184 DOCUMENT 129 OCTOBER 1915
Dies ist
gleichzeitig
die
Bedingung
dafür,
dass
QdV
eine Invariante
bezüglich
li-
nearer
Substitutionen
ist.[7]
Letzterer Umstand wäre
an
sich
gleichgültig.
Aber
er
erleichtert das
Aufsuchen
von
Q.
Es
ergibt
sich nämlich
aus
dieser
Invarianz
un-
mittelbar,
dass eine lineare
homog.
Funktion der
fünf
auf Seite 1075
unten
in
J-g
meiner
Abhandlung angegebenen
Ausdrücke
sein
muss.
Dass
von
mir
gleich
J-8
dem vierten
der dort
angegebenen
Ausdrücke
gesetzt
wurde,
lässt sich dadurch
rechtfertigen,
dass die Theorie
nur
bei
dieser
Wahl die Newton’sche als
Näherung
enthält.
Dass ich
glaubte,
diese Auswahl
auf
die
Gleichung
SAu
stützen
zu
können,
beruhte
auf
Irrtum.[8]
Es
grüsst
Sie
herzlich
Ihr
A. Einstein.
ALS
(NeHR,
Archief
H.
A.
Lorentz). [16 442].
There
are perforations
for
a
loose-leaf
binder at the
head
of
the document.
[1]Lorentz’s
function
Q
represents
the
Lagrangian
for
the
gravitational
field. It
was
introduced in
Lorentz
1915c,
p.
1086
(p.
763 in the
English
translation).
Two and
a
half
weeks
earlier,
Einstein had
mentioned this
paper
and
some
related research
of
his
own
to
Lorentz,
but had made
no
reference
to
the determination
of
Q (see
Doc.
122).
[2]The
paper
is
Einstein
1914o
(Vol.
6,
Doc.
9),
in which
an erroneous
derivation
of
the
explicit
form of
the
Lagrangian
for
the “Entwurf” field
equations was given,
which,
Einstein
claimed,
was
based
purely on
covariance considerations.
[3]The quantity I,
also written
as
J, is defined in
Einstein 1914o
(Vol. 6,
Doc.
9),
eq.
(61),
as
the
integral
of H
J^g
over
some
region
of
space-time.
The
following
two
equations are equivalent
to
eqs.
(76)
and
(76a), respectively,
of
this
paper.
[4]
This
expression,
which
transforms
as
a
tensor
density
under
the coordinate transformations
con-
sidered here,
gives
the
general
form
of
the left-hand side
of
the
gravitational
field
equations
CLV
=
kS
,
where
S"v
is the
energy-momentum density
for matter
(see
Einstein 1914o
[Vol.
6,
Doc.
9], eqs.
(73)-(74); Lorentz
1915c,
eq.
(49)).
[5]For
a
discussion
of
this
new
derivation
of
the
Lagrangian
for
the “Entwurf” field
equations, see
Norton
1984,
pp.
301-302.
[6]The
law
of
energy-momentum
conservation
can
be written
as
+
=
0.
Using
the
field
equations
(as
given
in note 4
above)
to rewrite the second
term,
one
arrives at the conservation
laws
given
here. This
method
for
finding an expression
for
the
gravitational energy-momentum
den-
sity
tAu
is
essentially
the
same as
the
one
used in Einstein
and
Grossmann 1913
(Vol. 4,
Doc.
13),
pp.
15-16.
[7]The
condition is in fact
equivalent
to
the invariance under linear transformations
of Q/ J-g
,
and thus
of
QdV (see
Einstein
1916o
[Vol.
6,
Doc.
41],
p.
1114).
[8]
The relation
S'7;
=
0 is satisfied
no
matter
which
linear combination
of
the five
quantities
at the
foot
of
p.
1075
of Einstein
1914o
(Vol. 6,
Doc.
9)
is chosen
for H
=
Q/ J-g
(see
Norton
1984,
pp.
296-297).