196

DOCUMENT 140

NOVEMBER

1915

Soweit ich Ihre

neue

Abh.

verstehe,

ist die

von

Ihnen

geg.

Lösung

von

der meinen

völlig

verschieden,

zumal die

es

bei mir auch

nothwendig

das

elektrische

Potential

enthalten müssen.[8]

ALS

displayed on

two

postcards.

[13 052].

There is

a

perforation

for

a

loose-leaf

binder at the

left

margin

of

the

cards. The first is addressed “Herrn

Professor

Einstein

Mitglied

der.

Kgl.

Akademie

der

Wissenschaften in

Berlin

Wittelsbacherstr

13,”

and

postmarked

“Göttingen

1

14.11.15. 6-7V[ormit-

tags].”

The author has added:

“Fortsetzung

auf Blatt

I

mit

der

Einladung

auf

Dienstg

6 Uhr

her

zu

kommen. Viele Grusse H.”

Presumably

Hilbert

was referring

to

the second

card,

which is addressed

“Herrn Professor

Einstein

Mitglied

der

Kgl.

Gesellschaft

der Wissenschaften in Berlin Wittelsbacher-

str

13,”

and

postmarked

“Göttingen

1

14.11.15.

6-7V[ormittags].”

[1]Dated by

Hilbert’s announcement

of

the lecture.

[2]Hilbert’s

lecture

on

16

November to the Mathematical

Society

of

Göttingen was

entitled

“Grundgleichungen

der

Physik” (see

Jahresbericht der

Deutschen

Mathematiker-Vereinigung

24

(1915)

part

2, p. 111).

[3]On

20

November,

Hilbert

presented

a

paper

on

a

unified field

theory

of

gravitation

and electro-

magnetism

to

the

Royal Society

in

Göttingen,

which

was published

after extensive revision

on

31

March 1916

(see

Corry

et

al.

1997)

as

Hilbert

1915.

In

the

published paper,

Hilbert

bases

his

theory

on

two

axioms:

(i)

the

field

equations can

be derived

through

a

variational

principle

from

a

“world

function”

(“Weltfunktion”),

that is

a

function

of

the

metric

tensor and its first and second

derivatives

and

of

the

electromagnetic

vector

potential

and its first

derivatives;

and

(ii)

the

“world function” is

invariant under

arbitrary

transformations of the

space-time

coordinates

or

“world

parameters”

(“Welt-

parameter”) as

Hilbert calls them. The first axiom is

adapted

from the

theory

of

matter

presented

in

Mie

1912a, 1912b,

1913. See

Mehra

1973 and

Earman and

Glymour

1978 for

historical

discussions

of

Hilbert’s

theory; see

also Einstein’s harsh

criticism of

Hilbert’s

approach

in Doc. 278.

[4]This

result is

on p.

397

of Hilbert

1915.

Hilbert

derives

it

from

a

theorem that is

a special case

of

what would

later

be known

as

Noether’s theorem.

[5]The

expression

for E

given

here

can

be found

(in a slightly

different

notation)

in

a

set

of

proofs

of Hilbert

1915,

marked “first

proofs

of

my

first note”

(“Erste

Korrektur

meiner

ersten Note”)

and

bearing

a

stamp

with

the

date, 6 December 1915

(see GyGöU,

Cod. Ms. D.

Hilbert

634).

For

a

dis-

cussion

of

these

page proofs, see Corry et al.

1997.

The

quantity

E,

called the

“energy

form”

(“Ener-

gie

Form”),

is

constructed

out

of

the

tensor

JgPg

H,

where

Pg

is

an

operator

(see

Doc.

222,

note

13,

for

its

definition) acting

on

the world function H. In

these

page proofs,

Hilbert

argues,

in

a way

that

is

reminiscent of

Einstein’s “hole

argument”

(see

Doc.

43,

note

2,

for

discussion),

that

the

generally

covariant

Euler-Lagrange

equations

for the world

function

H

cannot

determine the fields

uniquely,

and that in order to avoid violation

of

what he calls the

“causality principle” (“Kausalitätsprinzip”)

four additional

equations are

needed that

are

not

generally

covariant. The

equations

added

by

Hilbert

are

es

=

0.

He then adds

a

third axiom to his

theory,

called the

“space-time

axiom”

(“Axiom von

Raum und

Zeit”).

This

axiom

restricts the covariance

of

the

theory

to

arbitrary

transformations

of

the

“space-time

coordinates”

(“Raum-Zeitkoordinaten”),

defined

as

those world

parameters

in terms

of

which the

“energy

theorem”

(“Energie

Satz”),

which

can

be written

as

es

=

0,

holds. This

way

of

avoiding a

conflict

with

the

causality principle

is similar

to

the

escape

from

the “hole

argument”

through

the introduction

of

adapted

coordinates

(see

Doc.

18,

note

5,

for

a

definition

of

this

notion)

presented

in Einstein 1914o

(Vol.

6,

Doc.

9), offprints

of

which Einstein

had

just

sent Hilbert

(see

the

preceding

document).

None

of

the above

considerations

appear

in the

published

version

of

Hilbert’s

paper. Instead, a generally

covariant

“energy

equation”

(“Energie Gleichung”)

is derived

(see

Hilbert

1915,

p. 402).

In

Hilbert

1917,

a new

definition

of

the

principle

of

causality

is introduced, which

re-

moves

the need to restrict the

general

covariance

of

the

theory

(see

Stachel

1992,

sec.

3,

for discussion

of

this

new definition).

[6]See

Hilbert

1915,

p.

404.