880 DOCUMENT 619 SEPTEMBER 1918
1918b in Lorentz
et
al. 1922,
p.
159).
The derivation
can
also be found
in Pauli
1921,
sec.
65
(d).
The
simple Lagrangian
used in
Weyl’s
derivation is the
sum
of
(F2/4)Jg
(where
F
is
the
cur
vature
scalar for the connection in
Weyl’s theory)
and the
Lagrangian
for the free Maxwell
equations.
Variation
of
the first
term
gives:
(F/2)8(F/g)
+
(F2/4)h4g. In
general,
this will lead
to
fourthorder field
equations. By introducing
the
gauge
condition
F
=
a, however,
where
a
is
a
pos
itive
constant,
Weyl
obtained secondorder
equations. Dividing by ot, one can
rewrite
the
expression
above
as
the variation
of
(F/2
+
a/4)Vg.
As had been shown
earlier in
the
paper
(Weyl
1919c,
p.
110),
F
Vg can
be written
as
the
sum
of
a divergence
term
(which
can
be
ignored
in the derivation
of
the field
equations)
and
{R

(3/2)(piqi}Vg
(where R
is the Riemann
curvature
scalar and
(pi
is the
electromagnetic
fourvector
potential).
Thus,
the field
equations
follow from variation
of
the
sum
of
{R/2
+
a/4

(3/4)pipi}
,/g and the
Lagrangian
for the free Maxwell
equations.
The first
two terms
give
the lefthand side
of
the Einstein field
equations, including
the
cosmological
term;
the
last two
give
the
source
terms
on
the
righthand
side. The third
term,
Weyl pointed
out
(Weyl
1919c,
p. 122),
is the
simplest possible
addition
to
the
Lagrangian
for the free Maxwell
equations
if
one
wants to account
for the existence
of
matter
along
the lines
of
the
theory
of
Mie
1912a, 1912b,
1913.
[12]This
point
is
emphasized
in the discussion
of
conservation laws in
Weyl
1919c,
pp.
114121
(included
almost verbatim
in
Weyl
1919d,
sec.
34,
and further
developed
in
Weyl
1921a,
sec.
35,
and
Weyl
1923a,
sec.
41).
This
discussion
provides a
much
more
elaborate and
improved
version
of
the
derivation
of
the
conservation
laws
given
in
Weyl
1918b
(pp.
475476),
using
the variational methods
of
Weyl
1917 and Klein,
F.
1917
(cited
in
Weyl
1918b,
p. 476),
which had meanwhile been
perfected
in
Klein, F.
1918a
(cited
in
Weyl
1919c, p.
114).
Weyl saw
two
advantages
of
his
new theory over
general relativity
for
understanding
the conservation laws
(Weyl
1919c,
pp.
120121). First,
charge
conservation follows
from
gauge
invariance
of
the
action
integral
in the
same way
that
energy
momentum conservation follows
from
invariance
under
general
coordinate transformations.
The
second
advantage
is the
one
alluded
to
in
this
document, namely
that the relation between the
conser
vation laws associated with the latter invariance and the
physical
laws
of
energymomentum conser
vation is
more perspicuous
in
the
new theory.
For
a
concise discussion
of
conservation laws in
Weyl’s
theory, see
Pauli
1921,
sec.
65
(d).
[13]The
first two terms
give
,Jg
times the covariant
divergence
tki;k,
where
tki
is the
energy
momentum
tensor for matter
consisting
of
charged particles;
the third term
gives
the force that these
particles experience
from the
electromagnetic
field
Fuv.
[14]The
argument showing
that this choice for
tki
is
incompatible
with
gauge
invariance
can
be
found in
Weyl
1919d,
p.
256
(see
also
Weyl
1921a,
sec. 36).
[15]In
Doc.
579,
Einstein
pointed
out
that
as long as one assumes
that the
trajectories
of
uncharged
particles are geodesics, Weyl’s
theory predicts
that the
electromagnetic
fourvector
potential
exerts
an
influence
on
such
particles.
[16]The
results
of
these
investigations
were
first
published
in
Weyl
1919c, pp.
124128,
in
a
sub
section entitled
"Mechanik,"
and
can
also be found in
Weyl
1919d,
sec.
35,
as
well
as
in
subsequent
editions
of
this book. For
discussion,
see
Doc.
657, note
3.
[17]The
results
of
these
investigations were published
in
Weyl
1919c,
pp.
128132, in
a
subsection
entitled
"Das
Problem
der Materie."
[18]See
Arnold
Sommerfeld to
Hermann
Weyl,
7
July
1918,
SzZE
Bibliothek,
Hs. 91:751.
Sommerfeld
had
high
praise
for
Weyl
1918b: "What
you are saying
there is
really
wonderful.
Just
as
Mie had
grafted
a gravitational
theory onto
his fundamental
theory
of
electrodynamics
that did
not
form
an organic
whole with
it
... Einstein
grafted a
theory
of
electrodynamics (i.e.,
the usual elec
trodynamics)
onto his fundamental
theory
of
gravitation
that had little
to
do with it.
You
have worked
out
a
true
unity"
("Was
Sie da
sagen,
ist wirklich wundervoll. So wie Mie seiner
konsequenten
Elek
trodynamik
eine Gravitation
angeklebt
hatte,
die nicht
organisch
mit
jener zusammenhing
...
ebenso
hat
Einstein
seiner
konsequenten
Gravitation eine
Elektrodynamik
(d.
h. die
gewöhnliche
Elektro
dynamik) angeklebt,
die mit
jener
nicht viel
zu
tun
hatte. Sie stellen eine
wi[r]kliche
Einheit
her.").
Sommerfeld also
praised Weyl’s theory
in
a
letter
to Einstein,
as can
be inferred from the latter’s
reply
(Doc.
592).
[19]Sommerfeld had
probably
referred
to
work done
by
Charles St.
John
(see
St.
John 1917 and the
discussion
of
his results
in Evershed
1918). Although
these
publications give an
account
of
findings
on
solar
rather
than stellar redshift,
St. John
discusses both
in the
introductory chapter.