D O C U M E N T 3 0 5 N O V E M B E R 1 9 2 1 3 5 9
ALS. [14 248].
[1]In Kaluza 1921 velocities are written with their indices in superscript. In his letter, super- and
subscripts are used interchangeably.
[2]The geodesic equation in five dimensions (with the fifth dimension, Latin indices running
from 0 to 4, ds the five dimensional line element, and )
(1)
can be reduced under the approximation , to the familiar equation of motion
for a charged particle:
; (2)
is the force acting on the particle in four dimensions, is the energy-momentum tensor, and
is the charged current; and are the usual gravitational and electromagnetic field terms;
Greek indices run from 1 to 4. It follows in this approximation that the specific charge is small,
whereas that is not the case for an electron. For the actual electron value of the specific charge,
would be so large that the interaction with the unknown -field would dominate in the particle’s
equation of motion (see also Doc. 270 and Kaluza 1921, pp. 969–971).
[3]With the above approximation (see note 2) and a linearized metric, Kaluza found for the charge
density :
, (3)
with the mass density and ( is the gravitational constant). When deriving equation
(2) from (1), Kaluza came across the following expression for the equation of motion:
; and d are the four-velocity and the line-element
in four dimensions, respectively (Kaluza 1921, pp. 969–971). The ratio of the magnitudes of the elec-
trodynamic and force terms is given by . For a charge (n an integer and e
the elementary charge) with a mass of g follows that the ratio is approximately , using (3)
and geometric units.
[4]
“ ”
should be
“ ”
.
[5]Felix Ehrenhaft claimed to have observed charges that were a fraction of the elementary charge
(in 1915 his smallest charge was esu, one twelfth of the elementary charge) on mercury
drops with radii down to the order of cm (Ehrenhaft 1914, 1915). Robert A. Millikan, however,
found that in his oil drop experiments (with drops with a radius of the order of cm) an exact
multiple of the elementary charge always occurred (Millikan 1913); for a historical account, see Hol-
ton 1978.
[6]Absent in Kaluza 1921.
[7]At this point in the text Kaluza indicates a phrase he has appended at the foot of the page: “Für
nicht allzu stürmisch bewegte Materie.”
[8]A similar argument was made in Kaluza 1921 on pp. 971–972.
[9]Weyl stated that “[it] is … very unlikely that the Einstein gravitational equations are strictly cor-
rect, particularly since the gravitational constant contained in them is quite out of place with respect
to the other natural constants, so that the gravitational radius of the charge and mass of an electron,
for example, is of a completely different order of magnitude (namely respectively times
smaller) than the radius of the electron itself” (“[es] ist … sehr unwahrscheinlich, daß die Einstein-
schen Gravitationsgleichungen streng richtig sind, vor allem deshalb, weil die in ihnen vorkommende
Gravitationskonstante ganz aus dem Rahmen der übrigen Naturkonstanten herausfällt, so daß der
Gravitationsradius der Ladung und Masse eines Elektrons z.B. von völlig anderer Größenordnung
(nämlich bezw. mal so klein) ist wie der Radius des Elektrons selber” [Weyl 1918a,
p. 477]).
[10]Wolfgang Pauli argued that the differential equations for a static field in Weyl’s theory
(Weyl 1918a) are invariant under time reversal. Under such a transformation the electric field changes
x0
ul
dxl
ds
------- =
ul
·
rs
l urus =
u0 u1 u2 u3 1« u4 1
T F I + =
T
I F
u0
g00
0
0
2
0
u0 =
0
2 =
dv
d
-------- v v + = 2 F u0v
1g00
2
-- - u0 2 – v
g00 2
1
2
--u0 -
0
ne =
10 6– 54 n
dxv
ds
------- -
dx0
ds
--------
2 10–11
10
6–
10
4–
1020 1040
1020 1040