Einstein became interested in Kaluza’s theory again due to O. Klein’s paper concerning a
relation between “quantum theory and relativity in five dimensions”
(see Klein 1926 [184], received by the
journal on 28 April 1926). Einstein wrote to his friend and
colleague Paul Ehrenfest on 23 August 1926: “Subject Kaluza, Schroedinger, general
relativity”, and, again on 3 September 1926: “Klein’s paper is
beautiful and impressive, but I find Kaluza’s principle too unnatural.”
However, less than half a year later he had completely reversed his
opinion:
“It appears that the union of gravitation and Maxwell’s theory is achieved in a completely satisfactory way by the five-dimensional theory (Kaluza-Klein-Fock).” (Einstein to H. A. Lorentz, 16 February 1927)
On the next day (17 February 1927), and ten days
later Einstein was to give papers of his
own in front of the Prussian Academy in which he pointed out the
gauge-group, wrote down the geodesic equation, and derived exactly
the Einstein-Maxwell equations - not
just in first order as Kaluza had done [81, 82]. He
came too late: Klein had already shown the same before [184]. Einstein himself acknowledged
indirectly that his two notes in the report of the Berlin Academy
did not contain any new material. In his second communication, he
added a postscript:
“Mr. Mandel brings to my attention that
the results reported by me here are not new. The entire content can
be found in the paper by O. Klein.” 132
He then referred to the papers of
Klein [184, 185] and to
“Fochs Arbeit” which is a paper by Fock
1926 [130
], submitted three
months later than Klein’s paper. That Klein had published another
important clarifying note in Nature,
in which he closed the fifth dimension, seems to have escaped Einstein133 [183
]. Unlike in his
paper with Grommer, but as in Klein’s, Einstein, in his notes, applied
the “sharpened cylinder condition”, i.e., dropped the scalar field. Thus, the
three of them had no chance to find out that Kaluza had made a mistake: For
, even in first approximation the new
field will appear in the four-dimensional Einstein-Maxwell equations ([145], p. 5). Mandel of Leningrad was not given
credit by Einstein although he also had
rediscovered by a different method some of O. Klein’s results [215
]. In a footnoote, Mandel stated that he had learned
of Kaluza’s (whom he spelled
“Kalusa”) paper only through Klein’s article. He started by
embedding space-time as a hypersurface
into
, and derived the field equations in space-time by
assuming that the five-dimensional curvature tensor vanishes; by
this procedure he obtained also a matter-energy tensor “closely
linked to the second fundamental form of this hypersurface”. From
the geodesics in
he derived the equations of motion of a
charged point particle. One of the two additional terms appearing
besides the Lorentz force could be removed by a weakness
assumption; as to the second, Mandel opinioned
“that the experimental discovery of the second
term appears difficult, yet perhaps not entirely impossible.”
([215], p. 145)
As to Fock’s paper, it is remarkable
because it contains, in nuce, the
coupling of the Schrödinger wave function and the electromagnetic potential by the gauge
transformation
, where
is Planck’s constant and
“a new parameter with the
unit of the quantum of action” [130
]. In Fock’s words:
“The importance of the additional coordinate
parameter seems to lie in the fact that it causes
the invariance of the equations [i.e.,
the relativistic wave equations] with respect to addition of an
arbitrary gradient to the 4-potential.” 134
([130
], p. 228)
Fock derived the general
relativistic wave equation and the equations of motion of a charged
point particle; the latter is identified with the null geodesics of
. Neither Mandel nor Fock used the “sharpened cylinder
condition” (110
).
A main motivation for Klein was to relate the fifth dimension with quantum physics. From a postulated five-dimensional wave equation
and by neglecting the gravitational field, he arrived at the four-dimensional Schrödinger equation after insertion of the quantum mechanical differential operatorsIn his papers, Einstein took over Klein’s
condition , which removed the additional scalar
field admitted by the theory. It was Reichenbächer who apparently first
tried to perform the projection into space-time of the most general
five-dimensional metric, and without using the cylinder
condition (109
):
“Now, a rather laborious calculation of the
five-dimensional curvature quantities in terms of a
four-dimensional submanifold contained in it has shown to me also
in the general case (, dependence of the components of the fundamental
[tensor] of
is admitted) that the c h a r a c t e r i s t i
c properties of the field equations are then conserved as well,
i.e., they keep the form
Here, in nuce, is already contained what more than a decade later Einstein and Bergmann worked out in detail [102].
It is likely that Reichenbächer had been led to this
excursion into five-dimensional space, an idea which he had
rejected before as unphysical, because his attempt to build a
unified field theory in space-time through the ansatz for the
metric with
the electromagnetic 4-potential, had failed. Beyond
incredibly complicated field equations nothing much had been
gained [274]. Reichenbächer’s ansatz is well
founded: As we have seen in Section 4.2, due to the violation of
covariance in
,
transforms as a tensor under
the reduced covariance group.
Even L. de Broglie became interested in Kaluza’s “bold but very beautiful theory” and rederived Klein’s results his way [46], but not without getting into a squabble with Klein, who felt misunderstood [187, 47]. He also suggested that one should not accept the cylinder condition, a suggestion looked into by Darrieus who introduced an electrical 5-potential and 5-current, and deduced Maxwell’s equations from the five-dimensional homogeneous wave equation and the five-dimensional equation of continuity [43].
In 1929 Mandel tried to “axiomatise” the
five-dimensional theory: His two axioms were the cylinder
condition (109) and its sharpening,
Equation (110
). He then weakened the
second assumption by assuming that “an objective meaning does not
rest in the
proper, but only in their quotients”,
an idea he ascribed to O. Klein and Einstein. He then discussed
conformally invariant field equations, and tried to relate them to
equations of wave mechanics [219].
Klein’s lure lasted for some years. In 1930,
N. R. Sen claimed to have investigated the
“Kepler-problem for the five-dimensional wave equation of Klein”.
What he did was to calculate the energy levels of the hydrogen atom
(as a one particle-system) with the general relativistic wave
equation in space-time (148) with
where
is
the metric on space-time following from the 5-metric
by
. For
he took the Reissner-Nordström solution and did not
obtain a discrete spectrum [323]. He continued
his approach by trying to solve Schrödinger’s wave
equation [324]
Presently, the different contributions of Kaluza and O. Klein are lumped together by most physicists into what is called “Kaluza-Klein theory”. An early criticism of this unhistorical attitude has been voiced in [209].
Four years later, Einstein returned to Kaluza’s idea. Perhaps, he had
since absorbed Mandel’s ideas which included a
projection formalism from the five-dimensional space to
space-time [215, 216, 217, 218].
In a paper with his assistant Mayer, Einstein now presented Kaluza’s approach in the form of
an implicit projective four-dimensional theory, although he did not
mention the word “projective” [107]:
“Psychologically, the theory presented here
connects to Kaluza’s well-known theory;
however, it avoids extending the physical continuum to one of five
dimensions.” 137
In the eyes of Einstein, by avoiding the
artificial cylinder condition (109), the new method
removed a serious objection to Kaluza’s theory.
Another motivation is also put forward: The
linearity of Maxwell’s equations “may not correspond to reality”;
thus, for strong electromagnetic fields, Einstein expected deviations from
Maxwell’s equations. After a listing of all the shortcomings of Kaluza’s theory, the new approach
is introduced: At every event a five-dimensional vector space is affixed to space-time
, and “mixed”
tensors
are defined linking the tangent space
of space-time
with a
such that
Einstein and Mayer introduced what they called “Fünferkrümmung” (5-curvature) via the three-index symbol given above by
It is related to the Riemannian curvatureIt turns out that .
The field equations put forward in the paper by Einstein and Mayer now are
and turn out to be exactly the Einstein-Maxwell vacuum field equations. Thus, by another formalism, Einstein and Mayer rederived what Klein had obtained in his first paper on Kaluza’s theory [184The authors’ conclusion is:
“From the theory presented here, the equations
for the gravitational and the electromagnetic fields follow
effortlessly by a unifying method; however, up to now, [the theory]
does not bring any understanding for the way corpuscles are built,
nor for the facts comprised by quantum theory.” 141
([107
], p. 19)
After this paper Einstein wrote to Ehrenfest in a
letter of 17 September 1931 that this theory “in my opinion
definitively solves the problem in the macroscopic domain” ([240], p. 333).
Also, in a lecture given on 14 October 1931 in the Physics
Institute of the University of Wien, he still was proud of the
5-vector approach. In talking about the failed endeavours to
reconcile classical field theory and quantum theory (“a cemetery of
buried hopes”) he is reported to have said:
“Since 1928 I also tried to find a bridge, yet
left that road again. However, following an idea half of which came
from myself and half from my collaborator, Prof. Dr. Mayer, a startlingly simple
construction became successful. [...] According to my and Mayer’s opinion, the fifth
dimension will not show up. [...] according to which relationships
between a hypothetical five-dimensional space and the
four-dimensional can be obtained. In this way, we succeeded to
recognise the gravitational and electromagnetic fields as a logical
unity.” 142
[96]
In his letter to Besso of 30 October 1931, Einstein seemed intrigued by the mathematics used in his paper with Mayer, but not enthusiastic about the physical content of this projective formulation of Kaluza’s unitary field theory:
“The only result of our investigation is the
unification of gravitation and electricity, whereby the equations
for the latter are just Maxwell’s equations for empty space. Hence,
no physical progress is made, [if at all] at most only in the sense
that one can see that Maxwell’s equations are not just first
approximations but appear on as good a rational foundation as the
gravitational equations of empty space. Electrical and mass-density
are non-existent; here, splendour ends; perhaps this already
belongs to the quantum problem, which up to now is unattainable
from the point of view of field [theory] (in the same way as
relativity is from the point of view of quantum mechanics). The
witty point is the introduction of 5-vectors in fourdimensional space, which are bound to space
by a linear mechanism. Let
be the 4-vector belonging to
; then such a relation
obtains. In
the theory equations are meaningful which hold independently of the
special relationship generated by
. Infinitesimal
transport of
in fourdimensional space is defined,
likewise the corresponding 5-curvature from which spring the field
equations.” 143
([327
],
pp. 274-275)
In his report for the Macy-Foundation, which appeared in Science on the very same day in October 1931, Einstein had to be more optimistic:
“This theory does not yet contain the
conclusions of the quantum theory. It furnishes, however, clues to
a natural development, from which we may anticipate further
developments in this direction. In any event, the results thus far
obtained represent a definite advance in knowledge of the structure
of physical space.” ([94], p. 439)
Unfortunately, as in the case of his previous
papers on Kaluza’s theory, Einstein came in only second: Veblen had already worked on
projective geometry and projective connections for a couple of
years [374, 376, 375]. One year
prior to Einstein’s and Mayer’s publication, with his
student Hoffmann
, he had suggested
an application to physics equivalent to the Kaluza-Klein theory [381, 162
]. However, according
to Pauli, Veblen and Hoffmann had spoiled the advantage
of projective theory:
“But these authors choose a formulation that,
due to an unnecessary specialisation of the coordinate system,
prefers the fifth coordinate relative to the remaining
[coordinates] in much the same way as this had happened in Kaluza-Klein theory by means of
the cylinder condition [...].” 144
([248
], p. 307)
By using the idea that an affine ()-space can be represented by a projective
-space [413], Veblen and Hoffmann avoided the five
dimensions of Kaluza: There is a one-to-one
correspondence between the points of space-time and a certain
congruence of curves in a five-dimensional space for which the
fifth coordinate is the curves’ parameter, while the coordinates of
space-time are fixed. The five-dimensional space is just a
mathematical device to represent the events (points) of space-time
by these curves. Geometrically, the theory of Veblen and Hoffmann is more transparent and
also more general than Einstein and Mayer’s: It can house the
additional scalar field inherent in Kaluza’s original approach. Thus,
Veblen and Hoffmann also gained the
Klein-Gordon equation in curved space, i.e., an equation with the Ricci scalar
appearing besides its mass term. Interestingly, the
curvature term reads as
([381], p. 821). In his
note, Hoffmann generalised the formalism
such as to include Dirac’s equations (without gravitation),
although some technical difficulties remained. Nevertheless,
Hoffman remained optimistic:
“There is thus a possibility that the complete system will constitute an improved unification within the relativity theory of the gravitational, electromagnetic and quantum aspects of the field.” ([162], p. 89)
In his book, Veblen emphasised
“[...] that our theory starts from a physical
and geometrical point of view totally different from KALUZA’s. In
particular, we do not demand a relationship between electrical
charge and a fifth coordinate; our theory is strictly
four-dimensional.” 145
[379]
Shortly after Einstein’s and Mayer’s paper had appeared, Schouten and van Dantzig also proved that the 5-vector formalism of this paper can be brought into a projective form [314].
In a second note, Einstein and Mayer extended the
5-vector-formalism to include Maxwell’s equations with a non-vanishing current density [109]. Of the three basic
assumptions of the previous paper, the second had to be given up.
The expression in the middle of Equation (153) is replaced by
In the last paragraph, the compatibility of the equations was proven, and at the end Cartan was acknowledged:
“We note that Mr. Cartan, in a general and very
illuminating investigation, has analysed more deeply the property
of systems of differential equations that has been termed by us
‘compatibility’ in this paper and in previous papers.” 146
[37
]
At about the same time as Einstein and Mayer wrote their second note, van Dantzig continued his work on
projective geometry [361, 362, 360].
He used homogeneous coordinates , with
, and the invariant
, and
introduced projectors and covariant differentiation
(cf. Section 2.1.3).
Together with him, Schouten wrote a series of papers
on projective geometry as the basis of a unified field
theory [293
, 316
, 315
, 317
]147
, which, according to
Pauli, combine
“all advantages of the formulations of Kaluza-Klein and Einstein-Mayer while avoiding all their
disadvantages.” ([248], p. 307)
Both the Einstein-Mayer theory and Veblen and Hoffmann’s approach turned out to be subcases of the more general scheme of Schouten and van Dantzig intending
“to give a unification of general relativity
not only with Maxwell’s electromagnetic theory but also with
Schrödinger’s and Dirac’s theory of material waves.” ([317], p. 271)
In this paper ([317], p. 311, Figure 2), we find an early graphical representation of the parametrised set of all possible theories of a kind148 . The formalism of Schouten and van Dantzig allows for taking the additional dimension to be timelike; in their physical applications the metric of space- time is taken as a Lorentz metric; torsion is also included in their geometry.
Pauli, with his student
J. Solomon149
, generalised Klein,
and Einstein and Mayer by allowing for an arbitrary
signature in an investigation concerning “the form that take
Dirac’s equations in the unitary theory of Einstein and Mayer” 150 [252
]. In a note added
after proofreading, the authors showed that they had noted Schouten and Dantzig’s
papers [293, 316
]. The authors
pointed out that
“[...] even in the absence of gravitation we
must pay attention to a difference between Dirac’s equation in the
theory of Einstein and Mayer, and Dirac’s equation as it
is written out, usually.” 151
([252
], p. 458)
The second order wave equation iterated from
their form of Dirac’s equation, besides the spin term contained a
curvature term , with the numerical factor different
from Veblen’s and Hoffmann’s. In a sequel to this
publication, Pauli and Solomon corrected an
error:
“We examine from a general point of view the
theory of spinors in a five-dimensional space. Then we discuss the
form of the energy-momentum tensor and of the current vector in the
theory of Einstein-Mayer.[...] Unfortunately, it
turned out that the considerations of §in the first part are marred by a
calculational error…This has made it necessary to introduce a new
expression for the energy-momentum tensor and [...] likewise for
the current vector [...].” 152
([253], p. 582)
In the California Institute of Technology, Einstein’s and Mayer’s new mathematical technique found an attentive reader as well; A. D. Michal and his co-author generalised the Einstein-Mayer 5-vector-formalism:
“The geometry considered by Einstein and Mayer in their ‘Unified field
theory’ leads to the consideration of an -dimensional Riemannian space
with a metric tensor
, to each point of
which is associated an
-dimensional linear vector
space
, (
), for which vector spaces a
general linear connection is defined. For the general case (
) we find that the calculation of the
‘exceptional directions’ is not unique, and that an
additional postulate on the linear connection is necessary. Several
of the new theorems give new results even for
,
, the Einstein-Mayer case.” [227]
Michal had come from Cartan and Schouten’s papers on group manifolds and the distant parallelisms defined on them [226]. H. P. Robertson found a new way of applying distant parallelism: He studied groups of motion admitted by such spaces, e.g., by Einstein’s and Mayer’s spherically symmetric exact solution [281] (cf. Section 6.4.3).
Cartan wrote a paper on the Einstein-Mayer theory as well ([29], an article
published only posthumously) in which he showed that this could be
interpreted as a five-dimensional flat geometry with torsion, in
which space-time is embedded as a totally geodesic subspace.