EINSTEIN

IN

COLLABORATION

WITH GROSSMANN

297

III

In

spite

of

his achievements,

Einstein

was

dissatisfied

with the lack of

general

covari-

ance,

which

he

later called "this

ugly

dark

spot,"

and

did

not

have much faith

in

the

new

theory.[15]

In

collaboration with Grossmann

he

continued

to

search for the

most

general

transformations under which the

gravitational

field

equations

of

the

"Entwurf"

theory

are

covariant,

but

by

mid-August

of

1913

he

was

still

unable

to

find

a

single

nonlinear transformation admitted

by

the field

equations.[16]

Around that time he

apparently gave up

these

attempts

because he found

two

argu-

ments

which convinced him that the restriction of

general

covariance

was a

necessity.

The

first

of these

arguments

follows from

a

consideration of conservation laws and

was

found

by

Einstein

on

15 August 1913.[17]

He had

suddenly

realized that the

conservation

law

for

matter

and the

gravitational

field

as

derived

in

Einstein and

Grossmann 1913

(Doc. 13), part I,

eq.

(19)

is

only

covariant under linear transfor-

mations because

it

involves

a

coordinate

divergence

and not

a

covariant

divergence.

The

implicit assumption

made here

is

that the

stress-energy

tensors

for

matter and

for

the

gravitational

field

are

both

generally

covariant. Einstein's conclusion

was

that

his

theory

could

only

be covariant under linear

transformations.[18] As he

later

pointed

out,

this

conclusion

was

incorrect. For

a theory

that

is not

generally

covariant,

the

stress-energy

tensor

need

not be

a

generally

covariant

tensor,

and

the

conservation

laws

do not

have

to be

covariant

only

under linear transformations

(see

Einstein and

Grossmann

1914b).

Einstein's second

argument against

general covariance,

probably

discovered

only

shortly

afterwards,

seems

to

imply

that the metrical

tensor

guv

cannot be

uniquely

determined

by generally

covariant

field

equations.

It has been

called

"the hole

argument."[19]

The "hole

argument"

has been

extensively

discussed

in the

secondary

literature because of

its

significance

for

reconstructing

Einstein's

early understanding

[15]See Einstein

to

H. A. Lorentz,

14 August

1913

(Vol. 5,

Doc. 467);

for Einstein's reference

to

"dieser hässliche dunkle

Fleck,"

see

Einstein to H.

A. Lorentz,

16 August

1913

(Vol. 5,

Doc.

470).

[16]See Einstein

to H. A. Lorentz,

14 August

1913

(Vol. 5,

Doc.

467).

[17]See

Einstein

to H. A. Lorentz,

16 August

1913

(Vol. 5,

Doc.

470).

See also

Einstein

to

Paul

Ehrenfest,

before

7

November

1913 (Vol.

5,

Doc.

481).

[18]This

argument

is

presented

in

detail in the lecture Einstein delivered

to the

meeting

of

the

Gesellschaft Deutscher

Naturforscher

und

Ärzte

in

Vienna

on

23

September

(see

Einstein

1913c

[Doc. 17], §6).

[19]"Lochbetrachtung"

(see

Einstein

to

Michele

Besso,

3 January

1916).

For

historical dis-

cussions of this

argument, see

Norton

1984,

sec.

5;

Norton

1987,

pp.

168-184; Kox

1988,

pp.

69,

72-73;

Stachel

1989, sec. 3;

and

Howard and Norton

1993.

The

argument

is

alluded

to in

Einstein 1913d

(Doc. 15);

Einstein

1914g (Doc. 16);

Einstein

1913c

(Doc. 17),

p.

1257,

fn.

2;

Einstein to

Ludwig Hopf,

2

November

1913

(Vol.

5,

Doc.

480);

and Einstein

to

Paul

Ehrenfest,

second half of November

1913

(Vol.

5,

Doc.

484).

It

was

first

published

in

January

1914,

in

a

supplement to the republication

of the "Entwurf"

paper

in the

Zeitschrift für

Mathematik und

Physik (Einstein

1914d

[Doc.

26]).