DOCUMENT 511 APRIL 1918 725

des

ds,

nicht

nur

des Verhältnisses

verschiedener,

von

einem Punkte

ausgehender

ds).

ALS

(SzZE Bibliothek,

Hs.

91:560). [24 022].

[1]Dated

by

the reference

to

this

document

in Doc. 513.

[2]Weyl

1918c,

which Einstein

was examining

in

proof (see

Doc.

476).

[3]Section

33

on cosmology.

[4]This

argument

for

an elliptic geometry,

in which

antipodal points are

identified,

can

be found

on

p.

226

of

page proofs

of

Weyl

1918c,

dated

April

1918

(see

[24 037]).

The

possibility

of

elliptic

geometry was

discussed in the

context

of

Einstein’s

cosmological

model in Docs. 300 and 319.

[5]In

sections

30, 31,

and 33

of

Weyl

1918c,

the

author,

following

the method first

developed

in

Weyl

1917,

secs.

5-6, derived various static

spherically symmetric

solutions

of

Einstein’s field

equa-

tions

(with

and without the

cosmological

term),

including

the exterior and interior Schwarzschild

solutions,

and the static form

of

the De Sitter

solution,

although

it is not

so

identified in

Weyl

1918c.

Weyl

started from

a

static line element

of

the form ds2

=

f2dt2-da2,

where

da2

=

yijdxidxj

(i,j

=

1,2,3). Because

of

the assumed

spherical symmetry,

the

spatial

metric

has the

form

Yij

=

dij

+

lxixj,

and

l

and

ƒ

are

functions

of

r

= Jx21+x22+x23

only.

The

spatial geometry

can

be

visualized

by embedding

the 3-dimensional

space

in

a

4-dimensional Euclidean

space.

The embed-

ding

will be

a hypersurface

of

revolution around the

x4

-axis.

For various

cases, Weyl

determined the functions ƒ and

l (or, equivalently, h2

= 1

+

lr2)

with the

help

of

a

variational

principle.

For the

case

without matter but with

a non-vanishing cosmological

constant y,

Weyl

found the solution

1/h2

=

f2

=

1-y/6r2

(Weyl

1918c,

p.

225),

which Einstein

gives

under "I" below. This is the static form

of

the

De Sitter solution

(see

Doc.

566, note 7,

for

a

discussion

of

how the solution found

by Weyl

is related

to

other forms

of

the De Sitter

solution).

For this solution

l

=

1/R2-r2

(with

R

=

J6/y).

The

hypersurface

of

revolution

representing

the

spatial part

of this

solution in the 4-dimensional

embedding space

is

a hypersphere

of

radius

R (see

Weyl

1918c,

pp.

211-

212, or

Einstein 1917b

[Vol.

6,

Doc.

43], pp.

149-150).

Weyl’s

definition

of

X

differs

by a

factor 2

from

Einstein’s, so

the relation between R and

X given by Weyl

is

equivalent

to the relation

y

=

3/R2

satisfied

by

the De Sitter solution

(see,

for

instance,

Doc.

313).

For

r

=

R

-the

equator

x4 =

0

on

the

hypersphere,

which is the horizon

for

an

observer at

one

of

the

poles corresponding

to

the

origin

of

the

spatial

coordinate

system-the

metric has

a

singularity:

1/h2

=

ƒ2

=

0.

From

this,

Weyl

concluded that it is

impossible

to

have

a completely empty

universe and that there must be

at

least

a zone

of

matter

between

two

parallels on opposite

sides

of

the

equator.

He

constructed

a non-

singular

solution

by piecing

together

the

part

of

the

vacuum

solution in the

region rr0

(where

r0

is

only a

little smaller than

R)

and the

part

of

the solution for the

case

of

an incompressible

fluid with

mass density

u0

and

pressure

p

in the small

region

r0rR

between these

two

parallels (Weyl

1918c,

p. 225).

The latter solution is the

one

Einstein

gives

under

"II"

below.

Continuity

in

r

=

r0

requires

that M

=

u0/6r30.

The

spatial

part

of

the

space-time

geometry

resulting

from

Weyl’s

combi-

nation

of

the

two

solutions

can

still be

mapped

onto

a hypersphere

of

radius

R,

even though

the metric

field will

not

be the metric

of

this

hypersphere

in the

region

r0rR. In accordance with

Weyl’s

general claim,

which is refuted

by

Einstein,

this

hybrid

solution is

symmetric

around the

plane

of

the

equator.

[6]The figure represents a hypersphere

in

4-dimensional

Euclidean

space,

which in

turn

represents