RESEARCH

NOTES

ON

RELATIVITY

195

arise from

an

interaction between that

mass

and

the

remaining

masses

of

the

universe.

This

project

had

presumably

advanced little

by

4

July 1912,

the time of submission

of Einstein 1912h

(Doc. 8). Hoping

that

the

theory's equations

would remain invariant

under acceleration

and

rotational

transformations,

Einstein concludes

(pp.

1063-1064)

by

calling

on

his

colleagues to

search for

the

general

form

of

the

space-time

trans-

formation

equations

of

a

relativity

theory incorporating gravitation. By 16 August,

however,

Einstein could write from Zurich

to

Ludwig

Hopf

that unless

he

was com-

pletely mistaken,

he

had found the

most

general

equations.[14]

III

Einstein's

insight

into the

analogy

to

Gauss's

theory

of surfaces

was

just

a

first

step.

That

insight

had

now

to

be converted into

a

comprehensive

theory

that would

do

justice

to the

kinematics

and

dynamics

of

gravitational

and

nongravitational phenom-

ena.

The

development

of such

a

theory

was

furthered

immeasurably

when Grossmann

alerted Einstein

to

the absolute differential calculus of Ricci and Levi-Civita

1901,

which

proved

to

contain

precisely

the mathematical devices

needed to

complete

the

general theory

of

relativity.

Part

I

of

Doc.

10 begins

at

an

early stage

of

this

devel-

opment.

On

[pp.

1-2],

Einstein

uses a

nonstandard

uppercase

Guv

to

represent

the

metric

tensor,

before

reverting

to

the standard

guv

in

the remainder of the document.

The

elementary

nature

of the calculation

and

the

writing

out term

by

term

of

simple

sums

also

suggest

that Einstein

was

not

yet

familiar

with

the material.

As late

as

[p.

6],

there

is

no

indication of

techniques

characteristic of

the

absolute differential

calculus,

suggesting

that

Einstein

may

not

yet

have been alerted

to its

existence

by

Grossmann. Ricci

and

Levi-Civita,

for

example, distinguished

contravariant

and

covariant

systems.

The

distinction

does

not

appear

until

[p. 8].[15]

On

[p. 10] we

find

Ricci and

Levi-Civita's covariant differentiation

operation.

On

[p.

6]

we

also

find

direct evidence of Einstein's

awareness

of Gauss's

theory

of

surfaces. He

recapitulates

a

standard

result:[16]

The

trajectory

of

a

particle

free of forces but constrained

to

a

curved surface

is

a

geodesic

of

the

surface.

By [p. 10],

Einstein

has

developed

the

basic

law

governing

the

dynamics

of distributed

matter: the

vanishing

of

the

covariant

divergence

of

its

stress-energy

tensor.

By

the close of Part

I

Einstein has undertaken

calculations of considerable

sophistication.

His

confidence

has

grown by

[p.

25]

to

the

point

that

he is

prepared

to

attempt

a

classification of

tensors of various orders.

Einstein's

use

of the

term

"tensor" for other than second-rank

quantities is

an

innovation of Einstein and Grossmann

1913

(Doc. 13)

and

replaces

Ricci

and

Levi-

[14]See

Einstein to

Ludwig Hopf, 16 August

1912

(Vol.

5,

Doc.

416).

[15]Note

that the dramatic increase in

sophistication

of

[pp.

8-9]

and

possibly

also

[p. 7]

suggests

that these

pages may

have

been written much later than

those

that follow

in

the

notebook.

[16]The

result

is

developed

in

Grossmann's lecture

notes

mentioned above.