Affine and mixed geometry
Already in July 1925 Einstein had laid aside his doubts concerning âthe deepening of the geometric foundationsâ. He modified Eddingtonâs approach to the extent that he now took both a nonsymmetric connection and a non-symmetric metric, i.e., dealt with a mixed geometry (metric-affine theory):
â[âŠ] Also, my opinion about my paper which appeared in these reports [i.e., Sitzungsberichte of the Prussian Academy, Nr. 17, p. 137, 1923], and which was based on Eddingtonâs fundamental idea, is such that it does not present the true solution of the problem. After an uninterrupted search during the past two years I now believe to have found the true solution.âFootnote 137 ([78], p. 414)
As in general relativity, he started from the Lagrangian \({\mathcal L} = {\hat g^{ik}}{R_{ik}}\), but now with Äik and the connection \({\Gamma _{kj}}^l\) being varied separately as independent variables. After some manipulations, the variation with regard to the metric and to the connection led to the following equations:
$$
\begin{array}{*{20}{c}}
{ - \frac{{\partial {g_{ik}}}}{{\partial {x^l}}} + {g_{rk}}{\Gamma _{il}}^r + {g_{ir}}{\Gamma _{lk}}^r + {g_{ik}}{\phi _l} + {g_{il}}{\phi _k} = 0,\;\;\;\;\;}&{{R_{ik}} = 0,}
\end{array}
$$
((134))
i.e., 64+16 equations for the same number of variables. Ïk is an arbitrary covariant vector. The asymmetric gik is related to Älm by
$$
\begin{array}{*{20}{c}}
{{{\hat g}_{ir}}{{\hat g}^{jr}} = {{\hat g}_{ri}}{{\hat g}^{rj}} = {\delta _i}^j,\;\;\;\;\;}&{{{\hat g}_{ik}} = \frac{{{g_{ik}}}}{{\sqrt { - g} }}.}
\end{array}
$$
((135))
The three equations (134) and
$$
\begin{array}{*{20}{c}}
{\frac{{\partial {{\hat g}^{ik}}}}{{\partial {x^k}}} - \frac{{\partial {{\hat g}^{ki}}}}{{\partial {x^k}}} = 0,\;\;\;\;\;}&{{R_{ik}} = 0,}
\end{array}
$$
((136))
were the result of the variation. In order to be able to interpret the symmetric part of gik as metrical tensor and its anti(skew)-symmetric part as the electromagnetic field tensor, Einstein put Ïk=0, i.e., overdetermined his system of partial differential equations. However, he cautioned:
âHowever, for later investigations (e.g., the problem of the electron) it is to be kept in mind that the HAMILTONian principle does not provide an argument for putting Ïk equal to zero.âFootnote 138
In comparing Equation (134) with Ïk=0 and Equation (47), we note that the expression does not seem to correspond to a covariant derivative due to the + sign where a â sign is required. But this must be due to either a calculational error, or to a printerâs typo because in the paper of J. M. Thomas following Einsteinâs by six months and showing that Einsteinâs
ânew equations can be obtained by direct generalisation of the equations of the gravitational field previously given by him. The process of generalisation consists in abandoning assumptions of symmetry and in adopting a definition of covariant differentiation which is not the usual one, but which reduces to the usual one in case the connection is symmetric.â ([346], p. 187)
J. M. Thomas wrote Einsteinâs Equation (134) in the form
$$
\begin{array}{*{20}{c}}
{{g_{ik/l}} = {g_{ik}}{\phi _l} + {g_{il}}{\phi _k},\;\;\;\;\;}&{{\phi _l} = - \frac{2}{{n - 1}}{\Omega _{rl}}^r,}
\end{array}
$$
((137))
with Ω being the skew-symmetric part of the asymmetric connection \({H_{ij}}^k = \Gamma _{(ij)}^{\;\;\;k} + \Omega _{[ij]}^{\;\;\;\;k}\), and gij being the symmetric part of the asymmetric metric \({h_{ij}} = {g_{(ij)}} + {\omega _{[ij]}}\). The two covariant derivatives introduced by J. M. Thomas are \({g_{ij,k}} = \tfrac{{\partial {g_{ij}}}}{{\partial {x^k}}} - {g_{rj}}{\Gamma _{ik}}^r - {g_{ir}}{\Gamma _{jk}}^r\) and \({h_{ij/k}} = \tfrac{{\partial {h_{ij}}}}{{\partial {x^k}}} - {h_{rj}}{H_{ik}}^r - {h_{ir}}{H_{jk}}^r\). J. M. Thomas then could reformulate Equation (137) in the form
$$
{g_{ij/l}} = {g_{[ri]}}{\Omega _{jl}}^r + {g_{[ir]}}{\Omega _{lj}}^r,
$$
((138))
and derive the result
$$
{g_{ij,l}} + {g_{jl,i}} + {g_{li,j}} = 0
$$
((139))
(see [346], p. 189).
After having shown that his new theory contains the vacuum field equations of general relativity for vanishing electromagnetic field, Einstein then proved that, in a first-order approximation, Maxwellâs field equations result cum grano salis: Instead of \({F_{ik,l}} + {F_{li,k}} + {F_{kl,i}} = 0\) he only obtained \(\Sigma \tfrac{\partial }{{\partial {x^l}}}({F_{ik,l}} + {F_{li,k}} + {F_{kl,i}}) = 0\).
This was commented on in a paper by Eisenhart who showed âmore particularly what kind of linear connection Einstein has employedâ and who obtained âin tensor form the equations which in this theory should replace Maxwellâs equations.â He then pointed to some difficulty in Einsteinâs theory: When identification of the components of the antisymmetric part Ïij of the metric \({a_{ij}} = {g_{ij}} + {\phi _{ij}}\) with the electromagnetic field is made in first order,
âthey are not the components of the curl of a vector as in the classical theory, unless an additional condition is added.â ([120], p. 129)
Toward the end of the paper Einstein discussed time-reversal; according to him, by it the sign of the magnetic field is changed, while the sign of the electric field vector is left unchangedFootnote 139. As he wanted to obtain charge-symmetric solutions from his equations, Einstein now proposed to change the roles of the magnetic fields and the electric fields in the electromagnetic field tensor. In fact, the substitutions \(\mathbf{\tilde E} \to \mathbf{\tilde B}\) and \({\mathbf{\tilde B}} \to - {\mathbf{\tilde E}}\) leave invariant Maxwellâs vacuum field equations (duality transformations)Footnote 140. Already Pauli had pointed to time-reflection symmetry in relation with the problem of having elementary particles with charge ±e and unequal mass ([246], p. 774).
At first, Einstein seems to have been proud about his new version of unified field theory; he wrote to Besso on 28 July 1925 that he would have liked to present him âorally, the egg laid recently, but now I do it in writingâ, and then explained the independence of metric and connection in his mixed geometry. He went on to say:
âIf the assumption of symmetryFootnote 141 is dropped, the laws of gravitation and Maxwellâs field laws for empty space are obtained in first approximation; the antisymmetric part of Äik is the electromagnetic field. This is surely a magnificent possibility which likely corresponds to reality. The question now is whether this field theory is consistent with the existence of quanta and atoms. In the macroscopic realm, I do not doubt its correctness.âFootnote 142 ([99], p. 209)
We have noted before that a similar suggestion within a theory with a geometry built from an asymmetric metric had been made, in 1917, by Bach alias Förster.
Yet, in the end, also this novel approach did not convince Einstein. Soon after the publication discussed, he found his argument concerning charge symmetric solutions not to be helpful. The link between the occurrence of solutions with both signs of the charge with time-symmetry of the field equations induced him to doubt, if only for a moment, whether the endeavour of unifying electricity and gravitation made sense at all:
âTo me, the insight seems to be important that an explanation of the dissimilarity of the two electricities is possible only if time is given a preferred direction, and if this is taken into account in the definition of the decisive physical quantities. In this, electrodynamics is basically different from gravitation; therefore, the endeavour to melt electrodynamics with the law of gravitation into one unity, to me no longer seems to be justified.âFootnote 143 [79]
In a paper dealing with the field equations
$$
{R_{ik}} - \frac{R}{4}{g_{ik}} = - \kappa {T_{ik}},
$$
((140))
which had been discussed earlier by Einstein [70], and to which he came back now after RainichFootnote 144âs insightful paper into the algebraic properties of both the curvature tensor and the electromagnetic field tensor ([263, 264, 265, 266]), Einstein indicated that he had lost hope in the extension of Eddingtonâs affine theory:
âThat the equations (140) have received only little attention is due to two circumstances. First, the attempts of all of us were directed to arrive, along the path taken by Weyl and Eddington or a similar one, at a theory melting into a formal unity the gravitational and electromagnetic fields; but by lasting failure I now have laboured to convince myself that truth cannot be approached along this path.âFootnote 145 (Einsteinâs italics; [80], p. 100)
The new field equation was picked up by R. N. Sen of Kalkutta who calculated âthe energy of an electric particleâ according to it [323].
In the same spirit as the one of his paper, Einstein said good bye to his theory in a letter to Besso on Christmas 1925 in words similar to those in his letter in June:
âRegrettably, I had to throw away my work in the spirit of Eddington. Anyway, I now am convinced that, unfortunately, nothing can be made with the complex of ideas by Weyl-Eddington. The equations
$$
{R_{ik}} - \frac{1}{4}R\;{g_{ik}} = - \kappa {T_{ik}}\;\;\;\;\;{\rm{electromagnetic}}
$$
I take as the best we have nowadays. They are 9 equations for the 14 variables gik and Îłik New calculations seem to show that these equations yield the motion of the electrons. But it appears doubtful whether there is room in them for the quanta.â
Footnote 146
([99], p. 216)
According to the commenting note by Tonnelat, the 14 variables are given by the 10 components of the symmetric part g(ik)Footnote 147 of the metric and the 4 components of the electromagnetic vector potential âthe rotation of which are formed by the Îł[ik]âFootnote 148.
But even âthe best we have nowadaysâ did not satisfy Einstein; half a year later, he expressed his opinion in a letter to Besso:
âAlso, the equation put forward by myselfFootnote 149,
$$
{R_{ik}} = {g_{ik}}{f_{lm}}{f^{lm}} - \frac{1}{2}{f_l}{f_{km}}{g^{lm}}
$$
gives me little satisfaction. It does not allow for electrical masses free from singularities. Moreover, I cannot bring myself to gluing together two items (as the l.h.s. and the r.h.s. of an equation) which from a logical-mathematical point of view have nothing to do with each other.â
Footnote 150 ([99], p. 230)
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 Weylâs, Einsteinâs, and Schoutenâs approaches, said about his own work:
âI show in the present paper that his [Einsteinâs] 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]. [âŠ] 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.â ([346], p. 187)
We met J. M. Thomasâ 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 Eddingtonâs and Einsteinâs theories a bit later was InfeldFootnote 151 of WarsawFootnote 152. In January 1928, he followed Einstein by using an asymmetric metric the symmetric part Îłik of which stood for the gravitational potential, the skew-symmetric part Ïik for the electromagnetic field. However, he set the non-metricity tensor (of the symmetric part Îł of the metric) \({Q_{ij}}^k = 0\), and assumed for the skew-symmetric part Ï,
$$
{\nabla _l}{\phi _{ij}} = {J_{ijl}},
$$
((141))
with an arbitrary tensor Jijl. The electric current vector then is defined by \({J^i} = {J^{il}}_l\) where the indices, as I assume, are moved with Îłik. In a weak-field approximation for the metric, Infeldâs connection turned out to be \({L_{ik}}^l = \{ \,_{ik}^l\,\} + \tfrac{1}{2}({\phi _{i,}}^l_k + {\phi _{k,}}^l_i + {\delta ^{ls}}{\phi _{ik,s}})\). 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., Einsteinâs and Maxwellâs 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
$$
{L_{ik}}^l = \{ \,_{ik}^l\,\} + \frac{\alpha }{2}({\phi _{i,}}^l_k + {\phi _{k,}}^l_i + {g^{ls}}{\phi _{ik,s}}),
$$
((142))
where α is âan extremely small numerical factorâ. By neglecting terms âŒÎ±2 he could gain both Einsteinâs field equation in empty space (94) and Maxwellâs equation, if the electric current vector is identified with \({\alpha ^{ - 1}}({L_{il}}^l - {L_{li}}^l)\). 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 \({h_{ik}} = {g_{(ik)}} + {f_{[ik]}}\). He defined an affine connection
$$
{L_{ik}}^j = \{ \,_{ik}^j\,\} + {g^{jl}}({f_{li,k}} + {f_{li,k}} - {f_{ik,l}}),
$$
((143))
where \({g^{il}}{g_{lk}} = {\delta ^i}_k\), 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 \({L_{ik}}^j\). By introducing the tensor \({f_{ijk}}: = \tfrac{{\partial {f_{ij}}}}{{\partial {x^k}}} + \tfrac{{\partial {f_{jk}}}}{{\partial {x^i}}} + \tfrac{{\partial {f_{ki}}}}{{\partial {x^j}}}\), he could write the (generalised) Ricci tensor as
$$
{K_{ik}} = {R_{ik}} + \frac{1}{4}{f_{im}}^n{f_{kn}}^m - {\nabla _l}{f_{ik}}^l,
$$
((144))
where the covariant derivative â is formed with the Levi-Civita connection of gij. The electromagnetic field tensor Fik now is introduced through a tensor potential by \({F_{ik}}: = {\nabla _l}{f_{ik}}^l\) and leads to half of âMaxwellâsâ equations. In the sequel, Hattori started from a Lagrangian \( {\mathcal L} = ({g^{ik}} + {\alpha ^2}{F^{ik}}){K_{ik}} \)
with the constant α2 and varied, alternatively, with respect to gij and fij. He could write the field equations in the form of Einsteinâs, with the energy-momentum tensor of the electromagnetic field Fik and a âmatterâ tensor Mik on the r.h.s., Mik 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 Maxwellâs equation, a further field equation resulted, which could be brought into the form
$$
{F^{ik}} = \frac{2}{3}{\nabla _l}{F^{ikl}},
$$
((145))
i.e., fiklâŒFikl Hattoriâs conclusion was:
âThe preceding equation shows that electrical charge and electrical current are distributed wherever an electromagnetic field exists.âFootnote 153
Thus, the same problem obtained as in Einsteinâs theory: A field without electric current or charge density could not exist [155]Footnote 154.
Infeld quickly reacted to Hattoriâs paper by noting that Hattoriâs voluminous calculations could be simplified by use of Schoutenâs Equation (39) of Section 2.1.2. As in Hattoriâs 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 Schoutenâs paper [165]. He also claimed that Hattoriâs Equation (145) is the same as the one that had been deduced from Eddingtonâs theory by Einstein in the Appendix to the German translation of Eddingtonâs book ([60], p. 367). All in all, Infeldâs critique tended to deny that Hattoriâs theory was more general than Einsteinâs, and to point out
âthat 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 Maxwellâs 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.âFootnote 155 ([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 Hattoriâs (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 Einsteinâs teleparallel geometry, but decided to take another route within mixed geometry; he started with a symmetric metric and the asymmetric connection
$$
{L_{ik}}^j = \{ \,_{ik}^j\,\} + 2{\delta ^j}_i{\psi _k}
$$
((146))
with both non-vanishing curvature tensor \({K^i}_{jkl} = {R^i}_{jkl} + 2\delta _j^i(\tfrac{{\partial {\psi _l}}}{{\partial {x^k}}} - \tfrac{{\partial {\psi _k}}}{{\partial {x^l}}})\) and torsion \({S_{ik}}^j = 2{\delta ^j}_{[i}{\psi _{k]}}\). 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.
$$
{K_{ik}} - \frac{1}{2}K{g_{ik}} = - \kappa {T_{ik}} + {\Psi _{ik}},
$$
((147))
where Tik is the symmetric and Κik the antisymmetric part of the l.h.s., do not fulfill Einsteinâs 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 \({\Psi _{ik}} = (\tfrac{{\partial {\psi _l}}}{{\partial {x^k}}} - \tfrac{{\partial {\psi _k}}}{{\partial {x^l}}})\); thus Κik is identified with the electromagnetic field tensor, and the electric current vector Ji defined by \({\Psi ^{il}}_l = {J^i}\). 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 KosambiFootnote 156 felt motivated to approach affine geometry from the system of curves solving áși+αi(x, x, t) with an arbitrary parameter t. He then defined two covariant âvector-derivationsâ 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 MayerFootnote 157 [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 â 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.
Kaluzaâs idea taken up again
Kaluza: Act I
Einstein became interested in Kaluzaâs theory again due to O. Kleinâs paper concerning a relation between âquantum theory and relativity in five dimensionsâ (see Klein 1926 [185], received by the journal on 28 April 1926). Einstein wrote to his friend and colleague Paul Ehrenfest on 23 August 1926: âSubject Kaluza, Schroedinger, general relativityâ, and, again on 3 September 1926: âKleinâs paper is beautiful and impressive, but I find Kaluzaâs principle too unnatural.â However, less than half a year later he had completely reversed his opinion:
âIt appears that the union of gravitation and Maxwellâs theory is achieved in a completely satisfactory way by the five-dimensional theory (Kaluza-Klein-Fock).â (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 â not just in first order as Kaluza had done [81, 82]. He came too late: Klein had already shown the same before [185]. 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:
âMr. 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.âFootnote 158
He then referred to the papers of Klein [185, 186] and to âFochs Arbeitâ which is a paper by FockFootnote 159 1926 [130], submitted three months later than Kleinâs paper. That Klein had published another important clarifying note in Nature, in which he closed the fifth dimension, seems to have escaped EinsteinFootnote 160 [184]. Unlike in his paper with Grommer, but as in Kleinâs, Einstein, in his notes, applied the âsharpened cylinder conditionâ, i.e., dropped the scalar field. Thus, the three of them had no chance to find out that Kaluza had made a mistake: For g55â const., even in first approximation the new field will appear in the four-dimensional Einstein.Maxwell equations ([145], p. 5).
MandelFootnote 161 of Leningrad was not given credit by Einstein although he also had rediscovered by a different method some of O. Kleinâs results [216]. In a footnoote, Mandel stated that he had learned of Kaluzaâs (whom he spelled âKalusaâ) paper only through Kleinâs article. He started by embedding space-time as a hypersurface x5=const. into M5, 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 âclosely linked to the second fundamental form of this hypersurfaceâ. From the geodesics in M5 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
âthat the experimental discovery of the second term appears difficult, yet perhaps not entirely impossible.â ([216], p. 145)
As to Fockâs paper, it is remarkable because it contains, in nuce, the coupling of the Schrödinger wave function Ï and the electromagnetic potential by the gauge transformation Ï=Ï0 e2Ïip/h, where h is Planckâs constant and p âa new parameter with the unit of the quantum of actionâ [130]. In Fockâs words:
âThe importance of the additional coordinate parameter p 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.âFootnote 162 ([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 M5. Neither Mandel nor Fock used the âsharpened cylinder conditionâ (110).
A main motivation for Klein was to relate the fifth dimension with quantum physics. From a postulated five-dimensional wave equation
$$
\begin{array}{*{20}{c}}
{{a^{ik}}\left( {\frac{{{\partial ^2}U}}{{\partial {x^i}\partial {x^k}}} - \{ \,_{\;r}^{ik}\,\} \frac{{\partial U}}{{\partial {x^r}}}} \right) = 0,\;\;\;\;\;}&{i,k, = \ldots ,5}
\end{array}
$$
((148))
and by neglecting the gravitational field, he arrived at the four-dimensional Schrödinger equation after insertion of the quantum mechanical differential operators \(- \tfrac{{ih}}{{2\pi }}\tfrac{\partial }{{\partial {x^i}}}\). It was Kleinâs papers and the magical lure of a link between classical field theory and quantum theory that raised interest in Kaluzaâs idea â seven years after Kaluza had sent his manuscript to Einstein. Klein acknowledged Mandelâs contribution in his second paper received on 22 October 1927, where he also gave further references on work done in the meantime, but remained silent about Einsteinâs papers [189]. Likewise, Einstein did not comment on Kleinâs new idea of âdimensional reductionâ as it is now called and which justifies Kleinâs name in the âKaluza-Kleinâ theories of our time. By this, the reduction of five-dimensional equations (as e.g., the five-dimensional wave equation) to four-dimensional equations by Fourier decomposition with respect to the new 5th spacelike coordinate x5, taken as periodic with period L, is understood:
$$
\psi (x,{x^5}) = \frac{1}{{\sqrt L }}{\Sigma _n}{\psi _n}(x),{e^{in{x^5}/{R^5}}}
$$
with an integer n. Klein had only the lowest term in the series. The 5th dimension is assumed to be a circle, topologically, and thus gets a finite linear scale: This is at the base of what now is called âcompactificationâ. By adding to this picture the idea of de Broglie waves, Klein brought in Planckâs constant and determined the linear scale of x5 to be unmeasurably small (âŒ10-30). From this, the possibility of âforgettingâ the fifth dimension arose which up to now has not been observed.
In his papers, Einstein took over Kleinâs condition g55=1, which removed the additional scalar field admitted by the theory. It was ReichenbĂ€cher 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):
âNow, 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 (g55â const., dependence of the components of the f u n d a m e n t a l [tensor] of x5 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
$$
\begin{array}{*{20}{c}}
{{R^{ik}} - \frac{1}{2}{g^{ik}}R = {T^{ik}},\;\;\;\;\;}&{\frac{{\partial \sqrt g {F^{ik}}}}{{\partial {x^k}}} = {s^i},}
\end{array}
$$
only the Tik contain further terms besides the electromagnetic energy tensor Sik, and the quantities collected in si do not vanish. [âŠ] The appearance of the new terms on the right hand sides could even be welcomed in the sense that now the field equations are obtained not only for a field point free of matter and chargeFootnote 163.â
Footnote 164 ([276], p. 426)
Here, in nuce, is already contained what more than a decade later Einstein and Bergmann worked out in detail [102].
It is likely that ReichenbĂ€cher 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 \({\gamma _{ik}} = {g_{ik}} - { \epsilon ^2}{\phi _i}{\phi _k}\) with Ïk the electromagnetic 4-potential, had failed. Beyond incredibly complicated field equations nothing much had been gained [275]. ReichenbĂ€cherâs ansatz is well founded: As we have seen in Section 4.2, due to the violation of covariance in M5, Îłik transforms as a tensor under the reduced covariance group.
Even L. de Broglie became interested in Kaluzaâs âbold but very beautiful theoryâ and rederived Kleinâs results his way [46], but not without getting into a squabble with Klein, who felt misunderstood [188, 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 Maxwellâs equations from the five-dimensional homogeneous wave equation and the fivedimensional equation of continuity [43].
In 1929 Mandel tried to âaxiomatiseâ 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 âan objective meaning does not rest in the gik proper, but only in their quotientsâ, 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 [220].
Kleinâs lure lasted for some years. In 1930, N. R. Sen claimed to have investigated the âKepler-problem for the five-dimensional wave equation of Kleinâ. 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 aik=Îłik, where \({\gamma _{ik}} = {g_{ik}} + {\alpha ^2}{\gamma _{55}}\) is the metric on space-time following from the 5-metric \({\gamma _{\alpha \beta }}\) by dx5=0. For gik he took the Reissner-Nordström solution and did not obtain a discrete spectrum [324]. He continued his approach by trying to solve Schrödingerâs wave equation [325]
$$
{g^{ik}}\left( {\frac{{{\partial ^2}u}}{{\partial {x^i}\partial {x^k}}} - \{ \,_{\;r}^{ik}\,\} \frac{{\partial u}}{{\partial {x^r}}}} \right) = - \frac{{4{\pi ^2}}}{{{h^2}}}m_0^2{c^2}u.
$$
Presently, the different contributions of Kaluza and O. Klein are lumped together by most physicists into what is called âKaluza-Klein theoryâ. An early criticism of this unhistorical attitude has been voiced in [210].
Kaluza: Act II
Four years later, Einstein returned to Kaluzaâs idea. Perhaps, he had since absorbed Mandelâs ideas which included a projection formalism from the five-dimensional space to space-time [216, 217, 218, 219].
In a paper with his assistant Mayer, Einstein now presented Kaluzaâs approach in the form of an implicit projective four-dimensional theory, although he did not mention the word âprojectiveâ [107]:
âPsychologically, the theory presented here connects to Kaluzaâs well-known theory; however, it avoids extending the physical continuum to one of five dimensions.âFootnote 165
In the eyes of Einstein, by avoiding the artificial cylinder condition (109), the new method removed a serious objection to Kaluzaâs theory.
Another motivation is also put forward: The linearity of Maxwellâs equations âmay not correspond to realityâ; thus, for strong electromagnetic fields, Einstein expected deviations from Maxwellâs equations. After a listing of all the shortcomings of Kaluzaâs theory, the new approach is introduced: At every event a five-dimensional vector space V5 is affixed to space-time V4, and âmixedâ tensors \({\gamma _\imath }^{\;k}\) are defined linking the tangent space of space-time V4 with a V5 such that
$$
{g_{\iota \kappa }}{\gamma ^\iota }_l{\gamma ^\kappa }_m = {g_{lm}},
$$
((149))
where glm is the metric tensor of V4, and \({g_{\iota \kappa }}\) a non-singular, symmetric tensor on V5 with Îč, Îș=1, ⊠, 5, and k, l=1, ⊠, 4Footnote 166. Indices are raised and lowered with the metrics of V5 or V4, respectively. There exists a âpreferred direction of V5â defined by \({\gamma _\iota }^{\;k}{A^\iota } = 0\), and which is the normal to a âpreferred planeâ \({\gamma _\iota }^{\;k}{\omega _k} = 0\)Footnote 167. A consequence then is
$$
{\gamma _{\sigma k}}{\gamma ^{\tau k}} = {\gamma _\sigma }^k{\gamma _k}^\tau = {\delta _\sigma }^\tau - {A_\sigma }{A^\tau }.
$$
((150))
A covariant derivative for five-vectors in V4 is defined with a âthree-index-symbolâ \({\Gamma ^\iota }_{\pi l}\) with two indices in V5, and one in V4 standing in for the connection coefficients:
$$
{\mathop \nabla \limits^ + _l}{X^\iota } = \frac{{\partial {X^\iota }}}{{\partial {x^l}}} + {\Gamma _\pi }^\iota {X^\pi }.
$$
((151))
The covariant derivative of 4-vectors is defined as usual,
$$
{\nabla _l}{X^i} = \frac{{\partial {X^i}}}{{\partial {x^l}}} + \{ \,_{jl}^i\,\} {X^j},
$$
((152))
where {
ijl
} is calculated from the metric of V4 as given in Equation (149). Both covariant derivatives are abbreviated by the same symbol A;k. The covariant derivative of tensors with both indices referring to V5 and those referring to V4, is formed correspondingly. In this context, Einstein and Mayer mention an extension of absolute differential calculus by âWAERDEN and BARTOLOTTIâ without giving any reference to their respective papers. They may have had in mind van der Waerdenâs [368] and BortolottiFootnote 168âs [24] papers. The autoparallels of V5 lead to the exact equations of motion of a charged particle, not the geodesics of V4.
Einstein and Mayer made three basic assumptions:
$$
\begin{array}{*{20}{l}}
{{g_{\iota \kappa ;\;l}} = 0,}\\
{\,{\gamma _\iota }\,_;^k{\,_l} \;= {A^\iota }{F_{kl}},}\\
{\;\;{F_{kl}} = - {F_{lk}},}
\end{array}
$$
((153))
where \({A^\iota }\) is the preferred direction and Fkl an arbitrary 2-form, later to be interpreted as the electromagnetic field tensor. From them \({A_{\sigma ;l}} = {\gamma _\sigma }^k{F_{lk}}\) follows. They also noted that a symmetric tensor Fkl could have been interpreted as the second fundamental form, and the formalism would then be the same as local isometric embedding of V4 into V5.
Einstein and Mayer introduced what they called âFĂŒnferkrĂŒmmungâ (5-curvature) via the three-index symbol given above by
$$
{P^\sigma }_{\iota kl} = {\partial _k}{\Gamma _{\iota l}}^\sigma - {\partial _l}{\Gamma _{\iota k}}^\sigma + {\Gamma _{\tau k}}^\sigma {\Gamma _{\iota l}}^\tau - {\Gamma _{\tau l}}^\sigma {\Gamma _{\iota k}}^\tau .
$$
((154))
It is related to the Riemannian curvature \({R^r}_{mlk}\) of V4 by
$$
{P^\sigma }_{\iota kl}\;{\gamma _{\sigma m}} = {A_\iota }({F_{mk;l}} - {F_{ml;k}}) + {\gamma _{\iota r}}({R^r}_{mlk} + {F_{mk}}{F_l}^{\;r} - {F_{ml}}{F_k}^r),
$$
((155))
and
$$
{P^\sigma }_{\iota kl}{A_\sigma } = {\gamma _{\iota r}}(F_{k\;;\;l}^{\;r} - F_{l\;;\;k}^{\;r}).
$$
((156))
From (154), by transvection with \({\gamma ^{\tau k}}\), the 5-curvature itself appears:
$$
{P^\tau }_{\iota kl} = {\gamma _{\iota r}}{A^\tau }(F_{k;l}^{\;r} - F_{l;k}^{\;r}) + {\gamma ^{\tau r}}{A_\iota }({F_{rk;l}} - {F_{rl;k}}) + {\gamma _{\iota r}}{\gamma ^{\tau s}}({R^r}_{slk} + {F_{sk}}{F_l}^{\;r} - {F_{sl}}{F_k}^{\;r}).
$$
((157))
By contraction, \({P_{\iota k}}: = {\gamma _\tau }^r{P^\tau }_{\iota rk}\) and \(P: = {\gamma ^{\iota k}}{P_{\iota k}}\). Two new quantities are introduced:
-
(1)
\({U_{\iota k}}: = {P_{\iota k}} - \tfrac{1}{4}(P + R)\), where R is the Ricci scalar of the Riemannian curvature tensor of V4, and
-
(2)
the tensor \({N_{klm}}: = {F_{\{ kl;m\} }}\)Footnote 169.
It turns out that \(P = R - {F_{kp}}{F^{kp}}\).
The field equations put forward in the paper by Einstein and Mayer now are
$$
\begin{array}{*{20}{c}}
{{U_{\iota k}} = 0,\;\;\;\;\;}&{{N_{klm}} = 0,}
\end{array}
$$
((158))
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 Kaluzaâs theory [185].
The authorsâ conclusion is:
âFrom 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.âFootnote 170 ([107], p. 19)
After this paper Einstein wrote to Ehrenfest in a letter of 17 September 1931 that this theory âin my opinion definitively solves the problem in the macroscopic domainâ ([241], 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 (âa cemetery of buried hopesâ) he is reported to have said:
âSince 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. [âŠ] According to my and Mayerâs opinion, the fifth dimension will not show up. [âŠ] 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.âFootnote 171 [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 Kaluzaâs unitary field theory:
âThe only result of our investigation is the unification of gravitation and electricity, whereby the equations for the latter are just Maxwellâs equations for empty space. Hence, no physical progress is made, [if at all] at most only in the sense that one can see that Maxwellâs 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 \({a^\sigma }\) in fourdimensional space, which are bound to space by a linear mechanism. Let as be the 4-vector belonging to \({a^\sigma }\); then such a relation \({a^s} = \gamma _\sigma ^s{a^\sigma }\) obtains. In the theory equations are meaningful which hold independently of the special relationship generated by \(\gamma _\sigma ^s\). Infinitesimal transport of \({a^\sigma }\) in fourdimensional space is defined, likewise the corresponding 5-curvature from which spring the field equations.âFootnote 172 ([99], pp. 274â25)
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:
âThis 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.â ([94], p. 439)
Unfortunately, as in the case of his previous papers on Kaluzaâs 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 Einsteinâs and Mayerâs publication, with his student HoffmannFootnote 173, he had suggested an application to physics equivalent to the Kaluza-Klein theory [381, 163]. However, according to Pauli, Veblen and Hoffmann had spoiled the advantage of projective theory:
âBut 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 [âŠ].âFootnote 174 ([249], p. 307)
By using the idea that an affine (n+1)-space can be represented by a projective n-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 curvesâ 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 Mayerâs: It can house the additional scalar field inherent in Kaluzaâs original approach. Thus, Veblen and Hoffmann also gained the Klein.Gordon equation in curved space, i.e., an equation with the Ricci scalar R appearing besides its mass term. Interestingly, the curvature term reads as 5/27R ([381], p. 821). In his note, Hoffmann generalised the formalism such as to include Diracâs equations (without gravitation), although some technical difficulties remained. Nevertheless, Hoffman remained optimistic:
âThere 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.â ([163], p. 89)
In his book, Veblen emphasised
â[âŠ] that our theory starts from a physical and geometrical point of view totally different from KALUZAâs. In particular, we do not demand a relationship between electrical charge and a fifth coordinate; our theory is strictly four-dimensional.âFootnote 175 [379]
Shortly after Einsteinâs and Mayerâs 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 Maxwellâs 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
$$
{\gamma ^\iota }_{k;l} = {A^\iota }{F_{kl}} + {\gamma ^{\iota r}}{V_{rlk}},
$$
((159))
where, again, Fkl=-Flk, and the new Vrlk=Vrkl are arbitrary tensors. The field equations were set up according to the method of the first paper; now the 5-curvature scalar was \(P = R - {F_{kp}}{F^{kp}} - {V_{rqp}}{V^{rpq}}\). It also turned out that \({V^{rpq}} = { \epsilon ^{lrpq}}{\phi _l}\) with \({\phi _l} = \tfrac{{\partial \phi }}{{\partial {x^l}}}\), i.e., that the introduction of Vrpq brought only one additional variable. The electric current density became âŒVVprqFrq.
In the last paragraph, the compatibility of the equations was proven, and at the end Cartan was acknowledged:
âWe 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 âcompatibilityâ in this paper and in previous papers.âFootnote 176 [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 \({X^\alpha }\), with α= 1, âŠ, 5, and the invariant \({g_{\alpha \beta }}{X^\alpha }{X^\beta }\), 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 [316, 317, 315, 318]Footnote 177, which, according to Pauli, combine
âall advantages of the formulations of Kaluza-Klein and Einstein-Mayer while avoiding all their disadvantages.â ([249], p. 307)
Both the Einstein-Mayer theory and Veblen and Hoffmannâs approach turned out to be subcases of the more general scheme of Schouten and van Dantzig intending
âto give a unification of general relativity not only with Maxwellâs electromagnetic theory but also with Schrödingerâs and Diracâs theory of material waves.â ([318], p. 271)
In this paper ([318], p. 311, Figure 2), we find an early graphical representation of the parametrised set of all possible theories of a kindFootnote 178. The formalism of Schouten and van Dantzig allows for taking the additional dimension to be timelike; in their physical applications the metric of spacetime is taken as a Lorentz metric; torsion is also included in their geometry.
Pauli, with his student J. SolomonFootnote 179, generalised Klein, and Einstein and Mayer by allowing for an arbitrary signature in an investigation concerning âthe form that take Diracâs equations in the unitary theory of Einstein and MayerâFootnote 180 [253]. In a note added after proofreading, the authors showed that they had noted Schouten and Dantzigâs papers [316, 317]. The authors pointed out that
â[âŠ] even in the absence of gravitation we must pay attention to a difference between Diracâs equation in the theory of Einstein and Mayer, and Diracâs equation as it is written out, usually.âFootnote 181 ([253], p. 458)
The second order wave equation iterated from their form of Diracâs equation, besides the spin term contained a curvature term -1/4R, with the numerical factor different from Veblenâs and Hoffmannâs. In a sequel to this publication, Pauli and Solomon corrected an error:
âWe 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.[âŠ] 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 [âŠ] likewise for the current vector [âŠ].âFootnote 182 ([254], p. 582)
In the California Institute of Technology, Einsteinâs and Mayerâs new mathematical technique found an attentive reader as well; A. D. Michal and his co-author generalised the Einstein-Mayer 5-vector-formalism:
âThe geometry considered by Einstein and Mayer in their âUnified field theoryâ leads to the consideration of an n-dimensional Riemannian space Vn with a metric tensor gij, to each point of which is associated an m-dimensional linear vector space Vm, (m>n), for which vector spaces a general linear connection is defined. For the general case (mânâ 1) we find that the calculation of the mân âexceptional directionsâ is not unique, and that an additional postulate on the linear connection is necessary. Several of the new theorems give new results even for n=4, m=5, the Einstein-Mayer case.â [228]
Michal had come from Cartan and Schoutenâs papers on group manifolds and the distant parallelisms defined on them [227]. H. P. Robertson found a new way of applying distant parallelism: He studied groups of motion admitted by such spaces, e.g., by Einsteinâs and Mayerâs spherically symmetric exact solution [282] (cf. Section 6.4.3).
Cartan wrote a paper on the Einstein-Mayer theory as well ([39], 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.
Distant parallelism
The next geometry Einstein took as a fundament for unified field theory was a geometry with Riemannian metric, vanishing curvature, and non-vanishing torsion, named âabsolute parallelismâ, âdistant parallelismâ, âteleparallelism, or âFernparallelismusâ. The contributions from the Levi-Civita connection and from contorsionFootnote 183 in the curvature tensor cancel. In place of the metric, tetrads are introduced as the basic variables. As in Euclidean space, in the new geometry these 4-beins can be parallely translated to retain the same fixed directions everywhere. Thus, again, a degree of absoluteness is re-introduced into geometry in contrast to Weylâs first attempt at unification which tried to soften the ârigidityâ of Riemannian geometry.
The geometric concept of âfields of parallel vectorsâ had been introduced on the level of advanced textbooks by Eisenhart as early as 1925â1927 [119, 121] without use of the concept of a metric. In particular, the vanishing of the (affine) curvature tensor was given as a necessary and sufficient condition for the existence of D linearly independent fields of parallel vectors in a D-dimensional affine space ([121], p. 19).
Cartan and Einstein
As concerns the geometry of âFernparallelismâ, it is a special case of a space with Euclidean connection introduced by Cartan in 1922/23 [31, 30, 32]. Pais let Einstein âinventâ and âdiscoverâ distant parallelism, and he states that Einstein âdid not know that Cartan was already aware of this geometryâ ([241], pp. 344â345). However, when Einstein published his contributions in June 1928 [84, 83], Cartan had to remind him that a paper of his introducing the concept of torsion had
âappeared at the moment at which you gave your talks at the CollĂšge de France. I even remember having tried, at Hadamardâs place, to give you the most simple example of a Riemannian space with Fernparallelismus by taking a sphere and by treating as parallels two vectors forming the same angle with the meridians going through their two origins: the corresponding geodesics are the rhumb lines.âFootnote 184 (letter of Cartan to Einstein on 8 May 1929; cf. [50], p. 4)
This remark refers to Einsteinâs visit in Paris in March/April 1922. Einstein had believed to have found the idea of distant parallelism by himself. In this regard, Pais may be correct. Every researcher knows how an idea, heard or read someplace, can subconsciously work for years and then surface all of a sudden as his or her own new idea without the slightest remembrance as to where it came from. It seems that this happened also to Einstein. It is quite understandable that he did not remember what had happened six years earlier; perhaps, he had not even fully followed then what Cartan wanted to explain to him. In any case, Einsteinâs motivation came from the wish to generalise Riemannian geometry such that the electromagnetic field could be geometrized:
âTherefore, the endeavour of the theoreticians is directed toward finding natural generalisations of, or supplements to, Riemannian geometry in the hope of reaching a logical building in which all physical field concepts are unified by one single viewpoint.âFootnote 185 ([84], p. 217)
In an investigation concerning spaces with simply transitive continuous groups, Eisenhart already in 1925 had found the connection for a manifold with distant parallelism given 3 years later by Einstein [118]. He also had taken up Cartanâs idea and, in 1926, produced a joint paper with Cartan on âRiemannian geometries admitting absolute parallelismâ [40], and Cartan also had written about absolute parallelism in Riemannian spaces [33]. Einstein, of course, could not have been expected to react to these and other purely mathematical papers by Cartan and Schouten, focussed on group manifolds as spaces with torsion and vanishing curvature ([41, 34], pp. 50â54). No physical application had been envisaged by these two mathematicians.
Nevertheless, this story of distant parallelism raises the question of whether Einstein kept up on mathematical developments himself, or whether, at the least, he demanded of his assistants to read the mathematical literature. Against his familiarity with mathematical papers speaks the fact that he did not use the name âtorsionâ in his publications to be described in the following section. In the area of unified field theory including spinor theory, Einstein just loved to do the mathematics himself, irrespective of whether others had done it before â and done so even better (cf. Section 7.3).
Anyhow, in his response (Einstein to Cartan on 10 May 1929, [50], p. 10), Einstein admitted Cartanâs priority and referred also to Eisenhartâs book of 1927 and to Weitzenböckâs paper [393]. He excused himself byWeitzenböckâs likewise omittance of Cartanâs papers among his 14 references. In his answer, Cartan found it curious that Weitzenböck was silent because
â[âŠ] he indicates in his bibliography a note by Bortolotti in which he several times refers to my papers.âFootnote 186 (Cartan to Einstein on 15 May 1929; [50], p. 14)
The embarrassing situation was solved by Einsteinâs suggestion that he had submitted a comprehensive paper on the subject to Zeitschrift fĂŒr Physik, and he invited Cartan to add his description of the historical record in another paper (Einstein to Cartan on 10 May 1929). After Cartan had sent his historical review to Einstein on 24 May 1929, the latter answered three months later:
âI am now writing up the work for the Mathematische Annalen and should like to add yours [âŠ]. The publication should appear in the Mathematische Annalen because, at present, only the mathematical implications are explored and not their applications to physics.âFootnote 187 (letter of Einstein to Cartan on 25 August 1929 [50, 35, 89])
In his article, Cartan made it very clear that it was not Weitzenböck who had introduced the concept of distant parallelism, as valuable as his results were after the concept had become known. Also, he took Einsteinâs treatment of Fernparallelism as a special case of his more general considerations. Interestingly, he permitted himself to interpet the physical meaning of geometrical structuresFootnote 188:
âLet us say simply that mechanical phenomena are of a purely affine nature whereas electromagnetic phenomena are essentially metric; therefore it is rather natural to try to represent the electromagnetic potential by a not purely affine vector.âFootnote 189 ([35], p. 703)
Einstein explained:
âIn particular, I learned from Mr. Weitzenböck and Mr. Cartan that the treatment of continua of the species which is of import here, is not really new.[âŠ] In any case, what is most important in the paper, and new in any case, is the discovery of the simplest field laws that can be imposed on a Riemannian manifold with Fernparallelismus.âFootnote 190 ([89], p. 685)
For Einstein, the attraction of his theory consisted
âFor me, the great attraction of the theory presented here lies in its unity and in the allowed highly overdetermined field variables. I also could show that the field equations, in first approximation, lead to equations that correspond to the Newton-Poisson theory of gravitation and to Maxwellâs theory. Nevertheless, I still am far from being able to claim that the derived equations have a physical meaning. The reason is that I could not derive the equations of motion for the corpuscles.âFootnote 191 ([89], p. 697)
The split, in first approximation, of the tetrad field hab according to \({h_{ab}} = {\eta _{ab}} + {\bar h_{ab}}\) lead to homogeneous wave equations and divergence relations for both the symmetric and the antisymmetric part identified as metric and electromagnetic field tensors, respectively.
How the word spread
Einstein in 1929 really seemed to have believed that he was on a good track because, in this and the following year, he published at least 9 articles on distant parallelism and unified field theory before switching off his interest. The press did its best to spread the word: On 2 February 1929, in its column News and Views, the respected British science journal Nature reported:
âFor some time it has been rumoured that Prof. Einstein has been about to publish the results of a protracted investigation into the possibility of generalising the theory of relativity so as to include the phenomena of electromagnetism. It is now announced that he has submitted to the Prussian Academy of Sciences a short paper in which the laws of gravitation and of electromagnetism are expressed in a single statement.â
Nature then went on to quote from an interview of Einstein of 26 January 1929 in a newspaper, the Daily Chronicle. According to the newspaper, among other statements Einstein made, in his wonderful language, was the following:
âNow, but only now, we know that the force which moves electrons in their ellipses about the nuclei of atoms is the same force which moves our earth in its annual course about the sun, and it is the same force which brings to us the rays of light and heat which make life possible upon this planet.â [2]
Whether Einstein used this as a metaphorical language or, whether he at this time still believed that the system ânucleus and electronsâ is dominated by the electromagnetic force, remains open.
The paper announced by Nature is Einsteinâs âZur einheitlichen Feldtheorieâ, published by the Prussian Academy on 30 January 1929 [88]. A thousand copies of this paper had been sold within 3 days, so the presiding secretary of the Academy ordered the printing of a second thousand. Normally, only a hundred copies were printed ([183], Dokument Nr. 49, p. 136). On 4 February 1929, The Times (of London) published the translation of an article by Einstein, âwritten as an explanation of his thesis for readers who do not possess an expert knowledge of mathematicsâ. This article then became reprinted in March by the British astronomy journal The Observatory [86]. In it, Einstein first gave a historical sketch leading up to the introduction of relativity theory, and then described the method that guided him to the new theory of distant parallelism. In fact, the only formulas appearing are the line elements for two-dimensional Riemannian and Euclidean space. At the end, by one figure, Einstein tried to convey to the reader what consequence a Euclidean geometry with torsion would have â without using that name. His closing sentences areFootnote 192:
âWhich are the simplest and most natural conditions to which a continuum of this kind can be subjected? The answer to this question which I have attempted to give in a new paper yields unitary field laws for gravitation and electromagnetism.â ([86], p. 118)
A few months later in that year, again in Nature, the mathematician H. T. H. Piaggio gave an exposition for the general reader of âEinsteinâs and other Unitary Field Theoriesâ. He was a bit more explicit than Einstein in his article for the educated general reader. However, he was careful to end it with a warning:
âOf course the ultimate test of the theory must be by experiment. It may succeed in predicting some interaction between gravitation and electromagnetism which can be confirmed by observation. On the other hand, it may be only a âgraphâ and so outside the ken of the ordinary physicist.â ([258], p. 879)
The use of the concept âgraphâ had its origin in Eddingtonâs interpretation of his and other peoplesâ unified field theories to be only graphs of the world; the true geometry remained the Riemannian geometry underlying Einsteinâs general relativity.
Even the French-Belgian writer and poet Maurice Maeterlinck had heard of Einsteinâs latest achievement in the area of unified field theory. In his poetic presentation of the universe âLa grande fĂ©erieâFootnote 193 we find his remark:
âEinstein, in his last publications comments to which are still to appear, again brings us mathematical formulae which are applicable to both gravitation and electricity, as if these two forces seemingly governing the universe were identical and subject to the same law. If this were true it would be impossible to calculate the consequences.âFootnote 194 ([214], p. 68)
Einsteinâs research papers
We are dealing here with Einsteinâs, and Einstein and Mayerâs joint papers on distant parallelism in the reports of the Berlin Academy and Mathematische Annalen, which were taken as the starting point by other researchers following suit with further calculations. Indeed, there was a lot of work to do, only in part because Einstein, from one paper to the next, had changed his field equationsFootnote 195.
In his first note [84], dynamics was absent; Einstein made geometrical considerations his main theme: Introduction of a local ân-bein-fieldâ \(h_{\hat \imath}^k\) at every point of a differentiable manifold and the related object \({h_{k\hat \imath}}\) defined as the collection of the ânormed subdeterminants of the \(h_{\hat \imath}^k\)âFootnote 196 such that \(h_{i\hat l}h_{\hat l}^k = \delta _i^k\). As we have seen before, the components of the metric tensor are defined by
$$
{g_{lm}} = {h_{l\hat \jmath}}{h_{m\hat \jmath}},
$$
((160))
where summation over \(\hat \jmath = 1, \ldots ,n\) is assumedFootnote 197.
âFernparallelismâ now means that if the components referred to the local n-bein of a vector \({A^{\hat k}} = h_l^{\hat k}{A^l}\) at a point p, and of a vector \({B^{\hat k}}\) at a different point q are the same, i.e., \({A^{\hat k}} = {B^{\hat k}}\), then the vectors are to be considered as âparallelâ. There is an underlying symmetry, called ârotational invarianceâ by Einstein: joint rotations of each n-bein by the same angle. All relations with a physical meaning must be ârotationally invariantâ. Of course, in space-time with a Lorentz metric, the 4-bein-transformations do form the proper Lorentz group.
If parallel transport of a tangent vector A is defined as usual by \(d{A^k} = - {\Delta _{lm}}{A^l}d{x^m}\), then the connection components turn out to be
$$
{\Delta _{lm}}^k = {h^{k\hat \jmath}}\frac{{\partial {h_{l\hat \jmath}}}}{{\partial {x^m}}}.
$$
((161))
An immediate consequence is that the covariant derivative of each bein-vector vanishes,
$$
h_{\hat \jmath;l}^k: = h_{\hat \jmath;l}^k + {\nabla _{sl}}^kh_{\hat \jmath}^s = 0,
$$
((162))
by use of Equation (161). Also, the metric is covariantly constant
$${g_{ik;l}} = 0.$$
((163))
Neither fact is mentioned in Einsteinâs note. Also, no reference is given to Eisenhartâs paper of 1925 [118], in which the connection (161) had been given (Equation (3.5) on p. 248 of [118]), as noted above, its metric-compatibility shown, and the vanishing of the curvature tensor concluded.
The (Riemannian) curvature tensor calculated from Equation (161) turns out to vanish. As Einstein noted, by gij from Equation (160) also the usual Riemannian connection \({\Gamma _{lm}}^k(g)\) may be formed. Moreover, \({Y_{lm}}^k: = {\Gamma _{lm}}^k(g) - {\Delta _{lm}}^k\) is a tensor that could be used for building invariants. In principle, distant parallelism is a particular bi-connection theory. The connection \({\Gamma _{lm}}^k(g)\) does not play a role in the following (cf., however, de DonderFootnote 198âs paper [48]).
From Equation (161), obviously the torsion tensor \({S_{lm}}^k = \tfrac{1}{2}({\Delta _{lm}}^k - {\Delta _{ml}}^k) \ne 0\) follows (cf. Equation (21)). Einstein denoted it by \({\Lambda _{lm}}^k\) and, in comparison with the curvature tensor, considered it as the âformally simplestâ tensor of the theory for building invariants by help of the linear form \({\Lambda _{lj}}^jd{x^l}\) and of the scalars \({g^{ij}}{\Lambda _{im}}^l{\Lambda _{jl}}^m\) and \({g_{ij}}\;{g^{lr}}{g^{ms}}{\Lambda _{lm}}^i{\Lambda _{rs}}^{j.}\). He indicated how a Lagrangian could be built and the 16 field equations for the field variables hlj obtained.
At the end of the note Einstein compared his new approach to Weylâs and Riemannâs:
-
WEYL: Comparison at a distance neither of lengths nor of directions;
-
RIEMANN: Comparison at a distance of lengths but not of directions;
-
Present theory: Comparison at a distance of both lengths and directions.
In his second note [83], Einstein departed from the Lagrangian \({\mathcal L} = h\;{g^{ij}}{\Lambda _{im}}^l{\Lambda _{jl}}^m\), i.e., a scalar density corresponding to the first scalar invariant of his previous noteFootnote 199. He introduced \({\phi _k}: = {\Lambda _{kl}}^l\), and took the case Ï=0 to describe a âpurely gravitational fieldâ. However, as he added in a footnote, pure gravitation could have been characterised by \(\tfrac{{\partial {\phi _i}}}{{\partial {x^k}}} - \tfrac{{\partial {\phi _k}}}{{\partial {x^i}}}\) as well. In his first paper on distant parallelism, Einstein did not use the names âelectrical potentialâ or âelectrical fieldâ. He then showed that in a first-order approximation starting from \({h_{i\hat \jmath}} = {\delta _{i\hat \jmath}} + {j_{i\hat \jmath}} + \ldots\), both the Einstein vacuum field equations and Maxwellâs equations are surfacing. To do so he replaced \({h_{i\hat \jmath }}\) by \({g_{ij}} = {\delta _{ij}} + 2{k_{(ij)}}\) and introduced \({\phi _k}: = \tfrac{1}{2}\left( {\tfrac{{\partial {k_{kj}}}}{{\partial {x^j}}} - \tfrac{{\partial {k_{jj}}}}{{\partial {x^k}}}} \right)\). Einstein concluded that
âThe separation of the gravitational and the electromagnetic field appears artificial in this theory. [âŠ] Furthermore, it is remarkable that, according to this theory, the electromagnetic field does not enter the field equations quadratically.âFootnote 200 ([83], p. 6)
In a postscript, Einstein noted that he could have obtained similar results by using the second scalar invariant of his previous note, and that there was a certain indeterminacy as to the choice of the Lagrangian.
This shows clearly that the ambiguity in the choice of a Lagrangian had bothered Einstein. Thus, in his third note, he looked for a more reassuring way of deriving field equations [88]. He left aside the Hamiltonian principle and started from identities for the torsion tensor, following from the vanishing of the curvature tensorFootnote 201. He thus arrived at the identity given by Equation (29), i.e., (Einsteinâs equation (3), p. 5; his convention is \({\Lambda _{\hat k\hat l}}\,^{\hat \imath}: = 2{S_{\hat k\hat l}}\,^{\hat \imath}\))
$$
0 = 2{\nabla _{\{ \hat \jmath}}{S_{\hat k\hat l\} }}^{\hat \imath} + 4{S_{\hat m\{ \hat \jmath}}^{\hat \imath}{S_{\hat k\hat l\} }}^{\hat m}.
$$
((164))
By defining \({\phi _{\hat k}}: = {\Lambda _{\hat k\hat l}}^{\;\hat l} = 2\;{S_{\hat k\hat l}}^{\;\hat l}\), and contracting equation (164), Einstein obtained another identityFootnote 202:
$$
\nabla { * _j}{\hat V_{\hat k\hat l}}^j = 0,
$$
((165))
where the covariant divergence refers to the connection components \({\Delta _{\hat m\hat l}}^{\hat k}\), and the tensor density \({\hat V_{\hat k\hat l}}^{\hat \jmath}\) is given by
$$
{\hat V_{\hat k\hat l}}^{\hat \jmath }: = 2h({S_{\hat k\hat l}}^{\,\hat \jmath} + {\phi _{[\hat l}}{\delta _{\hat k]}}^{\hat \jmath}).
$$
For the proof, he used the formula for the covariant vector density given in Equation (16), which, for the divergence, reduces to \({\mathop \nabla \limits^ + _i}{X^i} = \tfrac{{\partial {{\hat X}^i}}}{{\partial {x^i}}} - 2{S_j}{\hat X^j}\).
The second identity used by Einstein follows with the help of Equation (27) for vanishing curvature (Einsteinâs equation (5), p. 5):
$$
{\mathop \nabla \limits^ + _{[j}}{\mathop \nabla \limits^ + _{k]}}\;{\hat A^{jk}} = - {S_{jk}}^r{\mathop \nabla \limits^ + _r}{\hat A^{jk}}.
$$
((166))
As we have seen in Section 2, if he had read it, Einstein could have taken these identities from Schoutenâs book of 1924 [300].
By replacing Ăjk by VÌkÌlÌj and using Equation (165), the final form of the second identity now is
$$
\nabla { * _j}({\nabla _{\hat l}}{\hat V^{\hat k\hat lj}} - 2{\hat V^{\hat klr}}{S_{lr}}^j) = 0.
$$
((167))
Einstein first wrote down a preliminary set of field equations from which, in first approximation, both the gravitational vacuum field equations (in the limit â=0, cf. below) and Maxwellâs equations follow:
$$
\begin{array}{*{20}{c}}
{\nabla { * _j}{{\hat U}_{\hat k\hat l}}^j = 0,\;\;\;\;\;}&{\nabla { * _r}\nabla { * _l}{{\hat V}_{\hat k}}^{lr} = 0.}
\end{array}
$$
((168))
Here,
$$
{\hat U_{\hat k\hat l}}^{\hat \jmath}: = {\hat V_{\hat k\hat l}}^{\hat \jmath} - 2 \epsilon h\;{\phi _{[\hat l}}\;\delta _{\hat k]}^{\hat \jmath}
$$
((169))
replaces \({\hat V_{kl}}^j\) such that the necessary number of equations is obtained. With this first approximation as a hint, Einstein, after some manipulations, postulated the 20 exact field equations:
$$
\begin{array}{*{20}{c}}
{\nabla { * _{\hat l}}{{\hat V}^{\hat k\hat lr}} - 2{{\hat V}^{\hat krs}}{S_{sr}}^{\;l} = 0,\;\;\;\;\;}&{\nabla { * _j}[h\;ph{i^{[k;j]}}] = 0,}
\end{array}
$$
((170))
among which 8 identities hold.
Einstein seems to have sensed that the average reader might be able to follow his path to the postulated field equations only with difficulty. Therefore, in a postscript, he tried to clear up his motivation:
âThe field equations suggested in this paper may be characterised with regard to other such possible ones in the following way. By staying close to the identity (167), it has been accomplished that not only 16, but 20 independent equations can be imposed on the 16 quantities h
kÌi
By âindependentâ we understand that none of these equations can be derived from the remaining ones, even if there exist 8 identical (differential) relations among them.âFootnote 203 ([88], p. 8)
He still was not entirely sure that the theory was physically acceptable:
âA deeper investigation of the consequences of the field equations (170) will have to show whether the Riemannian metric, together with distant parallelism, really gives an adequate representation of the physical qualities of space.âFootnote 204
In his second paper of 1929, the fourth in the series in the Berlin Academy, Einstein returned to the Hamiltonian principle because his collaborators Lanczos and MĂŒntzFootnote 205 had doubted the validity of the field equations of his previous publication [88] on grounds of their unproven compatibility. In the meantime, however, he had found a Lagrangian such that the compatibility-problem disappeared. He restricted the many constructive possibilities for \({\mathcal L} = {\mathcal L}(h_i^{\hat k},{\partial _l}h_i^{\hat k})\) by asking for a Lagrangian containing torsion at most quadratically. His Lagrangian is a particular linear combination of the three possible scalar densities, as follows:
-
(1)
\(\hat H = {\tfrac{1}{2}}{{\hat \jmath }_1} + {\tfrac{1}{4}}{{\hat \jmath }_2} - {{\hat \jmath }_3},\)
-
(2)
\(\hat H* = {\tfrac{1}{2}}{\hat \jmath _1} - {\tfrac{1}{4}}{\hat \jmath _2},\)
-
(3)
\(\hat H** = {{\hat \jmath }_3},\)
with \({{\hat \jmath }_1}: = h\;{S_{kl}}^m{S^k}{_m^l}\), \({{\hat \jmath }_2}: = h\;{S_{kl}}^m{S^{kl}}_m\), and \({{\hat \jmath }_3}: = h\;{S_j}{S^j}\). If â1, â2 are small parameters, then his final Lagrangian is \({\mathcal L} = \hat H + { \epsilon _1}\hat H* + { \epsilon _2}\hat H**\). In order to prove that Maxwellâs equations follow from his Lagrangian, Einstein had to perform the limit \(\sigma : = \tfrac{{{ \epsilon _1}}}{{{ \epsilon _2}}} \to 0\) in an expression termed \(\hat G*{\;^{\;ik}}\), which he assumed to depend homogeneously and quadratically on a linear combination of torsionFootnote 206.
In a Festschrift for his former teacher and colleague in ZĂŒrich, A. Stodola, Einstein summed up what he had reachedFootnote 207. He exchanged the definition of the invariants named \({{\hat \jmath}_2}\), and \({{\hat \jmath}_3}\), and stated that a choice of A=-B, C=0 in the Lagrangian \(\hat \jmath = A{{\hat \jmath }_1} + B{{\hat \jmath }_2} + C{{\hat \jmath }_3}\) would give field equations
â[âŠ] that coincide in first approximation with the known laws for the gravitational and electromagnetic field [âŠ]âFootnote 208
with the proviso that the specialisation of the constants A, B, C must be made only after the variation of the Lagrangian, not before. Also, together with MĂŒntz, he had shown that for an uncharged mass point the Schwarzschild solution again obtained [87].
Einsteinâs next publication was the note preceding Cartanâs paper in Mathematische Annalen [89]. He presented it as an introduction suited for anyone who knew general relativity. It is here that he first mentioned Equations (162) and (163). Most importantly, he gave a new set of field equations not derived from a variational principle; they areFootnote 209.
$${G^{ik}}: = S{_{\;.\;\;.}^{il\;k}{_{\left\| l \right.}}} - S_{.\;\;.}^{imn}S_{nm}^{\;\;\;k} = 0,$$
((171))
$$
{F_{ik}}: = \;\;\;\;\;S_{ik}^{..}\;^l\,_{\left\| l \right.}\;\;\;\;\; = 0,
$$
((172))
where Sikl=gimgknSmnl. There exist 4 identities among the 16+6 field equations
$$
{G^{il}}_{\left\| l \right.} - {F^{il}}_{\left\| l \right.} + S_{.\;\;.}^{imn}{F_{nm}} = 0.
$$
((173))
As Cartan remarked, Equation (172) expresses conservation of torsion under parallel transport:
âIn fact, in the new theory of Mr. Einstein, it is natural to call a universe homogeneous if the torsion vectors that are associated to two parallel surface elements are parallel themselves; this means that parallel transport conserves torsion.âFootnote 210 ([35], p. 703)
From Equation (173) with the help of Equation (171), (172), Einstein wrote down two more identities. One of them he had obtained from Cartan:
âBut I am very grateful to you for the identity
$$
{G^{ik}}_{\left\| i \right.} - {S_{lm}}^{\;k}{G^{lm}} = 0,
$$
which, astonishingly, had escaped me. [âŠ] In a new presentation in the Sitzungsberichten, I used this identity while taking the liberty of pointing to you as its source.â
Footnote 211 (letter of Einstein to Cartan from 18 December 1929, Document X of [50], p. 72)
In order to show that his field equations were compatible he counted the number of equations, identities, and field quantities (in n-dimensional space) to find, in the end, n2+n equations for the same number of variables. To do so, he had to introduce an additional variable Ï via \({F_k} = {\phi _k} - \tfrac{{\partial \log \psi }}{{\partial {x^k}}}\). Here, Fk is introduced by \({F_{ik}} = {\partial _k}{F_i} - {\partial _i}{F_k} = {\partial _k}{\phi _i} - {\partial _i}{\phi _k}\). Einstein then showed that \({\partial _l}{F_{ik}} + {\partial _k}{F_{li}} + {\partial _i}{F_{kl}} = 0\).
The changes in his approach Einstein continuously made, must have been hard on those who tried to follow him in their scientific work. One of them, ZaycoffFootnote 212, tried to make the best out of them:
âRecently, A. Einstein ([89]), following investigations by E. Cartan ([35]), has considerably modified his teleparallelism theory such that former shortcomings (connected only to the physical identifications) vanish by themselves.âFootnote 213 ([433], p. 410)
In November 1929, Einstein gave two lectures at the Institute Henri PoincarĂ© in Paris which had been opened one year earlier in order to strengthen theoretical physics in France ([14], pp. 263â272). They were published in 1930 as the first article in the new journal of this institute [92]. On 23 pages he clearly and leisurely outlined his theory of distant parallelism and the progress he had made. As to references given, first Cartanâs name is mentioned in the text:
âIt is not for the first time that such spaces are envisaged. From a purely mathematical point of view they were studied previously. M. Cartan was so amiable as to write a note for the Mathematische Annalen exposing the various phases in the formal development of these concepts.âFootnote 214 ([92], p. 4)
Note that Einstein does not say that it was Cartan who first âenvisagedâ these spaces before. Later in the paper, he comes closer to the point:
âThis type of space had been envisaged before me by mathematicians, notably by WEITZENBĂCK, EISENHART et CARTAN [âŠ].âFootnote 215 [92]
Again, he held back in his support of Cartanâs priority claim.
Some of the material in the paper overlaps with results from other publications [85, 90, 93]. The counting of independent variables, field equations, and identities is repeated from Einsteinâs paper in Mathematische Annalen [89]. For n=4, there were 20 field equations (Gik=0, Fl=0) for 16+1 variables \(h_{\hat \imath}^k\) and Ï, four of which were arbitrary (coordinate choice). Hence 7 identities should exist, four of which Einstein had found previously. He now presented a derivation of the remaining three identities by a calculation of two pagesâ length. The field equations are the same as in [89]; the proof of their compatibility takes up, in a slightly modified form, the one communicated by Einstein to Cartan in a letter of 18 December 1929 ([92], p. 20). It is reproduced also in [90].
Interestingly, right after Einsteinâs article in the instituteâs journal, a paper of C. G. Darwin, âOn the wave theory of matterâ, is printed, and, in the same first volume, a report of Max Born on âSome problems in Quantum Mechanics.â Thus, French readers were kept up-to-date on progress made by both parties â whether they worked on classical field theory or quantum theory [45, 21].
A. Proca, who had attended Einsteinâs lectures, gave an exposition of them in a journal of his native Romania. He was quite enthusiastic about Einsteinâs new theory:
âA great step forward has been made in the pursuit of this total synthesis of phenomena which is, right or wrong, the ideal of physicists. [âŠ] the splendid effort brought about by Einstein permits us to hope that the last theoretical difficulties will be vanquished, and that we soon will compare the consequences of the theory with [our] experience, the great stepping stone of all creations of the mind.âFootnote 216 [260, 261]
Einsteinâs next paper in the Berlin Academy, in which he reverts to his original notation h
kÌl
, consisted of a brief critical summary of the formalism used in his previous papers, and the announcement of a serious mistake in his first note in 1930, which made invalid the derivation of the field equations for the electromagnetic field ([90], p. 18). The mistake was the assumption on the kind of dependence on torsion of the quantity Ä*ik, which was mentioned above. Also, Einstein now found it better âto keep the concept of divergence, defined by contraction of the extension of a tensorâ and not use the covariant derivative \(\nabla { * _l}\) introduced by him in his third paper in the Berlin Academy [88].
Then Einstein presented the same field equations as in his paper in Annalen der Mathematik, which he demanded to be
-
(1)
covariant,
-
(2)
of second order, and
-
(3)
linear in the second derivatives of the field variable h
ki
.
while these demands had been sufficient to uniquely lead to the gravitational field equations (with cosmological constant) of general relativity, in the teleparallelism theory a great deal of ambiguity remained. Sixteen field equations were needed which, due to covariance, induced four identities.
âTherefore equations must be postulated among which identical relations are holding. The higher the number of equations (and consequently also the number of identities among them), the more precise and stronger than mere determinism is the content; accordingly, the theory is the more valuable, if it is also consistent with the empirical facts.âFootnote 217 ([90], p. 21)
He then gave a proof of the compatibility of his field equations:
âThe proof of the compatibility, as given in my paper in the Mathematische Annalen, has been somewhat simplified due to a communication which I owe to a letter of Mr. CARTAN (cf. §3, [16]).âFootnote 218
The reader had to make out for himself what Cartanâs contribution really was.
In linear approximation, i.e., for \({h_{ik}} = {\delta _{ik}} + {\bar h_{ik}}\), Einstein obtained dâAlembertâs equation for both the symmetric and the antisymmetric part of hÌik, identified with the gravitational and the electromagnetic field, respectively.
Einsteinâs next note of one and a half pages contained a mathematical result within teleparallelism theory: From any tensor with an antisymmetric pair of indices a vector with vanishing divergence can be derived [93].
In order to test the field equations by exhibiting an exact solution, a simple case would be to take a spherically symmetric, asymptotically (Minkowskian) 4-bein. This is what Einstein and Mayer did, except with the additional assumption of space-reflection symmetry [106]. Then the 4-bein contains three arbitrary functions of one parameter s:
$$
\begin{array}{*{20}{c}}
{h_{\hat \imath}^\alpha = \lambda (s)\delta _{\hat \imath}^\alpha ,\;\;\;}&{h_{\hat 4}^\alpha = \tau (s){x_\alpha },\;\;\;}&{h_{\hat \imath}^4 = 0,\;\;\;}&{h_{\hat 4}^{\hat 4} = u(s),}
\end{array}
$$
((174))
where α, \(\hat \imath = 1,2,3\). As an exact solution of the field equations (171, 172), Einstein and Mayer obtained \(\lambda = \tau = e{s^{ - 3}}{(1 - {e^2}/{s^4})^{ - \tfrac{1}{4}}})\) and \(u = 1 + m\;\int {ds\;{s^{ - 2}}(1 - {e^2}/{s^4})} {\;^{ - \tfrac{1}{4}}}\). The constants e and m were interpreted as electric charge and âponderomotive mass,â respectively. A further exact solution for uncharged point particles was also derived; it is static and corresponds to âtwo or more unconnected electrically neutral masses which can stay at rest at arbitrary distancesâ. Einstein and Mayer do not take this physically unacceptable situation as an argument against the theory, because the equations of motion for such singularities could not be derived from the field equations as in general relativity. Again, the continuing wish to describe elementary particles by singularity-free exact solutions is stressed.
Possibly, W. F. G. Swan of the Bartol Research Foundation in Swarthmore had this paper in mind when he, in April 1930, in a brief description of Einsteinâs latest publications, told the readers of Science:
âIt now appears that Einstein has succeeded in working out the consequences of his general law of gravity and electromagnetism for two special cases just as Newton succeeded in working out the consequences of his law for several special cases. [âŠ] It is hoped that the present solutions obtained by Einstein, or if not these, then others which may later evolve, will suggest some experiments by which the theory may be tested.â ([339], p. 391)
Two days before the paper by Einstein and Mayer became published by the Berlin Academy, Einstein wrote to his friend Solovine:
âMy field theory is progressing well. Cartan has already worked with it. I myself work with a mathematician (S. MayerFootnote 219 from Vienna), a marvelous chap [âŠ].âFootnote 220 ([98], p. 56)
The mentioning of Cartan resulted from the intensive correspondence of both scientists between December 1929 and February 1930: About a dozen letters were exchanged which, sometimes, contained long calculations [50] (cf. Section 6.4.6). In an address given at the University of Nottingham, England, on 6 June 1930, Einstein also must have commented on the exact solutions found and on his program concerning the elementary particles. A report of this address stated about Einsteinâs program:
âThe problem is nearly solved; and to the first approximations he gets laws of gravitation and electro-magnetics. He does not, however, regard this as sufficient, though those laws may come out. He still wants to have the motions of ordinary particles to come out quite naturally. [The program] has been solved for what he calls the âquasistatical motionsâ, but he also wants to derive elements of matter (electrons and protons) out of the metric structure of space.â ([91], p. 610)
With his âassistantâ Walther Mayer, Einstein then embarked on a very technical, systematic study of compatible field equations for distant parallelism [108]. In addition to the assumptions (1), (2), (3) for allowable field equations given above, further restrictions were made:
-
(4)
the field equations must contain the first derivatives of the field variable h
kÌi
only quadratically;
-
(5)
the identities for the left hand sides Gik of the field equations must be linear in Gik and contain only their first derivatives;
-
(6)
torsion must occur only linearly in Gik.
For the field equations, the following ansatz was made:
$$
0 = {G^{ik}} = pS_{.\;\;.\;\;\left\| l \right.}^{il\;\;k} + qS_{.\;\;.\;\;\left\| l \right.}^{kl\;\;i} + {a_1}\phi _{.\;\;.}^{i\left\| k \right.} + {a_2}\phi _{.\;\;.}^{k\left\| i \right.} + {a_3}g_{.\;\;.}^{ik}\phi _{\left\| l \right.}^l + {R^{ik}},
$$
((175))
where Rik is a collection of terms quadratic in torsion S, and p, q, a1, a2, a3 are constants. They must be determined in such a way that the âdivergence-identityâ
$$
{G^{ik}}_{;i} + {G^{ki}}_{;i} + {G^{lm}}({c_1}{S_{lm}}^k + {c_2}{S_l}^{\; \cdot m}\,_k + {c_3}{S_m}^{\; \cdot l}\,_k) + {c_4}{G^{kl}}{\phi _l} + {c_5}{G^{lk}}{\phi _l} + {c_6}G_l^ \cdot \phi _ \cdot ^k + B{G^{l \cdot }}\,_{l\left\| \cdot \right.}\,^k = 0
$$
((176))
is satisfied. Here, 8 new constants A, B, cr with r=1, âŠ, 6 to be fixed in the process also appear. After inserting Equation (175) into Equation (176), Einstein and Mayer reduced the problem to the determination of 10 constants by 20 algebraic equations by a lengthy calculation. In the end, four different types of compatible field equations for the teleparallelism theory remained:
âTwo of these are (non-trivial) generalisations of the original gravitational field equations, one of them being known already as a consequence of the Hamiltonian principle. The remaining two types are denoted in the paper by [âŠ].âFootnote 221
With no further restraining principles at hand, this ambiguity in the choice of field equations must have convinced Einstein that the theory of distant parallelism could no longer be upheld as a good candidate for the unified field theory he was looking for, irrespective of the possible physical contentFootnote 222. Once again, he dropped the subject and moved on to the next. While aboard a ship back to Europe from the United States, Einstein, on 21 March 1932, wrote to Cartan:
â[âŠ] In any case, I have now completely given up the method of distant parallelism. It seems that this structure has nothing to do with the true character of space [âŠ].â ([50], p. 209)
What Cartan might have felt, after investing the forty odd pages of his calculations printed in Debeverâs book, is unknown. However, the correspondence on the subject came to an end in May 1930 with a last letter by Cartan.
Footnote 223.
Reactions I: Mostly critical
About half a year after Einsteinâs two papers on distant parallelism of 1928 had appeared, ReichenbachFootnote 224, who always tended to defend Einstein against criticism, classified the new theory [268] according to the lines set out in his book [267] as âhaving already its precisely fixed logical position in the edifice of Weyl-Eddington geometryâ ([267], p. 683). He mentioned as a possible generalization an idea of Einsteinâs, in which the operation of parallel transport might be taken as integrable not with regard to length but with regard to direction: âa generalisation which already has been conceived by Einstein as I learned from himâ ([267], p. 687)Footnote 225.
As concerns parallelism at a distance, Reichenbach was not enthusiastic about Einsteinâs new approach:
â[âŠ] it is the aim of Einsteinâs new theory to find such an entanglement between gravitation and electricity that it splits into the separate equations of the existing theory only in first approximation; in higher approximation, however, a mutual influence of both fields is brought in, which, possibly, leads to an understanding of questions unanswered up to now as [is the case] for the quantum riddle. But this aim seems to be in reach only if a direct physical interpretation of the operation of transport, even of the immediate field quantities, is given up. From the geometrical point of view, such a path [of approach] must seem very unsatisfactory; its justifications will only be reached if the mentioned link does encompass more physical facts than have been brought into it for building it up.âFootnote 226 ([267], p. 689)
A first reaction from a competing colleague came from Eddington, who, on 23 February 1929, gave a cautious but distinct review of Einsteinâs first three publications on distant parallelism [84, 83, 88] in Nature. After having explained the theory and having pointed out the differences to his own affine unified field theory of 1921, he confessed:
âFor my own part I cannot readily give up the affine picture, where gravitational and electric quantities supplement one another as belonging respectively to the symmetrical and antisymmetrical features of world measurement; it is difficult to imagine a neater kind of dovetailing. Perhaps one who believes that Weylâs theory and its affine generalisation afford considerable enlightenment, may be excused for doubting whether the new theory offers sufficient inducement to make an exchange.â [62]
Weyl was the next unhappy colleague; in connection with the redefinition of his gauge idea he remarked (in April/May 1929):
â[âŠ] my approach is radically different, because I reject distant parallelism and keep to Einsteinâs general relativity. [âŠ] Various reasons hold me back from believing in parallelism at a distance. First, my mathematical intuition a priori resists to accept such an artificial geometry; I have difficulties to understand the might who has frozen into rigid togetherness the local frames in different events in their twisted positions. Two weighty physical arguments join in [âŠ] only by this loosening [of the relationship between the local frames] the existing gauge-invariance becomes intelligible. Second, the possibility to rotate the frames independently, in the different events, [âŠ] is equivalent to the s y m m e t r y o f t h e e n e r g y - m o m e n t u m t e n s o r, or to the validity of the conservation law for angular momentum.âFootnote 227 ([407], pp. 330â332.)
As usual, Pauli was less than enthusiastic; he expressed his discontent in a letter to Hermann Weyl of 26 August 1929:
âFirst let me emphasize that side of the matter about which I fully agree with you: Your approach for incorporating gravitation into Diracâs theory of the spinning electron [âŠ] I am as adverse with regard to Fernparallelismus as you are [âŠ] (And here I must do justice to your work in physics. When you made your theory with gâČik=λgik this was pure mathematics and unphysical; Einstein rightly criticised and scolded you. Now the hour of revenge has come for you, now Einstein has made the blunder of distant parallelism which is nothing but mathematics unrelated to physics, now you may scold [him].)âFootnote 228 ([251], pp. 518â519)
Another confession of Pauliâs went to Paul Ehrenfest:
âBy the way, I now no longer believe in one syllable of teleparallelism; Einstein seems to have been abandoned by the dear Lord.âFootnote 229 (Pauli to Ehrenfest 29 September 1929; [251], p. 524)
Pauliâs remark shows the importance of ideology in this field: As long as no empirical basis exists, beliefs, hopes, expectations, and rationally guided guesses abound. Pauliâs letter to Weyl from 1 July 1929 used non-standard language (in terms of science):
âI share completely your skeptical position with regard to Einsteinâs 4-bein geometry. During the Easter holidays I have visited Einstein in Berlin and found his opinion on modern quantum theory reactionary.âFootnote 230 ([251], p. 506)
while the wealth of empirical data supporting Heisenbergâs and Schrödingerâs quantum theory would have justified the use of a word like âuninformedâ or even ânot up to dateâ for the description of Einsteinâs position, use of âreactionaryâ meant a definite devaluation.
Einstein had sent a further exposition of his new theory to the Mathematische Annalen in August 1928. When he received its proof sheets from Einstein, Pauli had no reservations to criticise him directly and bluntly:
âI thank you so much for letting be sent to me your new paper from the Mathematische Annalen [89], which gives such a comfortable and beautiful review of the mathematical properties of a continuum with Riemannian metric and distant parallelism [âŠ]. Unlike what I told you in spring, from the point of view of quantum theory, now an argument in favour of distant parallelism can no longer be put forward [âŠ]. It just remains [âŠ] to congratulate you (or should I rather say condole you?) that you have passed over to the mathematicians. Also, I am not so naive as to believe that you would change your opinion because of whatever criticism. But I would bet with you that, at the latest after one year, you will have given up the entire distant parallelism in the same way as you have given up the affine theory earlier. And, I do not wish to provoke you to contradict me by continuing this letter, because I do not want to delay the approach of this natural end of the theory of distant parallelism.âFootnote 231 (letter to Einstein of 19 December 1929; [251], 526â527)
Einstein answered on 24 December 1929:
âYour letter is quite amusing, but your statement seems rather superficial to me. Only someone who is certain of seeing through the unity of natural forces in the right way ought to write in this way. Before the mathematical consequences have not been thought through properly, is not at all justified to make a negative judgement. [âŠ] That the system of equations established by myself forms a consequential relationship with the space structure taken, you would probably accept by a deeper study â more so because, in the meantime, the proof of the compatibility of the equations could be simplified.âFootnote 232 ([251], p. 582)
Before he had written to Einstein, Pauli, with lesser reservations, complained vis-a-vis Jordan:
âEinstein is said to have poured out, at the Berlin colloquium, horrible nonsense about new parallelism at a distance. The mere fact that his equations are not in the least similar to Maxwellâs theory is employed by him as an argument that they are somehow related to quantum theory. With such rubbish he may impress only American journalists, not even American physicists, not to speak of European physicists.âFootnote 233 (letter of 30 November 1929, [251], p. 525)
Of course, Pauliâs spells of rudeness are well known; in this particular case they might have been induced by Einsteinâs unfounded hopes for eventually replacing the Schrödinger-Heisenberg-Dirac quantum mechanics by one of his unified field theories.
The question of the compatibility of the field equations played a very important role because Einstein hoped to gain, eventually, the quantum laws from the extra equations (cf. his extended correspondence on the subject with Cartan ([50] and Section 6.4.6).
That Pauli had been right (except for the time span envisaged by him) was expressly admitted by Einstein when he had given up his unified field theory based on distant parallelism in 1931 (see letter of Einstein to Pauli on 22 January 1932; cf. [241], p. 347).
Bornâs voice was the lonely approving one (Born to Einstein on 23 September 1929)Footnote 234:
âYour report on progress in the theory of Fernparallelism did interest me very much, particularly because the new field equations are of unique simplicity. Until now, I had been uncomfortable with the fact that, aside from the tremendously simple and transparent geometry, the field theory did look so very involvedâFootnote 235 ([154], p. 307)
Born, however, was not yet a player in unified field theory, and it turned out that Einsteinâs theory of distant parallelism became as involved as the previous ones.
Einsteinâs collaborator Lanczos even wrote a review article about distant parallelism with the title âThe new field theory of Einsteinâ [201]. In it, Lanczos cautiously offers some criticism after having made enough bows before Einstein:
âTo be critical with regard to the creation of a man who has long since obtained a place in eternity does not suit us and is far from us. Not as a criticism but only as an impression do we point out why the new field theory does not house the same degree of conviction, nor the amount of inner consistency and suggestive necessity in which the former theory excelled.[âŠ] The metric is a sufficient basis for the construction of geometry, and perhaps the idea of complementing RIEMANNian geometry by distant parallelism would not occur if there were the wish to implant something new into RIEMANNian geometry in order to geometrically interpret electromagnetism.âFootnote 236 ([201], p. 126)
When Pauli reviewed this review, he started with the scathing remark
âIt is indeed a courageous deed of the editors to accept an essay on a new field theory of Einstein for the âResults in the Exact Sciencesâ [literal translation of the journalâs title]. His never-ending gift for invention, his persistent energy in the pursuit of a fixed aim in recent years surprise us with, on the average, one such theory per year. Psychologically interesting is that the author normally considers his actual theory for a while as the âdefinite solutionâ. Hence, [âŠ] one could cry out: âEinsteinâs new field theory is dead. Long live Einsteinâs new field theory!ââFootnote 237 ([248], p. 186)
For the remainder, Pauli engaged in a discussion with the philosophical background of Lanczos and criticised his support for Mieâs theory of matter of 1913 according to which
âthe atomism of electricity and matter, fully separated from the existence of the quantum of action, is to be reduced to the properties of (singularity-free) eigen-solutions of still-to-be-found nonlinear differential equations for the field variables.âFootnote 238
Thus, Pauli lightly pushed aside as untenable one of Einsteinâs repeated motivations and hoped-for tests for his unified field theories.
Lanczos, being dissatisfied with Einsteinâs distant parallelism, then tried to explain âelectromagnetism as a natural property of Riemannian geometryâ by starting from the Lagrangian quadratic in the components of the Ricci tensor: \({\mathcal L} = {R_{ik}}{R_{lm}}{g^{il}}{g^{km}} + C{({R_{ik}}{g^{ik}})^2}\) with an arbitrary constant C. He varied gik and Rik independently [202]. (For Lanczos see J. Stachelâs essay âLanczosâ early contributions to relativity and his relation to Einsteinâ in [330], pp. 499â518.)
Reactions II: Further research on distant parallelism
The first reactions to Einsteinâs papers came quickly. On 29 October 1928, de Donder suggested a generalisation by using two metric tensors, a space-time metric gik, and a bein-metric g
âik
, connected to the 4-bein components \({h_{l\hat \jmath }}\) by
$$
{g_{lm}} = g_\star^{\hat \jmath \hat k}{h_{l\hat \jmath}}{h_{m\hat k}}.
$$
((177))
In place of Einsteinâs connection (161), defined through the 4-bein only, he took:
$$
{\Delta _{lm}}^k = \frac{1}{2}{h^{k\hat \jmath}}({h_{l\hat \jmath \cdot m}} - {h_{m\hat \jmath \cdot l}}),
$$
((178))
where the dot-symbol denotes covariant derivation by help of the Levi-Civita connection derived from g
âik
. If the Minkowski metric is used as a bein metric gâ, then the dot derivative reduces to partial derivation, and Einsteinâs original connection is obtained [48].
Another application of Einsteinâs new theory came from Eugen Wigner in Berlin whose paper showing that the tetrads in distant-parallelism-theory permitted a generally covariant formulation of âDiracs equation for the spinning electronâ, was received by Zeitschrift fĂŒr Physik on 29 December 1928 [419]. He did point out that âup to now, grave difficulties stood in the way of a general relativistic generalisation of Diracâs theoryâ and referred to a paper of Tetrode [344]. Tetrode, about a week after Einsteinâs first paper on distant parallelism had appeared on 14 June 1928, had given just such a generally relativistic formulation of Diracâs equation through coordinate dependent Gamma-matricesFootnote 239; he also wrote down a (symmetric) energy-momentum tensor for the Dirac field and the conservation laws. However, he had kept the metric gik introduced into the formalism by
$$
{\gamma _i}{\gamma _k} + {\gamma _k}{\gamma _i} = 2{g_{ik}}
$$
((179))
to be conformally flat. For the matrix-valued 4-vector Îłi he prescribed the condition of vanishing divergence. Wigner did not fully accept Tetrodeâs derivations because there, implicitly and erroneously, it had been assumed that the two-dimensional representation of the Lorentz group (2-spinors) could be extended to a representation of the affine group. Wigner stated that such difficulties would disappear if Einsteinâs teleparallelism theory were used. Nevertheless, nowhere did he claim that the Dirac equation could only be formulated covariantly with the help of Einsteinâs new theory.
Zaycoff of the Physics Institute of the University in Sofia also followed Einsteinâs work closely. Half a year after Einsteinâs first two notes on distant parallelism had appeared [84, 83], i.e., shortly before Christmas 1928, Zaycoff sent off his first paper on the subject, whose arrival in Berlin was acknowledged only after the holidays on 13 January 1929 [429]. In it he described the mathematical formalism of distant parallelism theory, gave the identity (42), and calculated the new curvature scalar in terms of the Ricci scalar and of torsion. He then took a more general Lagrangian than Einstein and obtained the variational derivatives in linear and, in a simple example, also in second approximation. In his presentation, he used both the teleparallel and the Levi-Civita connections. His second and third papers came quickly after Einsteinâs third note of January 1929 [88], and thus had to take into account that Einstein had dropped derivation of the field equations from a variational principle. In his second paper, Zaycoff followed Einsteinâs method and gave a somewhat simpler derivation of the field equations. An exact, complicated wave equation followed:
$$
{D_l}{D^l}{S_k} - {F_{kj}}{S^j} - {X_k} = 0,
$$
((180))
where \({X_k}: = \tfrac{1}{2}{D_k}({V^{lmn}}{S_{lnm}}) + {S_{lkm}}{D_r}{S^{lrm}} - \tfrac{1}{2}{S_k}{V^{lmn}}{S_{lnm}} + {V^{lmn}}{S_{mn}}^r{S_{lkr}} + {S_{rkm}}{S^r}{S^m}\) with torsion \({S_{mn}}^r\) and the torsion vector Sk, and the covariant derivative \({D_l}: = {\nabla _l} - {S_l}\), âlbeing the teleparallel connection (161). In linear approximation, the Einstein vacuum and the vacuum Maxwell equations are obtained, supplemented by the homogeneous wave equation for a vector field [431]. In his third note, Zaycoff criticised Einstein âfor not having shown, in his most recent publication, whether his constraints on the world metric be permissible.â He then derived additional exact compatibility conditions for Einsteinâs field equations to hold; according to him, their effect would show up only in second approximation [430]. In his fourth publication Zaycoff came back to Einsteinâs Hamiltonian principle and rederived for himself Einsteinâs results. He also defended Einstein against critical remarks by Eddington [62] and Schouten [304], although Schouten, in his paper, had mentioned neither Einstein nor his teleparallelism theory, but only gave a geometrical interpretation of the torsion vector in a geometry with semi-symmetric connection. Zaycoff praised Einsteinâs teleparallelism theory in words reminding me of the creation of the world as described in Genesis:
âWe may say that A. Einstein built a plane world which is no longer waste like the Euclidean space-time-world of H. Minkowski, but, on the contrary, contains in it all that we usually call physical reality.âFootnote 240 ([428], p. 724)
A conference on theoretical physics at the Ukrainian Physical-Technical Institute in Charkow in May 1929, brought together many German and Russian physicists. Unified field theory, quantum mechanics, and the new quantum field theory were all discussed. Einsteinâs former calculational assistant Grommer, now on his own in Minsk, in a brief contribution stressed Einsteinâs path for getting an overdetermined system of differential equations: Vary with regard to the 16 beinquantities but consider only the 10 metrical components as relevant. He claimed that Einstein had used only the antisymmetric part of the tensor \({P_{lm}}^k: = \Gamma {(g)_{lm}}^k - {\Delta _{lm}}^k\), where both \(\Gamma {(g)_{lm}}^k\) and \({\Delta _{lm}}^k\) were mentioned above (in Einsteinâs first note) although Einstein never used \(\Gamma {(g)_{lm}}^k\). According to Grommer the anti-symmetry of P is needed, because its contraction leads to the electromagnetic 4-potential and because the symmetric part can be expressed by the antisymmetric part and the metrical tensor. He also played the true voice of his (former) master by repeating Einsteinâs program of deriving the equations of motion from the overdetermined system:
âIf the law of motion of elementary particles could be derived from the overdetermined field equation, one could imagine that this law of motion permit only discrete orbits, in the sense of quantum theory.âFootnote 241 ([153], p. 646)
Levi-Civita also had sent a paper on distant parallelism to Einstein, who had it appear in the reports of the Berlin Academy [207]. Levi-Civita introduced a set of four congruences of curves that intersect each other at right angles, called their tangents \(\lambda _{\hat \imath}^k\) and used Equation (160) in the form:
$$
{\delta _{lm}} = {h_{l\hat \jmath}}{h_{m\hat \jmath}}.
$$
((181))
He also employed the Ricci rotation coefficients defined by \({\gamma _{\hat \imath \hat k\hat l}} = {\nabla _\sigma }h_{\hat \imath}^\beta {h_{\hat k\beta }}h_{\hat l}^\sigma\), where the hatted indices are âbeinâ-indices; the Greek letters denote coordinates. They obey
$$
{\gamma _{\hat \imath \hat k\hat l}} + {\gamma _{\hat k\hat \imath \hat l}} = 0.
$$
((182))
The electromagnetic field tensor Fik was entered via
$$
{F_{ik}}\lambda _{\hat l}^i\lambda _{\hat m}^k = \lambda _{\hat s}^r\frac{\partial }{{\partial {x^r}}}{\gamma _{\hat l\hat m}}^{\hat s}.
$$
((183))
Levi-Civita chose as his field equations the Einstein-Maxwell equations projected on a rigidly fixed âworld-latticeâ of 4-beins. He used the time until the printing was done to give a short preview of his paper in Nature [206]. About a month before Levi-Civitaâs paper was issued by the Berlin Academy, Fock and Ivanenko [135] had had the same idea and compared Einsteinâs notation and the one used by Levi-Civita in his monograph on the absolute differential calculus [205]:
âEinsteinâs new gravitational theory is intimately linked to the known theory of the orthogonal congruences of curves due to Ricci. In order to ease a comparison between both theories, we may bring together here the notations of R i c c i and L e v i - C i v i t a [âŠ] with those of Einstein.âFootnote 242
A little after the publication of Levi-Civitaâs papers, Heinrich Mandel embarked on an application of Kaluzaâs five-dimensional approach to Einsteinâs theory of distant parallelism [218]. Einstein had sent him the corrected proof sheets of his fourth paper [85]. The basic idea was to consider the points of M4 as equivalent to the ensemble of congruences with tangent vector X
i5
in M5 (with cylindricity condition) werden.âFootnote 243. The space-time interval is defined as the distance of two lines of the congruence on \({M_5}:d{\tau ^2} = ({\gamma _{il}} - {X_{5i}}\;{X_{5l}})(\delta _k^l - {X_5}^l{X_{5k}})d{x^i}d{x^k}\). Mandel did not identify the torsion vector with the electromagnetic 4-potential, but introduced the covariant derivative \({\Delta _k}{A^i}: = \tfrac{{\partial {A^i}}}{{\partial {x^k}}} + \{ _{kj}^i\} {A^j} + \tfrac{e}{c}{X_{5k}}{\mathcal M}_l^i{A^l}\), where the tensor \({\mathcal M}\) is skew-symmetric. We may look at this paper also as a forerunner of some sort to the Einstein.Mayer 5-vector formalism (cf. Section 6.3.2).
Before Einstein dropped the subject of distant parallelism, many more papers were written by a bakerâs dozen of physicists. Some were more interested in the geometrical foundations, in exact solutions to the field equations, or in the variational principle.
One of those hunting for exact solutions was G. C. McVittie who referred to Einsteinâs paper [88]:
â[âŠ] we test whether the new equations proposed by Einstein are satisfied. It is shown that the new equations are satisfied to the first order but not exactly.â
He then goes on to find a rigourous solution and obtains the metric \(d{s^2} = {e^{a{x_1}}}dx_4^2 - {e^{ - 2a{x_1}}}dx_1^2 - {e^{ - a{x_1}}}dx_2^2 - {e^{ - a{x_1}}}dx_3^2\) and the 4-potential \({\phi _4} = \tfrac{1}{{2\sqrt \pi }}{e^{\tfrac{1}{2}a{x_1}}}\) [225]. He also wrote a paper on exact axially symmetric solutions of Einsteinâs teleparallelism theory [226].
Tamm and Leontowich treated the field equations given in Einsteinâs fourth paper on distant parallelism [85]. They found that these field equations did not have a spherically symmetric solution corresponding to a charged point particle at restFootnote 244. The corresponding solution for the uncharged particle was the same as in general relativity, i.e., Schwarzschildâs solution. Tamm and Leontowitch therefore guessed that a charged point particle at rest would lead to an axially-symmetric solution and pointed to the spin for support of this hypothesis [342, 342].
WienerFootnote 245 and VallartaFootnote 246 were after particular exact solutions of Einsteinâs field equations in the teleparallelism theory. By referring to Einsteinâs first two papers concerning distant parallelism, they set out to show that the
â[âŠ] electromagnetic field is incompatible in the new Einstein theory with the assumption of static spherical symmetry and symmetry of the past and the future. [âŠ] the new Einstein theory lacks at present all experimental confirmation.â
In footnote 4, they added:
âSince writing this paper the authors have learned from Dr. H. MĂŒntz that the new Einstein field equations of the 1929 paper do not yield the vanishing of the gravitational field in the case of spherical symmetry and time symmetry. In this case he has been able to obtain results checking the observed perihelion of mercuryâ ([416], p. 356)
MĂŒntz is mentioned in [88, 85].
In his paper âOn unified field theoryâ of January 1929, Einstein acknowledges work of a Mr. MĂŒntz:
âI am pleased to dutifully thank Mr. Dr. H. MĂŒntz for the laborious exact calculation of the centrally-symmetric problem based on the Hamiltonian principle; by the results of this investigation I was led to the discovery of the road following here.âFootnote 247
Again, two months later in his next paper, âUnified field theory and Hamiltonian principleâ, Einstein remarks:
âMr. Lanczos and MĂŒntz have raised doubt about the compatibility of the field equations obtained in the previous paper [âŠ].â
and, by deriving field equations from a Lagrangian shows that the objection can be overcome. In his paper in July 1929, the physicist Zaycoff had some details:
âSolutions of the field equations on the basis of the original formulation of unified field theory to first approximation for the spherically symmetric case were already obtained by MĂŒntz.â
In the same paper, he states: âI did not see the papers of Lanczos and MĂŒntz.â Even before this, in the same year, in a footnote to the paper of Wiener and Vallarta, we read:
âSince writing this paper the authors have learned from Dr. H. MĂŒntz that the new Einstein field equations of the 1929 paper do not yield the vanishing of the gravitational field in the case of spherical symmetry and time symmetry. In this case he has been able to obtain results checking the observed perihelion of mercury.â
The latter remark refers to a constant query Pauli had about what would happen, within unified field theory, to the gravitational effects in the planetary system, described so well by general relativityFootnote 248.
Unfortunately, as noted by Meyer Salkover of the Mathematics Department in Cincinatti, the calculations by Wiener and Vallarta were erroneous; if corrected, one finds the Schwarzschild metric is indeed a solution of Einsteinâs field equations. In the second of his two brief notes, Salkover succeeded in gaining the most general, spherically symmetric solution [288, 287]. This is admitted by the authors in their second paper, in which they present a new calculation.
âIn a previous paper the authors of the present note have treated the case of a spherically symmetrical statical field, and stated the conclusions: first, that under Einsteinâs definition of the electromagnetic potential an electromagnetic field is incompatible with the assumption of static spherical symmetry and symmetry of the past and future; second, that if one uses the Hamiltonian suggested in Einsteinâs second 1928 paper, the electromagnetic potential vanishes and the gravitational field also vanishes.â
And they hasten to reassure the reader:
âNone of the conclusions of the previous paper are vitiated by this investigation, although some of the final formulas are supplemented by an additional term.â ([417], p. 802)
Vallarta also wrote a paper by himself ([358], p. 784) whose abstract reads:
âIn recent papers Wiener and the author have determined the tensors \(^s{h_\lambda }\) of Einsteinâs unified theory of electricity and gravitation under the assumption of static spherical symmetry and of symmetry of past and future. It was there shown that the field equations suggested in Einsteinâs second 1928 paper [83] lead in this case to a vanishing gravitational field. The purpose of this paper is to investigate, for the same case, the nature of the gravitational field obtained from the field equations suggested by Einstein in his first 1929 paper [88].â
He also claims
âthat Wiener has shown in a paper to be published elsewhere soon that the Schwarzschild solution satisfies exactly the field equations suggested by Einstein in his second 1929 paper ([85]).â
Finally, Rosen and Vallarta [283] got together for a systematic investigation of the spherically symmetric, static field in Einsteinâs unified field theory of the electromagnetic and gravitational fields [93].
Further papers on Einsteinâs teleparallelism theory were written in Italy by Bortolotti in Cagliari, Italy [22, 23, 25, 24], and by Palatini [242].
In Princeton, people did not sleep either. In 1930 and 1931, T. Y. Thomas wrote a series of six papers on distant parallelism and unified field theory. He followed Einsteinâs example by also changing his field equations from the first to the second publication. After that, he concentrated on more mathematical problems , such as proving an existence theorem for the Cauchy-Kowlewsky type of equations in unified field theory, by studying the characteristics and bi-characteristic, the characteristic Cauchy problem, and Huygenâs principle. T. Y. Thomas described the contents of his first paper as follows:
âIn a number of notes in the Berlin Sitzungsberichte followed by a revised account in the Mathematische Annalen, Einstein has attempted to develop a unified theory of the gravitational and electromagnetic field by introducing into the scheme of Riemannian geometry the possibility of distant parallelism. [âŠ] we are led to the construction of a system of wave equations as the equations of the combined gravitational and electromagnetic field. This system is composed of 16 equations for the determination of the 16 quantities h
kÌi
and is closely analogous to the system of 10 equations for the determination of the 10 components gik in the original theory of gravitation. It is an interesting fact that the covariant components h
kÌi
of the fundamental vectors, when considered as electromagnetic potential vectors, satisfy in the local coordinate system the universally recognised laws of Maxwell for the electromagnetic field in free space, as a consequence of the field equations.â [350]
This looks as if he had introduced four vector potentials for the electromagnetic field, and this, in fact, T. Y. Thomas does: âthe components h
kÌi
will play the role of electromagnetic potentials in the present theory.â The field equations are just the four wave equations \(\sum {{e_{\hat k}}h_{\hat \jmath,\hat k\hat k}^{\hat \imath}}\) where the summation extends over kÌ, with kÌ= 1,âŠ4, and the comma denotes an absolute derivative he has introduced. The gravitational potentials are still gik. In his next note, T. Y. Thomas changed his field equations on the grounds that he wanted them to give a conservation law.
âThis latter point of view is made the basis for the construction of a system of field equations in the present note â and the equations so obtained differ from those of note I only by the appearance of terms quadratic in the quantities \(h_{j,k}^{\hat \imath}\). It would thus appear that we can carry over the interpretation of the h
kÌi
as electromagnetic potentials; doing this, we can say that Maxwellâs equations hold approximately in the local coordinate system in the presence of weak electromagnetic fields.â [351]
The third paper contains a remark as to the content of the concept âunified field theoryâ:
âIt is the objective of the present note to deduce the general existence theorem of the Cauchy-Kowalewsky type for the system of field equations of the unified field theory. [âŠ] Einstein (Sitzber. 1930, 18â23) has pointed out that the vanishing of the invariant \(h_{j,k}^{\hat \imath }\) is the condition for the four-dimensional world to be Euclidean, or more properly, pseudo-Euclidean. From the point of view of our previous notes this fact has its interpretation in the statement that the world will be pseudo-Euclidean only in the absence of electric and magnetic forces. This means that gravitational and electromagnetic phenomena must be intimately related since the existence of gravitation becomes dependent on the electromagnetic field. Thus we secure a real physical unification of gravitation and electricity in the sense that these concepts become but different manifestations of the same fundamental entity â provided, of course, that the theory shows itself to be tenable as a theory in agreement with experience.â [352].
In his three further installments, T. Y. Thomas moved away from unified field theory to the discussion of mathematical details of the theory he had advanced [353, 354, 355].
Unhindered by constraints from physical experience, mathematicians try to play with possibilities. Thus, it was only consequential that Valentin Bargmann in Berlin, after Riemann and Weyl, now engaged in looking at a geometry allowing a comparison âat a distanceâ of directions but not of lengths, i.e., only of the quotient of vector components, Ai/Ak [5]. In the framework of a purely affine theory he obtained a necessary and sufficient condition for this geometry,
$$
R_{jkl}^i(\Gamma ) = \frac{1}{D}{\delta ^i}_j{V_{kl}},
$$
((184))
with the homothetic curvature Vkl from Equation (31). Then Bargmann linked his approach to Einsteinâs first note on distant parallelism [84, 89], introduced a D-bein h
ki
, and determined his connection such that the quotients Ai/Ak of vector components with regard to the D-bein remained invariant under parallel transport. The resulting connection is given by
$$
{\Gamma _{lm}}^k = h_j^k\frac{{\partial h_l^j}}{{\partial {x^m}}} - \delta _l^k{\psi _m},
$$
((185))
where Ïm corresponds to \({\Gamma _{lm}}^k\).
Schouten and van Dantzig also used a geometry built on complex numbers, and on Hermitian forms:
â[âŠ] we were able to show that the metric geometry used by Einstein in his most recent approach to relativity theory [84, 83] coincides with the geometry of a Hermitian tensor of highest rank, which is real on the real axis and satisfies certain differential equations.â ([313], p. 319)
The Hermitian tensor referred to leads to a linear integrable connection that, in the special case that it âis real in the realâ, coincides with Einsteinâs teleparallel connection.
Distant parallelism was revived four decades later within the framework of Poincaré gauge theory; the corresponding theories will be treated in the second part of this review.
Overdetermination and compatibility of systems of differential equations
In the course of Einsteinâs thinking about distant parallelism, his ideas about overdetermined systems of differential equations gradually changed. At first, the possibility of gaining hold on the paths of elementary particles â described as singular worldlines of point particles â was central. He combined this with the idea of quantisation, although Planckâs constant h could not possibly surface by such an approach. But somehow, for Einstein, discretisation and quantisation must have been too close to bother about a fundamental constant.
Then, after the richer constructive possibilities (e.g., for a Lagrangian) became obvious, a principle for finding the correct field equations was needed. As such, âoverdeterminationâ was brought into the game by Einstein:
âThe demand for the existence of an âoverdeterminedâ system of equations does provide us with the means for the discovery of the field equationsâFootnote 249 ([90], p. 21)
It seems that Einstein, during his visit to Paris in November 1929, had talked to Cartan about his problem of finding the right field equations and proving their compatibility. Starting in December of 1929 and extending over the next year, an intensive correspondence on this subject was carried on by both men [50]. On 3 December 1929, Cartan sent Einstein a letter of five pages with a mathematical note of 12 pages appended. In it he referred to his theory of partial differential equations, deterministic and âin involution,â which covered the type of field equations Einstein was using and put forward a further field equation. He clarified the mathematical point of view but used concepts such as âdegree of generalityâ and âgenerality indexâ not familiar to Einstein
Footnote 250
. Cartan admittedFootnote 251:
âI was not able to completely solve the problem of determining if there are systems of 22 equations other than yours and the one I just indicated [âŠ] and it still astonishes me that you managed to find your 22 equations! There are other possibilities giving rise to richer geometrical schemes while remaining deterministic. First, one can take a system of 15 equations [âŠ]. Finally, maybe there are also solutions with 16 equations; but the study of this case leads to calculations as complicated as in the case of 22 equations, and I was not fortunate enough to come across a possible system [âŠ].â ([50], pp. 25â26)
Einsteinâs rapid answer of 9 December 1929 referred to the letter only; he had not been able to study Cartanâs note. As the further correspondence shows, he had difficulties in following Cartan:
âFor you have exactly that which I lack: an enviable facility in mathematics. Your explanation of the indice de gĂ©nĂ©ralitĂ© I have not yet fully understood, at least not the proof. I beg you to send me those of your papers from which I can properly study the theory.â ([50], p. 73)
It would be a task of its own to closely study this correspondence; in our context, it suffices to note that Cartan wrote a special noteFootnote 252
â[âŠ] edited such that I took the point of view of systems of partial differential equations and not, as in my papers, the point of view of systems of equations for total differentials [âŠ]âFootnote 253
which was better suited to physicists. Through this note, Einstein came to understand Cartanâs theory of systems in involution:
âI have read your manuscript, and this enthusiastically. Now, everything is clear to me. Previously, my assistant Prof. MĂŒntz and I had sought something similar â but we were unsuccessful.âFootnote 254 ([50], pp. 87, 94)
In the correspondence, Einstein made it very clear that he considered Maxwellâs equations only as an approximation for weak fields, because they did not allow for non-singular exact solutions approaching zero at spacelike infinity.
âIt now is my conviction that for rigourous field theories to be taken seriously everywhere a complete absence of singularities of the field must be demanded. This probably will restrict the free choice of solutions in a region in a far-reaching way â more strongly than the restrictions corresponding to your degrees of determination.âFootnote 255 ([50], p. 92)
Although Einstein was grateful for Cartanâs help, he abandoned the geometry with distant parallelism.