The third main idea that emerged was Eddington’s suggestion to forego
the metric as a fundamental concept and start right away with a
(general) connection, which he then restricted to a symmetric one
in order to avoid an “infinitely crinkled”
world [58
]. His motivation
went beyond the unification of gravitation and electromagnetism:
“In passing beyond Euclidean geometry,
gravitation makes its appearance; in passing beyond Riemannian
geometry, electromagnetic force appears; what remains to be gained
by further generalisation? Clearly, the non-Maxwellian binding
forces which hold together an electron. But the problem of the
electron must be difficult, and I cannot say whether the present
generalisation succeeds in providing the material for its solution”
([58], p. 104)
In the first, shorter, part of two, Eddington describes affine
geometry; in the second he relates mathematical objects to physical
variables. He distinguishes the affine geometry as the “geometry of
the world-structure” from Riemannian geometry as “the natural
geometry of the world”. He starts by calculating both the curvature
and Ricci tensors from the symmetric connection according to
Equation (39). The Ricci tensor
is asymmetric87
,
“The divergence of will vanish
identically if
is itself the divergence of any
antisymmetrical contravariant tensor.” ([64
], p. 223;
cf. also [58
], p. 113)
Now, by Equation (25),
Eddington introduces the metrical tensor by the definition
“introducing a universal constant“Our gauging-equation is therefore certainly
true wherever light is propagated, i.e., everywhere inside the electron. Who
shall say what is the ordinary gauge inside the electron?” ([58], p. 114)
While this remark certainly is true, there is no
guarantee in Eddington’s approach that thus defined is a
Lorentzian metric, i.e., that it could
describe light propagation at all. Only connections leading to a
Lorentz metric can be used if a physical interpretation is wanted.
Note also, that the interpretation of
as the metric
implies that
We must read Equation (118) as giving
if the only basic variable in affine geometry, i.e., the connection
, has been determined by help of some field
equations. Thus, in general,
is not metric-compatible; in
order to make it such, we are led to the differential equations
for
, an equation not considered by Eddington. In the absence of an
electromagnetic field, Equation (118
) looks like Einstein’s vacuum field equation
with cosmological constant. In principle, now a fictitious
“Riemannian” connection (the Christoffel symbol) can be written
down which, however, is a horribly complicated function of the
affine connection - as the only fundamental geometrical quantity
available. This is due to the expression for the inverse of the
metric, a function cubic in
. Eddington’s affine theory thus can
also be seen as a bi-connection theory. Note also that Eddington does not explicitly say
how to obtain the contravariant form of the electromagnetic field
from
; we must assume that he thought of
raising indices with the complicated inverse metric tensor.
In connection with cosmological considerations,
Eddington cherished the -term in Equation (118
):
“I would as soon think of reverting to Newtonian theory as of dropping the cosmic constant.” ([63], p. 35)
Now, Eddington was able to identify the
energy-momentum tensor of the electromagnetic field
by decomposing the Ricci tensor
formed from
Equation (51
) into a metric part
and the rest. The energy-momentum tensor
of the electromagnetic field is then defined by Einstein’s field equations with a
fictitious cosmological constant
.
Although Eddington’s interest did not rest
on finding a proper set of field equations, he nevertheless
discussed the Lagrangian
, and showed that a variation with regard to
did not lead to an acceptable field equation.
Eddington’s main goal in this paper was to include matter as an inherent geometrical structure:
“What we have sought is not the geometry of
actual space and time, but the geometry of the world-structure
which is the common basis of space and time and things.” ([58], p. 121)
By “things” he meant
His aim was reached in the sense that all three quantities were fixed entirely by the connection; they could no longer be given from the outside. As to the question of the electron, it is seen as “a region of abnormal world-curvature”, i.e., of abnormally large curvature.
While Pauli liked Eddington’s distinction between “natural geometry” and “world geometry” - with the latter being only “a graphical representation” of reality - he was not sure at all whether “a point of view could be taken from which the gravitational and electromagnetical fields appear as union”. If so, then it must be a purely phenomenological one without any recourse to the nature of the charged elementary particles (cf. his letter to Eddington quoted below).
Lorentz did not like the large number of variables in Eddington’s theory; there were 4 components of the electromagnetic potential, 10 components of the metric and 40 components of the connection:
“It may well be asked whether after all it would not be preferable simply to introduce the functions that are necessary for characterising the electromagnetic and gravitational fields, without encumbering the theory with so great a number of superfluous quantities.” ([210], p. 382)
Eddington’s publication early in
1921, generalising Einstein’s and Weyl’s theories started a new
direction of research both in physics and mathematics. At first, Einstein seems to have been
reserved (cf. his letters to Weyl in June and September 1921
quoted by Stachel in his article on Eddington and Einstein ([330], pp. 453-475;
here p. 466)), but one and a half years later he became
attracted by Eddington’s idea. To Bohr, Einstein wrote from Singapore on
11 January 1923:
“I believe I have finally understood the connection between electricity and gravitation. Eddington has come closer to the truth than Weyl.” ([139], p. 274)
He now tried to make Eddington’s theory work as a physical theory; Eddington had not given field equations:
“I must absolutely publish since Eddington’s idea must be thought
through to the end.” (letter of Einstein to Weyl of 23 May 1923;
cf. [240], p. 343)
And a few days later, he was still intrigued about this sort of unified field theory, in particular about its elusiveness:
“[...] Over it lingers the marble smile of
inexorable nature, which has bestowed on us more longing than
brains.” 88
(letter of Einstein to Weyl of 26 May 1923; cf.[240
], p. 343)
And indeed Einstein published fast, even
while still on the steamer returning from Japan through Palestine
and Spain: The paper of February 1923 in the reports of the Berlin
Academy carries, as location of the sender, the ship “Haruna Maru”
of the Japanese Nippon Yushen Kaisha line89
[77].
“In past years, the wish to understand the
gravitational and electromagnetic field as one in essence has
dominated the endeavours of theoreticians. [...] From a purely
logical point of view only the connection should be used as a
fundamental quantity, and the metric as a quantity derived thereof
[...] Eddington has done this.” 90
([77
], p. 32)
Like Eddington, Einstein used a symmetric connection and wrote down the equation91
whereFor a Lagrangian, Einstein used ; he claims that for vanishing
electromagnetic field the vacuum field equations of general
relativity, with the cosmological term included, hold. Einstein varied with regard to
and
, not, as one might have expected, with
regard to the connection
. If
, then the electric current density
is defined by
is interpreted as “the contravariant tensor of the
electromagnetic field”.
The field equations are obtained from the
Lagrangian by variation with regard to the connection and are (Einstein worked in space-time)
If no electromagnetic field is present, reduces to
; the definition of the metric
in Equation (119
) is reinterpreted by
Einstein as giving his vacuum
field equation with cosmological constant
. In order that this makes sense, the identifications
in Equation (119
) are always to be made
after the variation of the Lagrangian
is performed.
For non-vanishing electromagnetic field, due to
Equation (124) the Equation (120
) now becomes
“But the extraordinary smallness of implies that finite
are possible only
for tiny, almost vanishing current density. Except for singular
positions, the current density is practically vanishing.” 92
Einstein went on to show that
Maxwell’s vacuum equations are holding in first order
approximation. Up to the same order, . In
general however,
Also, the geometrical theory presented here is
energetically closed, i.e., the
current density
cannot be given arbitrarily as in the
usual Maxwell theory with external sources.
Einstein was not sure whether
“electrical elementary elements”, i.e., nonsingular electrons, are possible in
this theory; they might be. He found it remarkable “[...] that,
according to this theory, positive and negative electricity cannot
differ just in sign” 93
([77
], p. 38). His
final conclusion was:
“that EDDINGTON’S general idea in context with
the Hamiltonian principle leads to a theory almost free of
ambiguities; it does justice to our present knowledge about
gravitation and electricity and unifies both kinds of fields in a
truly accomplished manner.” 94
([77
], p. 38)
Until the end of May 1923, two further
publications followed in which Einstein elaborated on the theory.
In the second paper, he exchanged the Lagrangian for a new one, i.e., for
where
.
is to be varied with respect to
and
. The resulting equations for the
gravitational and electromagnetic fields are the symmetric and
skew-symmetric part, respectively, of
Although the theory offered, for every solution with positive charge, also a solution with negative charge, the masses in the two cases were the same. However, the only known particle with positive charge at the time (what is now called the proton) had a mass greatly different from the particle with negative charge, the electron. Einstein noted:
“Therefore, the theory may not account for the
difference in mass of positive and negative electrons.” 95
([74
], p. 77)
In the third paper [76], apart from
changing notations96
, Einstein set
. He also dropped the assumption (119
) and replaced it by
allowing his Lagrangian (Hamiltonian)
to be a function of
the two independent variables,
After a field rescaling, he then took a third expression to become his Lagrangian
While, in the meantime, mathematicians had taken over the conceptual development of affine theory, some other physicists, including the perpetual pièce de resistance Pauli, kept a negative attitude:
“[...] I now do not at all believe that the
problem of elementary particles can be solved by any theory
applying the concept of continuously varying field strengths which
satisfy certain differential equations to regions in the interior
of elementary particles. [...] The quantities cannot be measured directly, but must be obtained
from the directly measured quantities by complicated calculational
operations. Nobody can determine empirically an affine connection
for vectors at neighbouring points if he has not obtained the line
element before. Therefore, unlike you and Einstein, I deem the
mathematician’s discovery of the possibility to found a geometry on
an affine connection without a metric as meaningless for physics,
in the first place.” 99
(Pauli to Eddington on 20 September 1923;
[250
],
pp. 115-119)
Also Weyl, in the 5th edition of Raum-Zeit-Materie ([398], Appendix 4), in
discussing “world-geometric extensions of Einstein’s theory”, found Eddington’s theory not convincing.
He criticised a theory that keeps only the connection as a
fundamental building block for its lack of a guarantee that it
would also house the conformal
structure (light cone structure). This is needed for special
relativity to be incorporated in some sense, and thus must be an
independent fundamental input [405].
Likewise, Eddington himself did not appreciate much Einstein’s followership. In Note 14, § 100 appended to the second edition of his book, he laid out Einstein’s theory but not without first having warned the reader:
“The theory is intensely formal as indeed all such action-theories must be, and I cannot avoid the suspicion that the mathematical elegance is obtained by a short cut which does not lead along the direct route of real physical progress. From a recent conversation with Einstein I learn that he is of much the same opinion.” ([64], pp. 257-261)
In fact, when Eddington’s book was translated
into German in 1925 [60], Einstein wrote an appendix to it
in which he repeated, with minor changes, the results of his last
paper on the affine theory. His outlook on the state of the theory
now was rather bleak:
“For me, the final result of this consideration
regrettably consists in the impression that the deepening of the
geometrical foundations by Weyl-Eddington is unable to bring
progress for our physical understanding; hopefully, future
developments will show that this pessimistic opinion has been
unjustified.” 100
([60
], p. 371)
An echo of this can be found in Einstein’s letter to Besso of 5 June 1925:
“I am firmly convinced that the entire chain of
thought Weyl-Eddington-Schouten does not lead to
something useful in physics, and I now have found another,
physically better founded approach. To me, the quantum-problem
seems to require something like a special scalar, for the
introduction of which I have found a plausible way.” 101
([327
], p. 204)
This remark shows that Einstein must have taken some notice of Schouten’s work in affine geometry. What the “special scalar” was, remains an open question.
Einstein spent much time in thinking about the “quantum problem”, as he confessed to Born:
“I do not believe that the theory will be able
to dispense with the continuum. But I fail to succeed in giving my
pet idea a tangible form: to understand the quantum-structure
through an overdetermination by differential equations.” 102
([103
],
pp. 48-49)
In a paper from December 1923, Einstein not only stated clearly the necessary conditions for a unified field theory to be acceptable to him, but also expressed his hope that this technique of “overdetermination” of systems of differential equations could solve the “quantum problem”.
“According to the theories known until now the
initial state of a system may be chosen freely; the differential
equations then give the evolution in time. From our knowledge about
quantum states, in particular as it developed in the wake of Bohr’s
theory during the past decade, this characteristic feature of
theory does not correspond to reality. The initial state of an
electron moving around a hydrogen nucleus cannot be chosen freely;
its choice must correspond to the quantum conditions. In general:
not only the evolution in time but also the initial state obey
laws.” 103
([75],
pp. 360-361)
He then ventured the hope that a system of overdetermined differential equations is able to determine
“also the mechanical behaviour of singular points (electrons) in such a way that the initial states of the field and of the singular points are subjected to constraints as well. [...] If it is possible at all to solve the quantum problem by differential equations, we may hope to reach the goal in this direction.”
We note here Einstein’s emphasis on the very special problem of the quantum nature of elementary particles like the electron, as compared to the general problem of embedding matter fields into a geometrical setting.
One of the crucial tests for an acceptable unified field theory for him now was:
“The system of differential equations to be
found, and which overdetermines the field, in any case must admit
this static, spherically symmetric solution which describes,
respectively, the positive and negative electron according to the
equations given above [i.e the Einstein-Maxwell equations].”
104
This attitude can also be found in a letter to M. Besso from 5 January 1924:
“The idea I am wrestling with concerns the
understanding of the quantum facts; it is: overdetermination of the
laws by more field equations than field variables. In such a way,
the un-ambiguity of the initial conditions ought to be understood
without leaving field theory. [...] The equations of motion of
material points (electrons) will be given up totally; their motion
ought to be co-determined by the field laws.” 105
([327
], p. 197)
In his answer, Besso asked for more information concerning the quantum aspect of the concept of “overdetermination”, because:
“On the one hand, this seems to be connected
only formally with a field theory; on the other, it has not yet
dawned on me how in this manner something corresponding to the
discrete quantum orbits may be reached.” 106
([327
], p. 199)