130

DOC.

9

FORMAL

FOUNDATION

OF RELATIVITY

[32]See

Einstein

1916b

(Doc. 27)

for

an

alternative

treatment

of the

electromagnetic equa-

tions

in

which the

use

of dual six-vectors

is avoided.

[33]The

reference

is

to eq. (24)

on p.

1044.

[34]The argument presented

in

§12

is

known

as

the "hole

argument."

Earlier versions

are

given

in Einstein 1914d

(Vol. 4,

Doc.

26),

Einstein 1914e

(Vol. 4,

Doc.

25),

and Einstein and

Grossmann 1914b

(Doc.

2).

See

Norton

1984,

sec.

5,

Stachel

1989,

sec.

3,

and Vol.

4,

the edi-

torial

note,

"Einstein

on

Gravitation and

Relativity:

The Collaboration with Marcel Gross-

mann,"

pp. 297-298,

for historical discussions.

See

also Einstein's retraction of

the

conclusion

drawn

at

the end of the third

paragraph

below

in

Einstein

to

Paul

Ehrenfest, 26

December

1915.

[35]The

limitation

to

linear transformations

is

considered

in

Einstein and Grossmann

1913

(Vol. 4,

Doc.

13);

in

Einstein and Grossmann 1914b

(Doc.

2)

it is

shown that the Einstein-

Grossmann

theory

is

covariant under

a

wider

group

of transformations. The

argument

devel-

oped

in

this section elaborates

on

the latter

paper.

[36]In

the fall of

1915

Einstein realized that it

was a

mistake

to

assume

that H

is

invariant

under linear transformations

and

that the

use

of

eq.

(77)

on p.

1075 to fix

H

was

fallacious

as

well

(see

Einstein

to

H. A. Lorentz,

12

October

1915,

and Einstein

1915f

[Doc.

21],

p.

778;

see

also Einstein 19160

[Doc.

41],

eq.

(13),

from which

it follows that

linearity

of H alone

implies

condition

(77)).

He

later listed this

insight

as one

of the

key

steps

toward

revising

his

theory

of

gravitation (see

Einstein

to

Arnold

Sommerfeld, 28 November

1915,

and Einstein

to

H. A.

Lorentz,

1

January

1916).

See also Norton

1984,

secs. 6

and

7,

for

a

historical discussion.

[37]See

also Einstein's elaboration of the

argument

of the remainder of this section

in

Ein-

stein

to

H. A. Lorentz, 23 January 1915,

which

responds

to objections

raised

by

Lorentz

in

a

letter

to

Einstein,

between

1

and

23 January

1915.

[38]The

minus

sign

on

the

right-hand

side should

be

dropped;

a

summation

over \i

is

implied.

[39]The reasoning

in

§14

leading to

the

proof

of the

tensor

character of

smv on p.

1073

was

later criticized

by

Tullio Levi-Civita in

a

series of letters

to

Einstein.

In

his

first

reply to

Levi-

Civita,

Einstein called

this

proof

"the

most

important

proof

of the

theory,

achieved

at

the

cost

of

streams

of

perspiration"

("den wichtigsten,

mit Strömen

von

Schweiss erkauften Beweis der

Theorie." Einstein

to

Tullio

Levi-Civita,

5

March

1915).

See the

correspondence

between Ein-

stein and Levi-Civita from March

to May

1916 in Vol.

8

for

more

details;

see

also Cattani and

De Maria 1989 for

an

analysis.

In

a

letter

to

David Hilbert

of

30

March

1916,

Einstein hints

at

an error

discovered

by

Hil-

bert

in

this section

as

well and admits that his

assumption

of

commutativity

of the variation

5

and the derivative made

on p.

1073 is

erroneous

(see

also Norton

1984, note 62,

for

an

inter-

pretation

of Einstein's remark

to

Hilbert).

[40]"±"

should

be "V."

[41]"xa'"

in

the second

and

fourth

expressions

should

be

"x0,."

[42]Einstein later realized that

eq. (77) only expresses

the invariance of

H

under linear

trans-

formations and does

not fix

H

at

the form of

eq.

(78),

as

is

claimed here.

See

also

note

36.

[43]The

first

term

on

the

right-hand

side should

be

preceded

by a

minus

sign.

[44]A

minus

sign is missing

between the

two expressions

in

the

parentheses.

[45]The sign

of the second

term

in

the

parentheses

on

the

right-hand

side should

be

minus.

[46]The sign

of the third

term

on

the left-hand side should

be

minus.

[47]This problem

was

also raised

in

a

letter

to

Lorentz of

17 January

1916.

[48]In

1949

Kurt Gödel showed that

general relativity

admits solutions that include

closed

timelike

curves

(see

Gödel

1949).

See,

e.g.,

Geroch and

Horowitz

1979,

pp.

232-243,

for

a

discussion of

causality

in

such

spacetimes.

[49]"da" should

be

"d."

[50]"xa2"

should

be "dx2

."

[51]Both

the result for the redshift and the

one

for the

speed

of

light

in the

next equation

already

occur

in

Einstein's

first paper

on

gravitation,

Einstein

1911h

(Vol. 3,

Doc.

23),

where

they

were

directly

derived from the

equivalence principle.

See

also Einstein

1914k (Doc.

3),

note 4,

for

more on

the

attempts

to measure

these effects.