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-