4.3 Eddington’s affine theory

4.3.1 Eddington’s paper

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 Γ in order to avoid an “infinitely crinkled” world [58Jump To The Next Citation Point]. 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” ([58Jump To The Next Citation Point], 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 (39View Equation). The Ricci tensor Kij (Γ ) := ∗ Gij is asymmetric87,

∗Gkl = Rkl + Fkl, (114 )
with Rkl (Γ ) being the symmetric and Fkl(Γ ) the antisymmetric part. According to Equation (31View Equation) F kl derives from a “vector potential”, i.e., F = ∂ Γ − ∂ Γ kl k l l k with Γ := Γ r l lr, such that an immediate physical identification of Fkl with the electromagnetic field tensor is at hand. With half of Maxwell’s equations being satisfied automatically, the other half is used to define the electric charge current jk by jl := F lk∥k. By this, Eddington claims to guarantee charge conservation:

“The divergence of jk will vanish identically if jk is itself the divergence of any antisymmetrical contravariant tensor.” ([64Jump To The Next Citation Point], p. 223; cf. also [58Jump To The Next Citation Point], p. 113)

Now, by Equation (25View Equation),

F lk∥[l∥k] = K [rk]F rk + Sjks∇sF jk. (115 )
For a symmetric connection thus, unlike in Riemannian geometry,
k lk rk j ∥k = F ∥[l∥k] = FrkF ⁄= 0. (116 )
However, for a tensor density, due to Equation (16View Equation) we obtain
1 ˆjk∥k = ˆFlk∥[l∥k] = -(Vrk + 2K [rk])F ˆrk + Sjks∇s Fˆjk, (117 ) 2
and thus for a torsionless connection (cf. Equation (38View Equation)) jk∥k = 0.

Eddington introduces the metrical tensor by the definition

λg = R , (118 ) kl kl
“introducing a universal constant λ, for convenience, in order to remain free to use the centimetre instead of the natural unit of length”. This is called “Einstein’s gauge” by Eddington; he is delighted that

“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?” ([58Jump To The Next Citation Point], p. 114)

While this remark certainly is true, there is no guarantee in Eddington’s approach that gkl 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 Rkl as the metric implies that detR ⁄= 0. kl

We must read Equation (118View Equation) as giving gkl(Γ ) if the only basic variable in affine geometry, i.e., the connection Γ ijk, has been determined by help of some field equations. Thus, in general, gkl is not metric-compatible; in order to make it such, we are led to the differential equations Rij∥k = 0 for Γ ijk, an equation not considered by Eddington. In the absence of an electromagnetic field, Equation (118View Equation) 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 Rkl. 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 F ij from Fij; we must assume that he thought of raising indices with the complicated inverse metric tensor.

In connection with cosmological considerations, Eddington cherished the λ-term in Equation (118View Equation):

“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 T ik of the electromagnetic field by decomposing the Ricci tensor Kij formed from Equation (51View Equation) into a metric part Rik and the rest. The energy-momentum tensor Tik of the electromagnetic field is then defined by Einstein’s field equations with a fictitious cosmological constant ik ik 1 ik κT := G − 2g (G − 2λ).

Although Eddington’s interest did not rest on finding a proper set of field equations, he nevertheless discussed the Lagrangian √ --- ℒ = − g ∗Gik ∗Gik, and showed that a variation with regard to gik 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.” ([58Jump To The Next Citation Point], p. 121)

By “things” he meant

  1. the energy-momentum tensor of matter, i.e., of the electromagnetic field,
  2. the tensor of the electromagnetic field, and
  3. the electric charge-and-current vector.

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.” ([211], p. 382)

4.3.2 Einstein’s reaction and publications

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 ([330Jump To The Next Citation Point], 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. [241Jump To The Next Citation Point], 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.”88View original Quote (letter of Einstein to Weyl of 26 May 1923; cf.[241Jump To The Next Citation Point], 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 [77Jump To The Next Citation Point].

“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.”90View original Quote ([77Jump To The Next Citation Point], p. 32)

Like Eddington, Einstein used a symmetric connection and wrote down the equation91

2 λ Kkl = gkl + ϕkl, (119 )
where gkl = g(kl) and ϕkl = ϕ[kl], and λ is a “large number”. By this, the metric was defined as the symmetric part of the Ricci tensor. Due to
( ) 1 ∂Γ kjj ∂ Γljj ϕkl = -- ----l-− ----k- (120 ) 2 ∂x ∂x
one half of Maxwell’s equations is satisfied if ϕ kl is taken to be the electromagnetic field tensor. Let us note, however, that while j Γ kj transforms inhomogeneously, its transformation law
′ ′ j′ m ∂xl-- --∂2xl--- ∂xm-- Γk′j′ = Γlm ∂xk′ + ∂xk′∂xm ′ ⋅ ∂xl′
is not exactly the same as that of the electric 4-potential under gauge transformations.

For a Lagrangian, Einstein used ∘ --------- ℒ = 2 − detKij; 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 g kl and ϕ kl, not, as one might have expected, with regard to the connection Γ j kj. If fˆkl := δδℒϕkl-, then the electric current density jl is defined by l ∂fˆlk ˆj := -∂xk . ˆfkl 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 Γ l kj and are (Einstein worked in space-time)

kl 1- k l 1- l k kl k sˆ ∥m + 3 δmˆj + 3 δmˆj = 0, 3ˆs ∥l + 5 ˆj = 0, (121 )
with the definition of the current density k ˆj given before, and kl δℒ- ˆs = δgkl. Besides kl ˆs, Einstein also uses skl introduced by
kl kl∘ --------- kl k ˆs = s − detskl, s sml = δm. (122 )
From Equation (121View Equation) the connection can be obtained. If √ --------- ˆjl = − det skljl, and jk = skljl, then the affine connection may formally be expressed by
1 ( ∂s ∂s ∂s ) 1 1 Γkjl= --slr --kjr+ --jrk − --kjr- − -skjˆjl +--δl(kˆjj) (123 ) 2 ∂x ∂x ∂x 2 3
This equation is an identity if a solution of the field equations (121View Equation) is inserted. From Equation (123View Equation),
j -∂--∘ ------ 1- Γ kj = ∂xk det slr + 3jk. (124 )

If no electromagnetic field is present, ˆskl reduces to √--------- sˆkl = gkl − detgkl; the definition of the metric gij in Equation (119View Equation) is reinterpreted by Einstein as giving his vacuum field equation with cosmological constant λ −2. In order that this makes sense, the identifications in Equation (119View Equation) are always to be made after the variation of the Lagrangian is performed.

For non-vanishing electromagnetic field, due to Equation (124View Equation) the Equation (120View Equation) now becomes

( ) ˆϕkl = 1- ∂ˆjk-− ∂ˆjl- , (125 ) 6 ∂xl ∂xk
which means that for vanishing current density no electromagnetic field is possible. Einstein concluded:

“But the extraordinary smallness of 1- λ2 implies that finite ϕ kl are possible only for tiny, almost vanishing current density. Except for singular positions, the current density is practically vanishing.”92View original Quote

Einstein went on to show that Maxwell’s vacuum equations are holding in first order approximation. Up to the same order, ˆkl ˆ f ≃ ϕkl. In general however, ˆ ˆnm ϕkl ⁄= skmslnf . Also, the geometrical theory presented here is energetically closed, i.e., the current density ˆjl 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”93View original Quote ([77Jump To The Next Citation Point], 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.”94View original Quote ([77Jump To The Next Citation Point], 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 ∘ --------- ℒ = 2 − detKij for a new one, i.e., for ∘ ------- ℒ = − 2 det Kij + ˆR − 16ˆslmilim, where √ ------- ˆık = ik det skl. ℒ is to be varied with respect to ˆskl and fˆkl. The resulting equations for the gravitational and electromagnetic fields are the symmetric and skew-symmetric part, respectively, of

( ) 1 ∂ij ∂ik Kjk = Rjk + 6- ∂xk-− ∂xj-+ ijik . (126 )

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.”95View original Quote ([74Jump To The Next Citation Point], p. 77)

In the third paper [76Jump To The Next Citation Point], apart from changing notations96, Einstein set λ = 1. He also dropped the assumption (119View Equation) and replaced it by allowing his Lagrangian (Hamiltonian) ˆH to be a function of the two independent variables,

γij = K(ij), ϕij = K [ij]. (127 )
The logic of the subsequent derivations in his paper is quite involved. The first step consisted in the definition of tensor densities
ˆgkl := δHˆ-, fˆkl := δHˆ-. (128 ) δγkl δϕkl
In the second step, the variations δ γkl and δ ϕkl were expressed by l δΓik via (127View Equation) and inserted into δHˆ = 0. The ensuing equation could be solved for l Γik and led to Equation (123View Equation). In the third step, the Lagrangian Hˆ∗ is taken as a functional of the variables introduced in the first step, i.e., of ˆgkl, fˆkl such that in place of Equation (128View Equation) the relations
δHˆ ∗ δHˆ∗ γkl :=---kl , ϕkl := -----. (129 ) δˆg δfˆkl
hold. Einstein then took “the expression most natural vis-a-vis our present knowledge”, i.e., ˆ∗ √ --- β ˆik H = 2α − g − 2fikf. By using both Equation (127View Equation) and Equation (129View Equation), Einstein obtained the Einstein–Maxwell equations augmented by a term 1 − 6ikil on the side of the energy-momentum tensor of the electromagnetic field and Equation (125View Equation) with a changed l.h.s. now reading − βfik.

After a field rescaling, he then took a third expression to become his Lagrangian

√ --- [ ( 1 ) 1 ] H¯ = − g R − 2α + κ --fijfij − --ilil , 2 β
where α and β are arbitrary constants, and κ is the gravitational constant. i l is defined to be proportional to the electromagnetic 4-potential fk, i.e., 1 β il = − fk, and fij corresponds to √ --- ϕij, − g fij to ˆfij. After the field equations had been obtained by this longwinded procedure, it became obvious that they could also be derived from ¯H = H taken as an “effective” Lagrangian varied with respect to gik and fik. In Einstein’s words: “R is the Riemannian curvature scalar formed from gij97View original Quote. In the third paper as well, Einstein’s desire to create a unified field theory satisfying all his criteria still was not fulfilled: His equations, again, did not give a singularity-free electron. In a paper on Hilbert’s vision of a unified science, Sauer and Majer recently have found out from lectures of Hilbert given in Hamburg and Zürich in 1923, that Hilbert considered Einstein’s work in affine theory a return to his own results of 1915 by “[...] a colossal detour via Levi-Civita, Weyl, Schouten, Eddington [...]” [215]. It seems that, in this evaluation, Hilbert was influenced by Einstein’s proportionality between the 4-potential and the electrical current which Hilbert had assumed as early as in 1915 [161]98.

4.3.3 Comments by Einstein’s colleagues

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 Γ μνα 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.”99View original Quote (Pauli to Eddington on 20 September 1923; [251Jump To The Next Citation Point], pp. 115–119)

Also Weyl, in the 5th edition of Raum–Zeit–Materie ([398Jump To The Next Citation Point], 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 [60Jump To The Next Citation Point], 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 WeylEddington is unable to bring progress for our physical understanding; hopefully, future developments will show that this pessimistic opinion has been unjustified.”100View original Quote ([60Jump To The Next Citation Point], 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 WeylEddingtonSchouten 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.”101View original Quote ([99Jump To The Next Citation Point], 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.

4.3.4 Overdetermination of partial differential equations and elementary particles

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.”102View original Quote ([103Jump To The Next Citation Point], 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.”103View original Quote ([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].”104View original Quote

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.”105View original Quote ([99Jump To The Next Citation Point], 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.”106View original Quote ([99Jump To The Next Citation Point], p. 199)

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