DOC. 13

GENERALIZED

THEORY

OF RELATIVITY

341

[19]See Einstein 1912c

(Doc.

3), §3,

for

an

earlier discussion of

the

effect of

the

gravitational

field

on

time

measurement;

see

also Einstein's consideration of the effect of rotational motion

on

the

geometry

of

a

rigidly rotating

disk

in the

same

paper

on p.

356.

[20]An outline of the

following

calculations

is

found in Einstein's

research

notes

on a

gen-

eralized

theory

of

relativity (Doc. 10),

[p.

40].

[21]This

formula

appears on

[p.

10],

[p.

40],

and

[p.

51]

of Einstein's research

notes

on a

generalized theory

of

relativity (Doc. 10).

On

the

first two

pages

listed

one

also

finds

expressions

equivalent

to

the

momentum

and the force

density given

here.

[22]This

equation appears

on [p. 10]

and

on

[p.

51]

of Einstein's research

notes

on a

gener-

alized

theory

of

relativity

(Doc.

10).

See

also

[p. 26],

where Einstein

uses a

gravitational

field

equation

to replace

the

stress-energy tensor

by a

(not generally covariant) gravitation

tensor.

He discusses

a

general

formulation of

energy-momentum

conservation

in terms

of

a sum

of

tensors

in

his

manuscript

on

special relativity

(Doc.

1), [p.

63].

[23]In

a no

longer

available letter

to Einstein, H. A.

Lorentz

pointed

out

that

the

general

form

of

equation

(10)

is

only

covariant

in

the

case

of

a symmetric stress-energy

tensor.

See Einstein

to H. A. Lorentz,

16 August

1913

(Vol. 5,

Doc.

470),

for Einstein's

reply.

[24]See

the editorial

note,

"Einstein

on

Gravitation

and

Relativity:

The Collaboration with

Marcel

Grossmann,"

pp.

297-298,

for

a

discussion of the evolution of Einstein's views

on

the

lack of

general

covariance of the "Entwurf"

theory.

[25]Early

in

1914,

Einstein found "most

general

transformations"

("höchst

allgemeine

Transformationen")

which transform

the

gravitational equations

into

themselves

(see

Einstein

to

Paul

Ehrenfest,

before

10

March

1914

[Vol. 5,

Doc. 512]). See

also Einstein and Grossmann

1914b,

for

a more

detailed discussion of the covariance

properties

of the "Entwurf"

field

equations.

[26]Einstein's

unpublished manuscript

on

special relativity

(Doc. 1)

includes

an

outline of

tensor

calculus

on

[pp.

46-54];

differential

operations

are

discussed

on [pp.

53-54].

[27]This

expression played

an

important

role

in

Einstein's search for

a

gravitational

tensor.

For

a

historical

discussion,

see

Norton

1984,

sec.

3.

[28]The

above

expression, together

with

equation

(11),

generally

does

not

lead

to

the weak

field

equations as they

follow from Einstein's

final

theory

of

gravitation

(see

Einstein

1915e;

see

also Norton

1984,

sec. 4,

for

a

historical

discussion).

For

an

analysis

of

the

role of Einstein's

preconceptions

on

the static

case

for the

development

of the "Entwurf"

theory developed in

this

paper,

see

Norton

1984,

sec.

3,

and

also the editorial

note,

"Einstein's

Research Notes

on

a

Generalized

Theory

of

Relativity,"

sec.

IV.

[29]Various

examples

of the method

employed in

the

following paragraphs

are

found

in

Einstein's research

notes

on a

generalized

theory

of

relativity (Doc. 10)

(see,

e.g.,

[p.

41]).

See

also Einstein 1913c

(Doc. 17),

p.

1254,

for

an application

of

this

method

to

Nordström's

theory

of

gravitation.

[30]On

[pp.

38-41]

of Einstein's research

notes

on a

generalized theory

of

relativity

(Doc.

10),

this

method

is

applied

to

the weak

field

case.

An outline of the derivation of the

equation

below

is

found

on [p. 51]

of these

notes;

see

also

[pp.

49-50]

for related calculations. Einstein's

approach,

and

in

particular

the conditions under

which

it

leads

to

a

unique

solution,

are

dis-

cussed

in

Norton

1984,

sec.

4.

[31]The

divergence

of

this tensor is

evaluated for

the

special

case

of

a

rotating

frame of

reference

in

Einstein's research

notes

on a

generalized theory

of

relativity

(Doc. 10), [p. 48].

[32]Einstein

discussed the transformation

properties

of

this

equation

in

a

letter

to

Lorentz

(see

Einstein

to

H.

A. Lorentz, 14

August

1913

[Vol. 5,

Doc.

467]).

[33]See

also the discussion of

this

similarity

between

the two tensors in

Einstein 1913c

(Doc.

16),

p.

1259,

where it

is

interpreted

as

expressing

the

equality

of

inertial

and

gravitational mass.

See

Einstein 1914c

(Doc. 24)

for Einstein's further

analysis

of the role of

gravitational energy

in

his

theory.

[34]For

a

discussion of the role

this

equation

later

played

in

the

analysis

of the covariance

properties

of the "Entwurf"

field

equations, see

the editorial

note,

"Einstein

on

Gravitation

and

Relativity:

The Collaboration with Marcel

Grossmann,"

p.

297.