296

EINSTEIN

IN

COLLABORATION WITH GROSSMANN

"Entwurf"

theory

was

rather

limited.

At

some point

in his

search for

a

theory

of

gravitation

Einstein turned

to Grossmann,

who drew

his attention

to

the work of

Riemann,

Christoffel,

Ricci,

and Levi-Civita

on

the

absolute differential

calculus.[10]

Grossmann

systematically

expounded

the

generalized tensor

calculus he and Einstein

had extracted from earlier mathematical studies

in

the second

part

of Einstein

and Grossmann

1913

(Doc. 13),

the

only

part

for which

he

took

direct

responsibil-

ity.[11]

During

their work

on

the "Entwurf"

paper,

Grossmann

helped

Einstein

in

his search

for

a gravitational

tensor,

and

he

apparently suggested

the

fourth-rank

Riemann

tensor

as a

starting

point.[12]

An obvious candidate for

a

gravitational

tensor

was

the Ricci

tensor,

constructed from

the Riemann

tensor;

but

at

some

point

in

their research Ein-

stein and Grossmann

came

to

reject it.[13] They

abandoned

general

covariance

and set

out to

derive

a

differential

equation

for the

gravitational field, taking

tensors

that

are

only

covariant

with

respect

to

linear transformations

as

their

starting point.

In the

derivation,

which

is

presented

in

§5

of Einstein and Grossmann 1913

(Doc. 13), they

use

the

requirements

of

covariance under

general

linear

transformations,

of

energy-

momentum conservation,

and of

recovering

the

correct

Newtonian limit

to find

the

field

equations.[14]

[10]For

evidence

of

Grossmann's role in

solving

the mathematical

problems

Einstein

was

facing,

see

Einstein

to

Erwin

Freundlich, 27

October

1912

(Vol.

5,

Doc.

420),

and Einstein

to

Arnold Sommerfeld,

29

October 1912

(Vol.

5,

Doc.

421).

For

an

account

of

the collaboration

between Einstein and Grossmann

as

well

as

of their

personal relationship,

see

Pais

1982, pp.

210-225. The

generalized

definition of the

tensor

concept as

it

appears

in

Einstein and Gross-

mann

1913 (Doc.

13)

merges

two

traditions that

up

to

that time had remained

essentially

separated:

the

study

of differential forms

(absolute

differential

calculus)

and

vector

calculus.

For

a

historical

discussion,

see

Klein,

F.

1927,

pp.

43-45;

Reich

1994,

sec.

5.3; and

Norton

1992a,

pp.

302-310.

In

the

vector analysis

tradition,

tensors

were

defined

more

narrowly

as

what

is

now

described

as

second-rank,

symmetric

tensors

(see, e.g., Sommerfeld 1910a,

p.

767),

whereas

in

Ricci and Levi-Civita

1901

objects

such

as

the

ones

defined

by

Einstein and Gross-

mann

were not

referred

to

as

"tensors" but

as

"covariant

or

contravariant

systems" ("des

systemes

covariants

ou

des

systemes contravariants,"

Ricci and Levi-Civita

1901,

p.

131).

See

also the editorial

note,

"Einstein's Research Notes

on a

Generalized

Theory

of

Relativity,"

pp.

195-196,

for further discussion.

[11]See

also Grossmann

1913

for

a

brief

account

of the

generalized vector calculus,

essen-

tially

similar

to

the

presentation

given

in

Einstein and Grossmann 1913

(Doc. 13).

On

pp.

291-

292 of his

paper

Grossmann

emphasizes

the

necessity

of

an

approach

to vector analysis

from

the

perspective

of the

theory

of

invariants,

as

opposed

to

the

understanding

of

vector

concepts

as

it is

customary

in

physics.

[12]See Einstein's research

notes

on a

generalized

theory

of

gravitation (Doc. 10),

in

particular

[p.

27]

and

[p.

44],

where Grossmann's

name occurs

in

connection with the Riemann and the

Ricci

tensor,

respectively.

These

tensors

would

play a key

role

in

Einstein's later

development

of the

general theory

of

relativity.

[13]See the editorial

note,

"Einstein's Research Notes

on

a

Generalized

Theory

of

Relativity," pp.

197-199,

for

more

details.

[14]Contrary

to

what Einstein and Grossmann

claim,

their

procedure

does

not

lead

to

unique

field

equations;

see

Norton

1984,

sec.

4,

for

a

discussion.