968 DOCUMENT
669
DECEMBER 1918
ALS.
[24 053].
[1]A
week and
a
half
earlier,
Einstein wrote that the
manuscript
that
Weyl
had
sent
him
together
with Doc. 657 could
not
be
accepted
for
publication
in the
proceedings
of
the Prussian
Academy
because
of
a paper shortage
(see
Doc.
661).
Einstein then reiterated his main
objection
to
Weyl’s
uni-
fied
theory
of
gravitation
and
electromagnetism
and added several
new
ones,
aimed
specifically
at the
elaboration
of
the
physical aspects
of
the
theory
in
Weyl’s manuscript.
[2]Half
a year
earlier,
Weyl
had
already expressed
this
same worry
(see Doc.
544).
[3]Weyl
1918a.
[4]Weyl
1918c.
[5]Weyl
1918d.
Weyl explained
the
structure
of
the
presentation
of
his
generalization
of
Riemanni-
an geometry
in this
paper
in Doc. 619.
Essentially
the
same presentation can
be found in the third
revised edition
of
Weyl’s
book
(Weyl
1919d,
secs.
14-17).
Weyl
also added two sections
at
the end
of
the book
(Weyl
1919d,
secs.
34-35)
on
the unified
theory
of
gravity
and
electromagnetism
based
on
this
new geometry.
These
two
sections
closely
follow
Weyl
1919c,
the revised version
of
the
manu-
script
that he had
sent to
Einstein.
[6]The
fact that in
ordinary
Riemannian
geometry
directions
of
vectors
can only
be
compared
lo-
cally,
whereas
lengths can
be
compared globally
(see
Weyl
1918b,
pp.
466-467; for
a
discussion,
see
Doc.
472,
note
3).
See Docs. 544 and
551 for
Weyl’s
and Einstein’s
opposing
views
on
how
serious
a problem
this
"inconsequence" is.
[7]The manuscript
enclosed with Doc. 657.
The
triumphant
tone
can
still be discerned in
Weyl
1919c,
the revised version
of
this
manuscript,
in which the author
emphasizes
the
superiority
of
his
own theory over general relativity
not just
from
a
mathematical
point
of
view but also
on
such
phys-
ical issues
as
the conservation laws
(pp.
120-121),
cosmology,
and the
problem
of
matter
(p. 133).
[8]In
a
letter
to
Carl
Seelig
of
19 May
1952,
Weyl
reflected
on
his
discussions with Einstein
of
1918
about his
attempt
to
unify gravitation
and
electromagnetism,
employing
the notion
of
gauge
invari-
ance
(this portion
of
the letter is
quoted,
in
a
lightly
edited
form,
in
Seelig 1960,
pp.
274-275).
Weyl
recalled how
Einstein had
objected to
his
speculative
mathematical
approach to
physics
and had in-
sisted
on
starting
from
physical
principles
instead.
Weyl
noted that he and Einstein had since switched
positions
on
this issue.
Concerning
this
specific attempt
at
a
unified
theory, Weyl
reminded
Seelig
that
it had been
recognized
in the late twenties that
gauge
invariance ties the
electromagnetic
field
to
the
Dirac field
of
the electron and
not to
the
gravitational
field
as Weyl
had
originally thought (see
also
the
preface
to
the first American
printing
of
Weyl
1922).
[9]The
square
brackets
are
in the
original.
[10]At this
point
in the
original
text,
Weyl
indicates
a
note
that he has
appended
at
the foot
of
the
page: "diejenige Ladung,
deren "Gravitationsradius"
=
dem Weltradius ist."
Weyl
defined the
gravi-
tational radius
of
a charge
in the context
of
a
discussion
of
the metric field
of
a pointlike source
with
mass m
and
charge e.
For the
44-component
of
this
field,
he found
1 -
2Km/r +
Ke2/c2r2,
where
k
is Ein-
stein’s
gravitational
constant
and
c
is the
velocity
of
light.
In
analogy
with the
gravitational
radius
of
m,
defined
as km,
he defined the
gravitational
radius
of
e as e
,Jk/c
(Weyl
1917, p. 133,
and
Weyl
1918c,
p.
207).
[11]In
Doc.
661,
Einstein
argued
that
Weyl’s theory implies
that there is
a new
constant
of
nature,
which he
wrote
as
1/y,
with the dimension
of
a charge
and
satisfying
the relation l
~
Jk/y.
In this
relation,
k
is
Einstein’s
gravitational
constant and l is
a
characteristic
length,
which has
to
be
very
large
if
Weyl’s theory
is to be
compatible
with Coulomb’s law. Since
k
~
10-27,
this
means
that
1/y
must
also
be
very large.
Einstein
saw
this
as
a
serious
objection
to
Weyl’s
theory.
As
pointed out by
Weyl
in this
paragraph,
however,
a purely
dimensional consideration
of
the relation derived
by
Ein-
stein
suggests
that this
new
constant
1/y
is the natural unit
of
charge
in Einstein’s
own theory
if
the
radius
of
the universe in Einstein’s
cosmological
model is taken
as
the natural unit
of
length.
This
notion
of
a
natural unit
of
charge
related
to
the size
of
the universe
can
also be found in the revised
version
of
Weyl’s manuscript
(Weyl
1919c,
pp.
123-124) and in
subsequent
editions
of
Weyl’s
book
on general relativity (Weyl
1919d,
sec.
35,
Weyl
1921a,
sec. 36).
Since
Weyl’s theory immediately
gives
the
field
equations
with
cosmological
term (as
explained
in
Doc.
619, note 11),
whereas Ein-
stein
had to
add this
term later,
Weyl
presented
this
cosmological
unit
of
charge as a
strong argument
in favor
of
his
own
theory.