Volume 6: The Berlin Years: Writings, 1914-1917 (English translation supplement) Page 363 (375 of 462)

DOC.
42
SPECIAL AND GENERAL RELATIVITY
363
Cosmological Difficulties
of
Newton's
Theory
121
theoretical foundation. We
can
imagine
innumerable
laws
which would
serve
the
same
purpose,
without
our
being
able
to state
a reason
why
one
of
them
is
to
be
preferred
to
the
others;
for
any
one
of
these
laws
would be founded
just
as
little
on more general
theoretical
principles
as
is the
law
of
Newton.

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DOC.
42
SPECIAL AND GENERAL RELATIVITY
363
Cosmological Difficulties
of
Newton's
Theory
121
theoretical foundation. We
can
imagine
innumerable
laws
which would
serve
the
same
purpose,
without
our
being
able
to state
a reason
why
one
of
them
is
to
be
preferred
to
the
others;
for
any
one
of
these
laws
would be founded
just
as
little
on more general
theoretical
principles
as
is the
law
of
Newton.

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362
DOC.
42
SPECIAL AND
GENERAL RELATIVITY
120
Relativity
tiness.
The
stellar universe
ought
to
be
a
finite island
in
the
infinite
ocean
of
space.1
This
conception is
in
itself
not very satisfactory.
It
is
still
less
satisfactory
because
it
leads
to
the result that the
light
emitted
by
the
stars
and
also
individual
stars
of the stellar
system
are perpetually passing
out
into infinite
space,
never
to
return,
and
without
ever again
coming
into interaction with
other
objects
of
nature.
Such
a
finite
material universe would
be destined
to
become
gradually
but
systematically impover-
ished.
In order
to escape
this
dilemma,
Seeliger suggested
a
mod-
ification
of Newton's
law,
in
which he
assumes
that
for
great
distances the
force
of attraction between
two
masses
dimin-
ishes
more rapidly
than would result
from
the inverse
square
law.
In this
way
it
is possible
for
the
mean
density
of
matter
to
be
constant
everywhere,
even
to
infinity,
without
infinitely
large
gravitational
fields
being
produced.
We
thus free
our-
selves
from
the distasteful
conception
that the material uni-
verse
ought
to
possess something
of the
nature
of
a
centre.
Of
course
we
purchase
our
emancipation
from
the fundamental
difficulties
mentioned,
at
the
cost
of
a
modification and
com-
plication
of Newton's
law
which
has
neither
empirical
nor
[72]
[73]
1
Proof-According
to
the
theory
of
Newton,
the number of "lines of force" which
come
from
infinity
and terminate in
a mass m
is proportional
to
the
mass m. If, on
the
average,
the
mass
density
p0
is
constant
throughout
the
universe,
then
a
sphere
of volume
V
will
enclose the
average
mass
p0V.
Thus the number of lines of force
passing through
the surface F of the
sphere
into
its
interior
is
proportional to
p0V.
For unit
area
of the surface of the
sphere
the number of lines of
force which
enters
the
sphere
is
thus
proportional to p0
V/f
or to
p0R.
Hence the
intensity
of the
field
at
the surface would
ultimately
become infinite with
increasing
radius
R
of the
sphere,
which
is impossible.

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364
DOC. 42 SPECIAL AND GENERAL RELATIVITY
THIRTY-ONE
The
Possibility
of
a
"Finite"
and
Yet
"Unbounded"
Universe
But
speculations on
the
structure
of the
universe also
move
in
quite
another direction.
The
development
of
non-Euclidean
geometry
led
to
the
recognition
of the
fact,
that
we can
cast
doubt
on
the
infiniteness
of
our
space
without
coming
into conflict
with
the
laws
of
thought
or
with
experience
(Riemann, Helmholtz).
These
questions
have al-
ready
been treated
in
detail and with
unsurpassable lucidity
by
Helmholtz and
Poincare,
whereas
I
can
only
touch
on
them
[74]
briefly
here.
In the
first
place,
we
imagine
an
existence in
two-
dimensional
space.
Flat
beings
with
flat
implements,
and
in
particular
flat
rigid
measuring-rods,
are
free
to
move
in
a
plane.
For them
nothing
exists outside of
this
plane:
that which
they
observe
to happen to
themselves and
to
their
flat "things" is
the all-inclusive
reality
of their
plane.
In
particular,
the
con-
structions of
plane
Euclidean
geometry
can
be carried
out by
means
of the
rods,
e.g.
the lattice
construction,
considered
in
[75]
Section
24.
In
contrast to
ours,
the universe
of
these
beings is
two-dimensional; but,
like
ours,
it extends
to
infinity.
In their
122

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