968 DOCUMENT

669

DECEMBER 1918

ALS.

[24 053].

[1]A

week and

a

half

earlier,

Einstein wrote that the

manuscript

that

Weyl

had

sent

him

together

with Doc. 657 could

not

be

accepted

for

publication

in the

proceedings

of

the Prussian

Academy

because

of

a paper shortage

(see

Doc.

661).

Einstein then reiterated his main

objection

to

Weyl’s

uni-

fied

theory

of

gravitation

and

electromagnetism

and added several

new

ones,

aimed

specifically

at the

elaboration

of

the

physical aspects

of

the

theory

in

Weyl’s manuscript.

[2]Half

a year

earlier,

Weyl

had

already expressed

this

same worry

(see Doc.

544).

[3]Weyl

1918a.

[4]Weyl

1918c.

[5]Weyl

1918d.

Weyl explained

the

structure

of

the

presentation

of

his

generalization

of

Riemanni-

an geometry

in this

paper

in Doc. 619.

Essentially

the

same presentation can

be found in the third

revised edition

of

Weyl’s

book

(Weyl

1919d,

secs.

14-17).

Weyl

also added two sections

at

the end

of

the book

(Weyl

1919d,

secs.

34-35)

on

the unified

theory

of

gravity

and

electromagnetism

based

on

this

new geometry.

These

two

sections

closely

follow

Weyl

1919c,

the revised version

of

the

manu-

script

that he had

sent to

Einstein.

[6]The

fact that in

ordinary

Riemannian

geometry

directions

of

vectors

can only

be

compared

lo-

cally,

whereas

lengths can

be

compared globally

(see

Weyl

1918b,

pp.

466-467; for

a

discussion,

see

Doc.

472,

note

3).

See Docs. 544 and

551 for

Weyl’s

and Einstein’s

opposing

views

on

how

serious

a problem

this

"inconsequence" is.

[7]The manuscript

enclosed with Doc. 657.

The

triumphant

tone

can

still be discerned in

Weyl

1919c,

the revised version

of

this

manuscript,

in which the author

emphasizes

the

superiority

of

his

own theory over general relativity

not just

from

a

mathematical

point

of

view but also

on

such

phys-

ical issues

as

the conservation laws

(pp.

120-121),

cosmology,

and the

problem

of

matter

(p. 133).

[8]In

a

letter

to

Carl

Seelig

of

19 May

1952,

Weyl

reflected

on

his

discussions with Einstein

of

1918

about his

attempt

to

unify gravitation

and

electromagnetism,

employing

the notion

of

gauge

invari-

ance

(this portion

of

the letter is

quoted,

in

a

lightly

edited

form,

in

Seelig 1960,

pp.

274-275).

Weyl

recalled how

Einstein had

objected to

his

speculative

mathematical

approach to

physics

and had in-

sisted

on

starting

from

physical

principles

instead.

Weyl

noted that he and Einstein had since switched

positions

on

this issue.

Concerning

this

specific attempt

at

a

unified

theory, Weyl

reminded

Seelig

that

it had been

recognized

in the late twenties that

gauge

invariance ties the

electromagnetic

field

to

the

Dirac field

of

the electron and

not to

the

gravitational

field

as Weyl

had

originally thought (see

also

the

preface

to

the first American

printing

of

Weyl

1922).

[9]The

square

brackets

are

in the

original.

[10]At this

point

in the

original

text,

Weyl

indicates

a

note

that he has

appended

at

the foot

of

the

page: "diejenige Ladung,

deren "Gravitationsradius"

=

dem Weltradius ist."

Weyl

defined the

gravi-

tational radius

of

a charge

in the context

of

a

discussion

of

the metric field

of

a pointlike source

with

mass m

and

charge e.

For the

44-component

of

this

field,

he found

1 -

2Km/r +

Ke2/c2r2,

where

k

is Ein-

stein’s

gravitational

constant

and

c

is the

velocity

of

light.

In

analogy

with the

gravitational

radius

of

m,

defined

as km,

he defined the

gravitational

radius

of

e as e

,Jk/c

(Weyl

1917, p. 133,

and

Weyl

1918c,

p.

207).

[11]In

Doc.

661,

Einstein

argued

that

Weyl’s theory implies

that there is

a new

constant

of

nature,

which he

wrote

as

1/y,

with the dimension

of

a charge

and

satisfying

the relation l

~

Jk/y.

In this

relation,

k

is

Einstein’s

gravitational

constant and l is

a

characteristic

length,

which has

to

be

very

large

if

Weyl’s theory

is to be

compatible

with Coulomb’s law. Since

k

~

10-27,

this

means

that

1/y

must

also

be

very large.

Einstein

saw

this

as

a

serious

objection

to

Weyl’s

theory.

As

pointed out by

Weyl

in this

paragraph,

however,

a purely

dimensional consideration

of

the relation derived

by

Ein-

stein

suggests

that this

new

constant

1/y

is the natural unit

of

charge

in Einstein’s

own theory

if

the

radius

of

the universe in Einstein’s

cosmological

model is taken

as

the natural unit

of

length.

This

notion

of

a

natural unit

of

charge

related

to

the size

of

the universe

can

also be found in the revised

version

of

Weyl’s manuscript

(Weyl

1919c,

pp.

123-124) and in

subsequent

editions

of

Weyl’s

book

on general relativity (Weyl

1919d,

sec.

35,

Weyl

1921a,

sec. 36).

Since

Weyl’s theory immediately

gives

the

field

equations

with

cosmological

term (as

explained

in

Doc.

619, note 11),

whereas Ein-

stein

had to

add this

term later,

Weyl

presented

this

cosmological

unit

of

charge as a

strong argument

in favor

of

his

own

theory.