DOCUMENT 581 JULY 1918 827
Ihre
w.
Karte habe ich
gleich am
Montag
Hrn.
Hum[12]
einhändigen
können. Ich
lese
jetzt
mit
grösstem
Interesse
Weyl.[13]
Mit besten
Wünschen für
angenehmen
Landaufenthalt[14] Ihr
ergebener
Klein.
ALS.
[14 410].
There
are perforations
for
a
loose-leaf
binder at the left
margin
of
the document.
[1]Klein
gave a
lecture
on
Einstein
1918g
(Vol. 7,
Doc.
9)
to the Mathematical
Society
of
Göttingen
on Thursday,
4
July
(see
Jahresbericht der
Deutschen
Mathematiker-Vereinigung
27
(1919),
part
2,
p. 45).
[2]Einstein
provided
this
proof
in Doc. 561. In the
margins (see
Doc.
561,
notes 6 and
7)
and in
Doc.
566,
Klein noted
some problems
with
it, and,
in
a
postscript
to Doc.
566,
announced that he had
found
a way
to
solve them.
[3]This
same assumption
is
implicit
in the notion
of
an
isolated
system
used in Einstein
1918g
(Vol.
7,
Doc.
9) as
well
as
in Docs. 556 and 561. In
Klein,
F.
1918b,
p.
400,
the author introduces and
attributes
to
Einstein
1918g
(Vol.
7,
Doc.
9)
the notion that
a
closed
system
is
one
"which,
in
a
manner
of
speaking,
‘swims’ in
a
Minkowski
space-time,
i.e.,
a system
whose individual
particles
traverse
a
world
canal,
outside
of
which
a
ds2
of
vanishing
Riemannian curvature
prevails" ("welches
sozusagen
in
einer
Minkowskischen
Welt
"schwimmt",
d.
h. ein
System,
dessen Einzelteilchen
eine
Weltröhre
durchlaufen, außerhalb
deren ein ds2
von
verschwindendem
Riemannschen
Krümmungs-
maß
herrscht").
[4]For
the definition
of
the
quantities
Uvc
(or
Uvc)
and Ja,
see
Doc.
554,
notes
3
and 4.
[5]For
the
proof
that
A4c =
Ax4Ja, see
Doc. 561. For the
proof
that the
remaining components
vanish,
see
Doc.
556,
note 9.
Originally,
this had not
been clear
to
Klein
(see
Doc.
561, note 6),
per-
haps
because he
questioned a
crucial
assumption
needed
for
this
proof, namely
that there
are
isolated
systems
of
finite
spatial
extension
(see
Doc.
554).
[6]The
integral
jUvad4x
over some
fixed
space-time region
has the
same
transformation behavior
under Lorentz transformations
as
Uva/-g.
The
italicized
claim
(in
which
Uva
should be read
as
Uva/J-g)
follows
from
this observation and from the fact that in the limit considered
by
Klein the
cylindrical regions
used
to
define the
integrals Ava
and
Ava
are
the
same.
[7]The proof is
the
same
as
the
proof
that
A4a
=
Ax4Jc
given
in Doc.
561, except
that the
cylinder
bounded
by
x4 =
C and
x4
=
C'
has
to
be
replaced by
the
cylinder
bounded
by
x4 =
C and
x4 =
C'.
[8]This
can
be verified
by going through
the
same
calculation that Einstein
gave
in the derivation
of
eq. (3)
in Doc. 561. At this
point
in the
original
text,
Klein
appends a
footnote: "zunächst
für
Lorentztransformationen,
bez. solche
lineare
Koordinatentransformationen,
bei denen die
Ebenen
x4
=
C den
Zylinder
in
"Querschnitten" schneiden,
die
JG
also durch
Integrale
definiert werden
können. Für andere lineare Koordinatentransformationen wird
man
die Werte
der
JG
umgekehrt
durch den Vektorcharakter
der
Jc
definieren!"
[9]A simpler
and
more perspicuous
proof
of
the claim that
J0
transforms
as a
four-vector under
linear transformations is
given
in
Klein, F.
1918b. On
p.
404,
the author writes that "in extensive
cor-
respondence
with Einstein"
("[i]n
längerer
Korrespondenz
mit Einstein") he failed
to
see
that
J0
=
JU4ad3
x
transforms
as a
vector (see
Doc.
554),
until
he realized that
one can
define
such
three-
dimensional
spatial
integrations
in
four-dimensional
space-time
in
a
manifestly
covariant
way
as
Ia
= JeaK^uUaa
d'xK
d''xk
d'''x'A,
where
d'x^, d''x^,
and
d'''x^
are
unit
vectors
in the
directions
of
the
spatial axes
of
some
fixed coordinate
system,
and
where
eaK?l|J
is the
fully antisymmetric
Levi-
Civita
tensor (in
Klein’s
paper,
eaK^^
is not used
and the
integrand
is written
as
a
determinant in-
stead).
The
integrand can
also be written
as
Uv9nv,
where
nv
is
a
unit
vector perpendicular to
the
hypersurface
of
integration.
Vectors such
as I0 are
called
"free"
("freie") vectors to
distinguish
them
from
regular
"bound"
("gebundene") vectors,
which
are
located in
specific space-time points
(Klein,
F.
1918b,
pp.
398-399). Given
this
manifestly
covariant
definition
of
integrals over spatial hyper-
planes
and its natural
extension to
arbitrary
smooth
hypersurfaces,
Gauss’s theorem
can
be used
to
show
that
Ja transforms
as a
four-vector
under linear
transformations
(see
Klein, F. 1918b,
pp.
401-