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