Further work on (metric-) affine and mixed geometry

6.2 Further work on (metric-) affine and mixed geometry

Research on affine geometry as a frame for unified field theory was also carried on by mathematicians of the Princeton school. Thus J. M. Thomas, after having given a review of Weyl’s, Einstein’s, and Schouten’s approaches, said about his own work:

“I show in the present paper that his [Einstein’s] new equations can be obtained by a direct generalisation of the equations of the gravitational field previously given by him [ $gij;k = 0;Rij = 0$ ]. [...] In the final section I show that the adoption of the ordinary definition of covariant differentiation leads to a geometry which includes as a special case that proposed by Weyl as a basis for the electric theory; further that the asymmetric connection for this special case is of the type adopted by Schouten for the geometry at the basis of his electric theory.” ([346 ], p. 187)

We met J. M. Thomas’ paper before in Section 6.1.

During the period considered here, a few physicists followed the path of Eddington and Einstein. One who had absorbed Eddington’s and Einstein’s theories a bit later was Infeld of Warsaw¹²⁸ . In January 1928, he followed Einstein by using an asymmetric metric the symmetric part $γik$ of which stood for the gravitational potential, the skew-symmetric part $ϕik$ for the electromagnetic field. However, he set the non-metricity tensor (of the symmetric part $γ$ of the metric) $Q ijk= 0$ , and assumed for the skew-symmetric part $ϕ$ ,

with an arbitrary tensor $Jijl$ . The electric current vector then is defined by $Ji = Jill$ where the indices, as I assume, are moved with $γik$ . In a weak-field approximation for the metric, Infeld’s connection turned out to be $L l= {l} + 1(ϕ l + ϕ l + δlsϕ ). ik ik 2 i,k k,i ik,s$ For field equations Infeld postulated the (generalised) Einstein field equations in empty space, $Kij = 0$ . He showed that, in first approximation, he got what is wanted, i.e., Einstein’s and Maxwell’s equations [166 ]. Three months later, Infeld published a note in Comptes Rendus of the Parisian Academy in which he now presented the exact connection as

where $α$ is “an extremely small numerical factor”. By neglecting terms $2 ∼ α$ he could gain both Einstein’s field equation in empty space (94

) and Maxwell’s equation, if the electric current vector is identified with $α −1(Lill− L lil).$ Thus, he is back at vector torsion treated before by Schouten [298

The Japanese physicist Hattori embarked on a metric-affine geometry derived purely from an asymmetric metrical tensor $hik = g(ik) + f[ik]$ . He defined an affine connection

where $gilglk = δi k$ , and the Christoffel symbol is formed from g. The electromagnetic field was not identified with $fik$ by Hattori, but with the skew-symmetric part of the (generalised) Ricci tensor formed from $L j ik$ . By introducing the tensor $fijk := ∂fijk + ∂fjki + ∂fkij- ∂x ∂x ∂x$ , he could write the (generalised) Ricci tensor as

where the covariant derivative $∇$ is formed with the Levi-Civita connection of $gij.$ The electromagnetic field tensor $Fik$ now is introduced through a tensor potential by $Fik := ∇lf ik l$ and leads to half of “Maxwell’s” equations. In the sequel, Hattori started from a Lagrangian $ℒ = (gik + α2F ik)Kik$ with the constant $α2$ and varied, alternatively, with respect to $gij$ and $fij$ . He could write the field equations in the form of Einstein’s, with the energy-momentum tensor of the electromagnetic field $Fik$ and a “matter” tensor $ik M$ on the r.h.s., $ik M$ being a complicated, purely geometrical quantity depending on $Kik, K, fikl$ , and $Fikl$ . $Fikl$ is formed from $Fik$ as $fikl$ from $fik$ . From the variation with regard to $fik$ , in addition to Maxwell’s equation, a further field equation resulted, which could be brought into the form

i.e., $fikl ∼ Fikl$ . Hattori’s conclusion was:

“The preceding equation shows that electrical charge and electrical current are distributed wherever an electromagnetic field exists.”¹²⁹

Thus, the same problem obtained as in Einstein’s theory: A field without electric current or charge density could not exist [155 ]¹³⁰ .

Infeld quickly reacted to Hattori’s paper by noting that Hattori’s voluminous calculations could be simplified by use of Schouten’s Equation (39 ) of Section 2.1.2. As in Hattori’s theory two connections are used, Infeld criticised that Hattori had not explained what his fundamental geometry should be: Riemannian or non-Riemannian? He then gave another example for a theory allowing the identification of the electromagnetic field tensor with the antisymmetric part of the Ricci tensor: He displayed again the well-known connection with vector torsion used by Schouten [298 ] without referring to Schouten’s paper [165 ]. He also claimed that Hattori’s Equation (145 ) is the same as the one that had been deduced from Eddington’s theory by Einstein in the Appendix to the German translation of Eddington’s book ([60 ], p. 367). All in all, Infeld’s critique tended to deny that Hattori’s theory was more general than Einstein’s, and to point out

“that the problem of generalising the theory of relativity cannot be solved along a purely formal way. At first, one does not see how a choice can be made among the various non-Riemannian geometries providing us with the gravitational and Maxwell’s equations. The proper world geometry which ought to lead to a unified theory of gravitation and electricity can only be found by an investigation of its physical content.”¹³¹ ([165 ], p. 811)

Infeld could as well have applied this admonishment to his own unified field theory discussed above. Perhaps, he became irritated by comparing his expression for the connection (142 ) with Hattori’s (145 ).

In June 1931, von Laue submitted a paper of the Genuese mathematical physicist Paolo Straneo to the Berlin Academy [331 ]. In it Straneo took note of Einstein’s teleparallel geometry, but decided to take another route within mixed geometry; he started with a symmetric metric and the asymmetric connection

with both non-vanishing curvature tensor $Ki = Ri + 2δi(∂ψl− ∂ψk) jkl jkl j ∂xk ∂xl$ and torsion $S j= 2 δjψ ik [i k]$ . Thus, Straneo suggested a unified field theory with only vector torsion as Schouten had done 8 years earlier [298 , 142 ]) without referring to him. The field equations Straneo wrote down, i. e.

where $Tik$ is the symmetric and $Ψik$ the antisymmetric part of the l.h.s., do not fulfill Einstein’s conception of unification: Straneo kept the energy-momentum tensor of matter as an extraneous object (including the electromagnetic field) as well as the electric current vector. The antisymmetric part of (147

) just is $Ψik = (∂ψlk − ∂ψkl ) ∂x ∂x$ ; thus $Ψik$ is identified with the electromagnetic field tensor, and the electric current vector $Ji$ defined by $Ψil= Ji l$ . Straneo wrote further papers on the subject [332 , 333 ].

By a remark of Straneo, that auto-parallels and geodesics have to be distinguished in an affine geometry, the Indian mathematician Kosambi felt motivated to approach affine geometry from the system of curves solving $i i ¨x + α (x, ˙x,t)$ with an arbitrary parameter $t$ . He then defined two covariant “vector-derivations” along an arbitrary curve and arrived at an (asymmetric) affine connection. By this, he claimed to have made superfluous the five-vectors of Einstein and Mayer [107 ]. This must be read in the sense that he could obtain the Einstein–Mayer equations from his formalism without introducing a connecting quantity leading from the space of 5-vectors to space-time [195 ]. Einstein, in his papers, did not comment on the missing metric compatibility in his theory and its physical meaning. Due to this complication – for example even a condition of metric compatibility would not have the physical meaning of the conservation of the norm of an angle between vectors under parallel transport, and the further difficulty that much of the formalism was very clumsy to manipulate; essential work along this line was done only much later in the 10940s and 1950s (Einstein, Einstein and Strauss, Schrödinger, Lichnerowicz, Hlavaty, Tonnelat, and many others). In this work a generalisation of the equation for metric compatibility, i.e., Equation (47 ), will play a central role. The continuation of this research line will be presented in Part II of this article.