768
DOCUMENT 544 MAY 1918
ALS.
[24 036].
[1]Weyl
had revised the
page proofs
of
Weyl
1918c in
response to
an
objection by
Einstein
(see
Doc.
525).
In Doc.
535,
Einstein
objected
to
these revisions.
Weyl’s new
considerations outlined here
are only briefly
referred
to
in the
published
version
of
the book
(see
Weyl
1918c,
p.
225).
They are
presented
in detail in
Weyl
1919b,
submitted in
October
1918. For
a
discussion
of
various earlier
results referred to here and definitions
of
most
of
the
symbols
used,
see
Doc.
511, note
5.
Page
and
equation
numbers
presumably
refer
to
the
new page proofs
announced in Doc. 525. The
equation
numbers
are
the
same as
in the
published
version
of
the book.
Eqs.
64 and 65 cited below also
occur
(but
are
not numbered)
on pp.
225226 of
the earlier
page proofs
([24 037]).
[2]Einstein had written that he
did
not
understand
this
passage
(see
Doc.
535).
The
case
of
a
"fluid
cap"
("Flüssigkeitskalotte") is
referred
to
in Doc. 525 and is discussed
more fully
in
Weyl
1918c,
p.
226
(see
Doc.
525, note 2,
for further
details).
The
spatial geometry
in the fluid
cap
of
density
p0
for
this
case
is
given by
1/h2
=
1

2m0
+
A/6r2.
This
is
the
geometry
of
a
hypersphere
r2 +
z2 =
b20
(with
b02
=
6/
2u0
+
A
)
in
a
4dimensional
Euclidean
space.
It follows that
h
can
be rewritten
as
b0/z.
[3]This comment
and the
following
considerations
refer
to
a
static
spherically symmetric
solution
of
the field
equations
with
cosmological
term obtained
by piecing
together
parts
of
two
such solu
tions,
a
matterfree
solution,
which
is,
in
fact,
the De Sitter solution in static
formand,
contrary
to
Weyl
1918c,
Weyl
1919b does
so identify
itand
a
solution for
an
incompressible
fluid. The latter is
used in
the
vicinity
of
the
surface
were
the
former
is
singular.
On
the
hypersphere
representing
the
spatial
part
of
the solution in 4dimensional
space,
this
is
the
region
between two
parallels
equidistant
from the
equator.
[4]In
the
equation
below,
which
can
be found
on p.
211 in
Weyl
1918c, A
=
/g
=
fh and
v
=
(u0 +
p)f,
where the functions
ƒ
and h characterize the metric field and where
u0
and
p are
the
mass density
and the
pressure
of
an incompressible
fluid.
[5]See
Weyl
1919b,
pp.
3133, for
a
detailed
derivation
of
the relation between the
density
u0
and
the thickness
(determined by
r0)
of
the
mass
horizon in
Weyl's
combination
of
the static form
of
the
De Sitter solution and the solution for
an incompressible
fluid. The
spatial geometry
of
the latter
so
lution
is
determined
by
eq.
(64)
referred
to
below:
1/h2
= 1
+ 2M/r 
2M0/6 +
r2.
[6]The equation
below is
eq. (8)
in
Weyl
1919b in different notation. In the
paper, a,
u,
p, and
p0
are
used
instead
of a*,
u0,
y,
and
x, respectively; moreover,
u
and
v
(for
p
=
p0)
are
used
instead
of 1/h
and
p, respectively.
The
expression
for
1/h2
above is
not
equivalent
to
eq. (64) given
in
the
preceding
note
(it
would be
if
x were
defined
as
r/a*
rather than
as
r0/a*
and the
righthand
side
were
divided
by
1

y).
[7]For
an infinitely
thin
mass zone, x
=
1.
The
equation
for
x
above then reduces
to
3Y/2 +
y
=
1.
This
means
that
y
= 1
and that
A/2M0 =
1/y

1
=
0,
from
which
Weyl’s
claim
immediately
follows
(see
Weyl
1919b,
p.
33).
[8]The
corresponding expressions
in
Weyl
1919b,
p.
33,
differ
by a
factor
of
8nk,
reflecting a
dif
ferent
choice
of
units. The units used here
are
those used in
Weyl
1918c,
p.
224.
[9]Weyl’s
result
seems
to vindicate the
suggestion
in Einstein 1918c
(Vol.
7,
Doc.
5) (cited
in
Weyl
1919b)
that the De Sitter solution has
an
intrinsic
singularity indicating
the
presence
of
matter,
and
that the
way
to
prove
this is to
show
that
one
arrives
at
the De Sitter solution with
singularity by
con
sidering
the
type
of
combined solution constructed
by Weyl
(see note 3 above)
and
letting
the thick–