𝐐𝐮𝐚𝐧𝐭𝐮𝐦 𝐆𝐫𝐚𝐯𝐢𝐭𝐲 eddington’s ‘affine geometry’ 1923 𝐫𝐞𝐬𝐞𝐚𝐫𝐜𝐡𝐞𝐫 & 𝐥𝐞𝐜𝐭𝐮𝐫𝐞𝐫: james ni (phd theoretical physics, new mexico state university, las cruces, new mexico, usa) The post-higher-dimensional era began with Arthur Eddington suggesting to forego the metric as a fundamental concept and start right away with a (general) connection, which he then restricted to a symmetric one Г in order to avoid an “infinitely crinkled” world. 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” 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. The Ricci tensor Kij(Г) := Gij (‘ij’ are indices in subscript) is asymmetric, Gkl = Rkl + Fkl (‘kl’ indices in subscript) With Rkl(Г) being the symmetric and Fkl(Г) the anti-symmetric part. Fkl derives from a “vector potential”, i.e., Fkl = ∂kГl − ∂lГk (all indices in subscript) with Гl := Гʳlr (here ‘lr’ in subscript), 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 jᶫ := Fᶫᵏ ∥ 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.” Now, we have Fᶫᵏ∥[l∥k] = K[rk]Fʳᵏ + Sˢjk∇sFʲᵏ (‘lk’ in subscript; ‘[rk]’ in subscript; ‘jk’ in subscript; ‘s’ in ‘∇s’ in subscript) For a symmetric connection thus, unlike in Riemannian geometry, jᵏ∥k = Fᶫᵏ∥[l∥k] = FrkFʳᵏ ≠ 0 However, for a tensor density, we obtain ℐᵏ∥k = ℱᶫᵏ∥[l∥k] = ½ (Vrk + 2K[rk]) ℱʳᵏ + Sjkˢ∇s ℱʲᵏ (‘rk’ in ‘V’ and ‘K’ in subscript) and thus for a torsionless connection jᵏ∥k = 0. Eddington introduces the metrical tensor by the definition λgkl = Rkl (‘kl’ in subscript) “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?” 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 Rkl (‘kl’ in subscript) as the metric implies that det Rkl ≠ 0. We must read Equation λgkl = Rkl as giving gkl(Г) if the only basic variable in affine geometry, i.e., the connection Гijᵏ (‘ij’ in subscript), 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 Гijᵏ, an equation not considered by Eddington. In the absence of an electromagnetic field, Equation λgkl = Rkl looks like Einstein’s vacuum field equation with a 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 Fᶦʲ from Fij (‘ij’ in subscript); 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 λgkl = Rkl: “I would as soon think of reverting to Newtonian theory as of dropping the cosmic constant.” Now, Eddington was able to identify the energy-momentum tensor Tᶦᵏ of the electromagnetic field by decomposing the Ricci tensor Kᶦᵏ into a metric part Rik (‘ik’ in subscript) and the rest. The energy-momentum tensor Tᶦᵏ of the electromagnetic field is then defined by Einstein’s field equations with a fictitious cosmological constant κTᶦᵏ := Gᶦᵏ − 1/2gᶦᵏ(G − 2λ). Although Eddington’s interest did not rest on finding a proper set of field equations, he nevertheless discussed the Lagrangian Ŀ = √−g Gik (‘ik’ in subscript), 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.” 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. 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.” Like Eddington, Einstein used a symmetric connection and wrote down the equation λ²Kkl = gkl + φkl (‘kl’ in subscript) 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 Φkl = ½(∂Гkjʲ/∂xᶫ − ∂Гljʲ/∂xᵏ one half of Maxwell’s equations is satisfied if φkl is taken to be the electromagnetic field tensor. Let us note, however, that while Гkjʲ transforms inhomogeneously, its transformation law Гk’j’ʲ’ = Гlmᵐ ∂xᶫ/∂xᵏ’ + ∂²xᶫ’/∂xᵏ’∂xᵐ’ is not exactly the same as that of the electric 4-potential under gauge transformations. For a Lagrangian, Einstein used Ŀ = 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 gkl and φkl, not, as one might have expected, with regard to the connection Гkjʲ. If ℱᵏᶫ := δĿ/δφkl, then the electric current density jᶫ is defined by ℐᶫ := ∂ℱᶫᵏ/∂x ᵏ ℱᵏᶫ is interpreted as “the contravariant tensor of the electromagnetic field”. Of course the field equations are obtained from the Lagrangian by variation with regard to the connection Гkjᶫ. 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 [the Г’s] 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.” Also Weyl, in the 5th edition of Raum–Zeit–Materie, 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. Likewise, Eddington himself did not appreciate much Einstein’s followership. “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.” In fact, when Eddington’s book was translated into German in 1925, 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.” 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.” 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 over-determination by differential equations.” 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 “over-determination” 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.” He then ventured the hope that a system of over-determined 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:
Posted on: Tue, 04 Nov 2014 00:33:11 +0000
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