INTRODUCTION

TO

VOLUME

8

li

nection

between

general

covariance and

energy-momentum

conservation

(Doc.

269).[35]

Although

Einstein

did not

use

this

argument

in his

response

to

Kretschmann in Einstein

1918f(Vol. 7,

Doc.

4),

this connection

suggests

that the

general

covariance

of

general relativity

is not

physically vacuous

after

all. The

exact

nature

of

the

relationship

between

general

covariance and

energy-momentum

conservation

was

subsequently

clarified

by

Hermann

Weyl,

and the

Göttingen

group

of

Hilbert and Felix Klein and their

assistants,

foremost

among

whom

was

Emmy

Noether. Einstein

was

involved

in

this clarification

mainly

through

his

cor-

respondence

with

Klein in

1918.[36]

Given

this

general background,

one

can

under-

stand

why

Einstein showed

no

interest in the

algebraic

identities known

as

the

Bianchi identities

or

in the related differential identities known

as

the

contracted

Bianchi

identities,

of

which at least two

of

his

correspondents

availed themselves

to derive

energy-momentum

conservation from the field

equations.[37]

Einstein

was

interested in how these identities

are

related

to

the invariance

of

the

action

integral.

Because

of

physical

considerations,

Einstein insisted

that

the

right-hand

side

of

the field

equations

can

be written

as

the

sum

of

quantities representing

the

energy-

momentum

of

matter and the

energy-momentum

of

the

gravitational

field,

and

that

the law

of

energy-momentum

conservation

can

be written

as

the

vanishing

of

the

ordinary

coordinate

divergence

of

that

sum.

This meant

that

he had to

represent

gravitational energy-momentum by a nongenerally

covariant

quantity. Virtually

all

his

correspondents

objected

to this

pseudo-tensor

since it

marred

the

general

covariance

of

the

theory.[38]

The

alternative-suggested

independently

by Lorentz,

Levi-Civita,

and

Klein[39]-was

to

define the

left-hand

side

of

the field

equations

as

the

gravitational energy-momentum

tensor and

to

accept

that the law

of

energy-

momentum conservation cannot be written in the

simple

form

on

which Einstein

insisted. Einstein

staunchly

defended his

usage

of

the

pseudo-tensor

tuv,

both in

correspondence

and in

print.[40]

From

a

modem

point

of

view,

Einstein

was right

in

this debate. Gravitational

energy-momentum

is

represented by a

pseudo-tensor

because

it

cannot

be

localized.[41]

It is thus

perfectly

acceptable

that,

at

any given

point,

tuv

can

be

zero or non-zero depending on

one’s

choice

of

coordinates.

This

interpretation

in terms

of

nonlocalizability

of

gravitational energy-momen-

tum

was

not articulated

at

the time and Einstein

himself

struggled

with

some

of

the

pseudo-tensor’s

counterintuitive

properties.

In Einstein

1916g (Vol. 6,

Doc.

32),

he

arrived at the

strange

result

that his

theory seems

to

allow

gravitational waves

that

do

not

transport energy.

He noticed that these

spurious waves can

be transformed

away by switching

to coordinates

satisfying

the

determinant

condition

V-g

=

1.

Initially,

he

thought

that this shows that such coordinates

are physically privileged

(Doc. 227).

The

following year,

however,

Nordstrom

showed that these coordinates

engender some

counterintuitive results

of their

own.

In coordinates

satisfying