18

DOC.

2

COVARIANCE PROPERTIES

Published

in

Zeitschrift

für

Mathematik und

Physik

63

(1914):

215-225. Published

29

May

1914.

[1]Einstein

and

Grossmann 1913

(Vol. 4,

Doc.

13).

[2]Einstein

and

Grossmann 1914a.

[3]See

also

Vol.

4,

the editorial

note,

“Einstein

on

Gravitation and

Relativity:

The Collabo-

ration with Marcel

Grossmann,”

pp.

294-301,

for

a

discussion of Einstein’s

early

work

on

gen-

eral

relativity, including

this

paper.

[4]See

Einstein 1914d

(Vol. 4,

Doc.

26).

[5]In

a

letter

to

Besso of

ca.

10

March

1914

(Vol. 5,

Doc.

514)

Einstein claimed that

the

transformations for which the

gravitational

field

equations

were

covariant included

transfor-

mations

to

rotating systems.

In

the fall of

1915,

however, he realized that this

was

not

the

case

(see

Einstein

to

Erwin

Freundlich, 30

September

1915).

Einstein later listed this realization

as

one

of

the key insights

that led him

to modify

his

theory

of

gravitation

(see

Einstein

to

Arnold

Sommerfeld, 28

November

1915,

Einstein

to

H. A.

Lorentz,

1

January

1916,

and Einstein

to

Willem de

Sitter, 23 January

1917;

see

also the Introduction

to

this volume for

more

historical

background).

[6]The

right-hand

side of this

equation

should be

preceded by

a

minus

sign. "youTuv"

should be

"yuvTuo,"

both here and

in

eq.

(1).

[7]Einstein

and Grossmann 1913

(Vol. 4,

Doc.

13), pp.

23-27.

[8]Summation

over p

is

implied

in

eq.

(I).

These

equations

first

occur

in

Einstein 1913c

(Vol.

4,

Doc.

17).

[9]The

argument

that follows

is

known

as

the "hole

argument."

Earlier versions

are

given

in Einstein 1914d

(Vol.

4,

Doc.

26)

and Einstein 1914e

(Vol.

4,

Doc.

25).

See Norton

1984,

sec.

5,

Stachel

1989,

sec.

3,

and

Vol.

4,

the editorial

note,

“Einstein

on

Gravitation and Rela-

tivity:

The Collaboration with Marcel

Grossmann,”

pp.

297-298,

for

historical discussions.

See also Einstein 1914o

(Doc. 9), §§12-14,

for

a

later version of the

argument

and

a

revised

discussion of the covariance

properties

of the

theory.

[10]"yu"

should be

"yuv."

[11]"x"

should

be

"xv."

[12]See

Einstein 1914d

(Vol. 4,

Doc.

26).

[13]That

equation (III) implied

covariance under linear transformations

was

first discussed

in print in

Einstein

1913c

(Vol. 4,

Doc.

17), p.

1258. See

also the discussions

in

Einstein

to

H. A. Lorentz,

16 August

1913

[Vol. 5,

Doc.

470],

Einstein

to

Paul

Ehrenfest,

before

7

Novem-

ber

1913

[Vol. 5,

Doc.

481],

and Einstein

to

Paul

Ehrenfest,

second half of November

1913

[Vol. 5,

Doc.

484]).

[14]Paul

Bernays

(1888-1977)

was

Privatdozent

at

the

University

of Zurich.

He

was

also

the author

of

a

critique

of

special relativity (Bernays 1913).

[15]"2k" should be

"k."

[16]In

both

terms

one

"gTp"

should

be

a

"yTp."

[17]H

should be 2H.

[18]The

factor

-gyikgmndytn/dxk"

should be differentiated with

respect to

xr.

[19]The

symbol {ik, lm}

denotes the Riemann-Christoffel

tensor Rkiml. See

Einstein

and

Grossmann 1913

(Vol. 4,

Doc.

13),

pp.

35-36.