196 RESEARCH

NOTES

ON RELATIVITY

Civita's

terms

contravariant and covariant

"systems."[17]

Einstein also deviates from

Ricci and Levi-Civita's

notation,

where

covariant

systems

are

represented

by

lowered

indices and contravariant

systems

by

raised indices. Instead

he

follows

the

convention

of Einstein and Grossmann

1913

(Doc. 13).

All

indices

are

lowered;

covariant

quan-

tities

are

represented

by

Latin letters

(e.g.,

guv)

and the

corresponding

contravariant

quantities

are

represented

by

the

corresponding

Greek letters

(e.g.,

yuv).

Summation

over

repeated

indices

is not

routinely

assumed,

although

the

summation

symbol is

sometimes

suppressed.[18]

The

purpose

of

many

of

the

calculations of Parts

I

and II

remains unclear.

Repeated

themes

are

the

investigation

of coordinate

transformations,

including

infinitesimal

transformations and unimodular

transformations, and the

construction of

quantities by

differential

operations

that remain invariant under these transformations. In

this task,

Einstein

employs

the

four-dimensional

analogs

of Beltrami's

first

and second

operators

A

and

A2

which

are

given

by[19]

A4) =

7^4v4,v A((j),

4»)

=

7^4),A«*

A24

=

7^4».^

for

the

scalar

fields

Q

and

W.

In

several

places,

Einstein

constructs

quantities solely

from

the

metric

tensor and its derivatives.[20]

This

suggests

that

Einstein

was

seeking

gravitational

field

equations.

In

Einstein and Grossmann

1913

(Doc. 13), §5,

the

prob-

lem of

finding

these

field

equations

is

posed.

The

equations

were

to generalize

Pois-

son's

equation

Acp

=

4ukp

of Newtonian

gravitation

theory[21]

and

were

expected

to

have

the

form

K()

=

r

uv

a uv'

where

k

is

a

constant,

0uv

the

stress-energy tensor,

and the

gravitation

tensor,

Tuv,

a

second-rank covariant

tensor, is

constructed from

the

metric

tensor

and its

derivatives

up

to

second order.

It

became

clear, furthermore,

that

the

authors

expected

the

grav-

itation

tensor to

have

the

form

Edxr(dxb)

+

further

terms

that vanish

on

taking

the first

(1)

approximation.

[17]See

also the

editorial

note,

"Einstein

on

Gravitation

and

Relativity:

The Collaboration

with Marcel Grossmann,"

p.

296.

[18]In

this note

and the

notes to

individual

pages,

we

shall

follow the Einstein and Grossmann

convention.

Summation

over

repeated

indices

will,

however,

often

be

assumed for

compactness.

A

comma

will

represent

differentiation with

respect

to

a

coordinate.

A

semicolon will

represent

covariant

differentiation.

[19]For

a

discussion of Beltrami's

invariants,

see

Darboux

1887-1896, vol.

3,

pp.

193-217,

Wright

1908,

pp.

52-57,

and Bianchi

1896, chap.

2.

[20]See,

for

example,

[p. 3], [pp.

12-14],

[pp.

16-18],

[pp.

22-23],

and

[pp.

25-26].

[21]Q

is

the

gravitational potential,

p

the

mass

density,

and

k a

constant.