DOCUMENT 382

SEPTEMBER

1917

523

expression

for the

gravitational energy-momentum

pseudo-tensor

density.

In this

document,

a

roman

character

is used for

this

quantity.

[11]The

equation

referred

to

is:

^ggFvj ~

-(^o

+

to)

(Einstein

1916o

[Vol.

6,

Doc.

41]).

[12]In Nordstrom

1918a,

p.

1095

(p.

1078 in the

English translation),

the author writes:

“From

a

private correspondence

with

Einstein

I

learned that he has

proved

that these two

vectors

are really

identical,

at least when the

system

of

coordinates is thus chosen that

V-g

= 1

•”

[13]According

to

a

result derived for so-called

complete

static

systems

in Laue

1911a,

pp.

540-541,

the terms with

a

=

1,2,3

in

eq.

(8)

below vanish.

[14]See

note

10

above.

[15]See

note

11

above.

[16]See

Nordstrom

1918a,

sec.

3.

[17]Einstein

1916g (Vol.

6,

Doc.

32).

[18]At this

point

in the

original

text,

Nordstrom indicates

a

note

that he has

appended

at the foot

of

the

page:

“Als dies schon

geschrieben

war,

habe ich eine

Möglichkeit

gefunden

den

Widerspruch zu

erklären. Siehe Seite

5.”

See

“Fortsetzung,

28

Sept.”

below.

[19]The following

calculation

and the

(partial)

resolution

of

the

discrepancy

with Einstein’s

result

in

“Fortsetzung”

below,

can

be

found

in

sec.

2

of

Nordstrom

1918b,

submitted

to

the Amsterdam

Academy

in

January

1918

as a

sequel to Nordstrom 1918a.

[20]Droste 1916b,

eq.

(28).

[21]Eq.

(16)

below

gives

the

general

form

of

a

spherically

symmetric

metric in Cartesian coordi-

nates.

[22]The

expression

for

,.f-g

below follows from

eq.

(16)

by exploiting

the

spherical symmetry

of

the metric and

setting

x1

=

r,

x2 =

x3 =

0

(see

Nordstrom

1918a,

p.

1105

[p.

1087 in

the

English

translation]).

[23]Inserting

eq.

(17)

into

eq.

(16)

and

transforming

from

Cartesian to

spherical

coordinates, one

arrives at the metric

of

eq.

(15).

In Droste

1916b,

eq.

(18), v2

is the coefficient

of dû2

in

the

general

form

of

a spherically symmetric

metric in

spherical

coordinates.

[24]For

a

simple

proof

that the coordinates used in Einstein

1916g

(Vol.

6,

Doc.

32)

do

indeed

have

this

property, see

Einstein 1918a

(Vol.

7,

Doc.

1),

p.

159.

These

coordinates have

become

known

as

isotropic

coordinates.

[25]This example

of

how

gravitational

field

energy can

be transformed

away

is also

given

in Ein-

stein 1918a

(Vol.

7,

Doc.

1),

p. 159,

with

an acknowledgment

to

correspondence

with Nordstrom.

[26]In

Nordstrom

1918b,

sec.

2,

the author shows that

t4

vanishes

in

a

coordinate

system satisfying

V-g

=

1

and mentions without derivation that t4

does not vanish in

isotropic

coordinates. Nord-

strom writes that this makes Einstein’s choice

of

a

pseudo-tensor

for

representing gravitational ener-

gy-momentum

“somewhat less

sympathie”

and that it

supports

H. A. Lorentz’s alternative

choice of

a generally

covariant tensor

(see

Doc.

253,

note

3,

and Doc.

368,

note

6,

for

more

on

this

idea,

which

was championed by

Lorentz and Tullio

Levi-Civita).

[27]From

eq.

(11)

in Einstein

1916g (Vol.

6,

Doc.

32)

for

t^v

in

terms

of

Y'uv an expression

for

f44

results which is

equal

to

-2/k

times

t44

in

eq.

(28)

above

(in

Einstein’s

paper,

Ktuv

denotes what in

this

document

is called t,,,,). To

verify

this

result,

one

needs the relation

a

=

-kM/4^,

which is found

by

comparing

Einstein’s

approximate expression

for the metric in this

case (eq.

(14)

in

the

paper)

to

Droste’s exact

expression (see eqs. (16)

and

(24)

above).

[28]The

contradiction is,

in

fact,

due

to an

error

in

eq.

(11)

for

in Einstein

1916g

(Vol. 6,

Doc.

32),

which

was

corrected in

Einstein

1918a

(Vol.

7,

Doc.

1)

(see

especially

the footnote

on p.

157).

In the latter

paper,

the corrected

expression

for

t^v

is evaluated

explicitly

for the metric field

of

a

point

mass

in

isotropic

coordinates.

The result found

for

kí44

(Einstein

1918a

[Vol.

7,

Doc.

1], eq.

(13))

is

equal

to

the

expression

for

tf

in

eq. (28)

above

with

the

opposite sign.

Since

tßv

=

-tuv

in