen (1918),
p.
9), was
not
published (see
Doc.
540).
Klein
explained
the basic idea
of
Runge’s
lecture to Hilbert in the letter mentioned in
note
4
above. Hilbert
thereupon
sent Klein
page proofs
of
Hilbert 1915 to show Klein that he had
already
considered
Runge’s
idea but
had in the end
decided
not to include it in the
paper
(see
David Hilbert
to
Felix
Klein,
7 March
1918,
GyGöU,
Cod. Ms.
22B;
published
in Frei
1985,
p.
144).
See Doc.
140, note 5,
for
a
brief
discussion
of
the role
of
energy-momentum
conservation in these
page
proofs.
[21]Neither
paper was
published (see note 10
and the
preceding note).
[22]Einstein
1917b
(Vol. 6,
Doc.
43).
[23]The
last three sentences
of
this
paragraph
are
not
in the draft.
Instead,
the draft has
a postscript
with
an
outline
of
the
calculations,
which form the basis
for
the remarks in these sentences:
"Ps. "Zur
Kosmologie".