634

DOCUMENT 456 FEBRUARY 1918

curvature

R,

and

the

density p

of

the static uniform matter distribution

of

the model

are

related via

X

=

Kp/2

=

1/R2

(Einstein

1917b

[Vol.

6,

Doc.

43], eq. (14)).

Einstein

concluded that

X

deter-

mines both

p

and R.

In

eq. (15),

he used the relation between

p

and R to eliminate either

one

of

these

quantities

from the

expression

M

=

2ttpR2

for the total

mass

in the universe in his model.

[7]In

the

draft,

various alternative formulations

are

deleted at this

point:

"ist

aber in

gewisser

Weise

nur

ein

Schein;"

"kann leicht missverstanden

werden;"

and

"sein

Inhalt ist

etwas

weniger

merkwür-

dig,

als

man

auf den

ersten

Blick meinen möchte."

[8]The

metric field

of

Einstein’s cosmological model

can

be

multiplied by an arbitrary

constant C

and still be

a

solution

of

the field

equations

with

cosmological constant,

provided

that

C,

X,

R,

and

p

satisfy

the condition XC

=

C2Kp/2

=

1/R2.

The second

part

of

this

condition,

which

can

be

writ-

ten

as p

=

p1/C2

with

p1 =

2/kR2,

and the fact that the value

of

C

can

be

freely

adjusted

suggest

that

any

value

of

p

is

compatible

with

a given

value of

R.

Since

R

has be

to

equal

to

1/JXC,

however,

it follows that different values

of

C lead

to

different values

of

p

as

well

as

R.

[9]In

Mie

1917c,

the third

part

of

the

published

version

of

Mie’s Wolfskehl lectures

of

June

1917,

Mie

suggested

the

projection

of

the curved

space-time

of

the universe onto

a

flat Minkowski

space-

time,

which would coincide with the curved

space-time

in

regions

far removed from matter

(see

Doc.

416, note 2,

for

more

details).

[10]In

Einstein 1917b

(Vol.

6,

Doc.

43),

p.

148,

this

analogy

is used

to justify

the

use

of

a

matter

distribution,

whose

density

varies

slowly

and

continuously

with

position,

to

approximate

the

compli-

cated actual matter distribution

of

the universe

at

large.

Einstein

goes

on

to

propose a

cosmological

model with constant

density

and

spherical spatial geometry.

This

suggests

the further

analogy

be-

tween the surface

of

the earth

being approximated by an ellipsoid

and the

spatial geometry

of

the uni-

verse being approximated by a hypersphere.

Einstein

explicitly

used this

type

of

analogy

in Doc. 356.

[11]The

further elaboration

of

Einstein’s

analogy

in this last

sentence

is Mie’s and is neither stated

nor implied

in Einstein 1917b

(Vol.

6,

Doc.

43).

It is

only

with this elaboration that Mie

can use

Ein-

stein’s

analogy

to

support

his insistence

on using a

flat Minkowski

space-time

for

the

description

of

the universe

even

though

the

metric field

of

the universe will not be Minkowskian.

[12]What

Mie

had in

mind

is

probably a

passage

at

the end of

chap.

4

of

Poincaré 1902

(pp.

83-

87),

where Poincaré considered

a

Euclidean

space

with

some

temperature

field. In this

case,

the

tem-

perature dependence

of

the

length

of

measuring

rods would make the

geometry

appear

to

be

non-

Euclidean.

In

the context

of

this

example,

Poincaré also considered the effect

of

a

variable

speed

of

light

on

the

observed

geometry.

[13]In

Mie 1917c,

p.

599,

Mie

gave

the

example

of

a

rod,

which is

straight

and at

rest

in

one coor-

dinate

system

while curved and

moving

like

a

snake in

another, to

argue against

the

equivalence

of

all coordinate

systems.

The

same example

is discussed

at

greater length

in Mie

1921,

pp.

62-64.

Two

years

earlier,

Mie had

already

raised the

objection

that

if

Einstein

were right,

"one

could,

for

instance,

give no

absolute criterion for whether

a

wire is

straight or

bent"

("könnte

man

ja

z.B. kein absolutes

Kriterium dafür

angeben,

ob ein Draht

gerade

oder

gebogen

ist." See Gustav Mie

to

Wilhelm

Wien,

6

February

1916,

GyMDM,

NL

056/009).

[14]See

Einstein 1916e

(Vol. 6,

Doc.

30),

pp.

771-772.

The

argument

in this

passage

is

as

follows.

Consider

two

identical bodies

S1

and

S2

in relative rotation around the line

connecting

their

centers

of

mass. Suppose

S1

is

spherical,

whereas

S2

is flattened

at

the

poles.

What

causes

this difference in

shape?

In

Newtonian

theory

the

answer

is that the frame

in

which

S1

is at

rest

is

an

inertial

frame,

whereas the frame in which

S2

is

at rest

is not. This

answer,

Einstein

argued,

is

unsatisfactory,

since

it makes

something

unobservable

(the

inertial

frame)

the

cause

of

the observable difference in

shape

between

the

two

bodies. The

only satisfactory answer,

he

continued

in

a

Machian

spirit,

is that the

difference in

shape

is caused

by

the effects

of

other bodies in the

universe,

effects

one

should be able

to

describe in

arbitrary

frames

of

reference

using

the

same physical

laws in all

of

them. Einstein’s

argument

was

criticized in Mie

1917c,

p.

598.

[15]In

Mie

1917c,

pp.

599-602, Mie

contrasted

a "‘physical’

world

picture"

(""physikalisches"

Weltbild")

to what he called "Hilbert’s

‘geometrical’

world

picture" ("Hilbertschem

"geometrischen"

Weltbild").

In the

geometrical

world

picture,

the metric field

represents

the non-Euclidean

geometry

of

the

universe,

which

is

directly

measurable with rods and clocks. In the

physical

world

picture,

the

metric field

represents a gravitational

field in

a

Minkowski

space-time, or, more generally

(as

Mie is