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In the period considered, mutual reservations may have existed between the followers of the new quantum mechanics and those joining Einstein in the extension of his general relativity. Lanczos , in , was well aware of his being out of tune with those adherent to quantum mechanics:.

On the other hand, those working in quantum theory may have frowned upon the wealth of objects within unified field theories uncorrelated to a convincing physical interpretation and thus, in principle, unrelated to observation. Einstein belonged to those who regarded the idea of unification as more fundamental than the idea of field quantisation [ 95 ]. Section 7. During the time span considered here, there also were those whose work did not help the idea of unification, e.

Matter is characterised by a bivectordensity […]. A methodological weak point in the process of the establishment of field equations for unified field theory was the constructive weakness of alternate physical limits to be taken:. A main hindrance for an eventual empirical check of unified field theory was the persistent lack of a worked out example leading to a new gravito-electromagnetic effect. In the following Section 2 , a multitude of geometrical concepts affine, conformal, projective spaces, etc.

In Section 4 , the main ideas are developed. After a short excursion to the world of mathematicians working on differential geometry see Section 5 , the research of Einstein and his assistants is studied see Section 6. The appearance of spinors in a geometrical setting, and endeavours to link quantum physics and geometry in particular, the attempt to geometrize wave mechanics are also discussed see Section 7.

We have included this topic although, strictly speaking, it only touches the fringes of unified field theory. In Section 9 , particular attention is given to the mutual influence exerted on each other by the Princeton Eisenhart , Veblen , French Cartan , and the Dutch Schouten, Struik schools of mathematicians, and the work of physicists such as Eddington, Einstein, their collaborators, and others.

In Section 10 , the reception of unified field theory at the time is briefly discussed. As a rule, the point of departure for unified field theory was general relativity. In this review, we will encounter essentially five different ways to include the electromagnetic field into a geometric setting:.

In the period considered, all four directions were followed as well as combinations between them like e. Nevertheless, we will almost exclusively be dealing with the extension of geometry and of the number of space dimensions. It is very easy to get lost in the many constructive possibilities underlying the geometry of unified field theories. We briefly describe the mathematical objects occurring in an order that goes from the less structured to the more structured cases.

In the following, only local differential geometry is taken into account. At each point, D linearly independent vectors linear forms form a linear space, the tangent space cotangent space of M D. We will assume that the manifold M D is space- and time-orientable. On it, two independent fundamental structural objects will now be introduced. The first is a prescription for the definition of the distance ds between two infinitesimally close points on M D , eventually corresponding to temporal and spatial distances in the external world.

For ds , we need positivity, symmetry in the two points, and the validity of the triangle equation. We know that ds must be homogeneous of degree one in the coordinate differentials dx i connecting the points. This condition is not very restrictive; it still includes Finsler geometry [ , , ] to be briefly touched, below.

In the following, ds is linked to a non-degenerate bilinear form g X, Y , called the first fundamental form ; the corresponding quadratic form defines a tensor field, the metrical tensor , with D 2 components g ij such that. From this we note that an antisymmetric part of the metrical tensor does not influence distances and norms but angles. With the metric tensor having full rank, its inverse g ik is defined through.

We are used to g being a symmetric tensor field, i. In the following this need not hold, so that the decomposition obtains :. An asymmetric metric was considered in one of the first attempts at unifying gravitation and electromagnetism after the advent of general relativity. We also note that.

In a manifold with Lorentzian metric, a non-trivial real conformal structure always exists; from the equation. In view of the physical interpretation of the light cone as the locus of light signals, a causal structure is provided by the equivalence class of metrics [ 67 ].

A special case of a space with a Lorentzian metric is Minkowski space, whose metrical components, in Cartesian coordinates, are given by. A geometrical characterization of Minkowski space as an uncurved, flat space is given below. Let be the Lie derivative with respect to the tangent vector X ; then holds for the Lorentz group of generators X p. The metric tensor g may also be defined indirectly through D vector fields forming an orthonormal D -leg -bein.

From the geometrical point of view, this can always be done cf. By introducing 1-forms , Equation 11 may be brought into the form. The fibre at each point of the manifold contains, in the case of an orthonormal D -bein tetrad , all D -beins tetrads related to each other by transformations of the group O D , or the Lorentz group, and so on. In Finsler geometry , the line element depends not only on the coordinates x i of a point on the manifold, but also on the infinitesimal elements of direction between neighbouring points dx i :.

The second structure to be introduced is a linear connection L with D 3 components L ij k ; it is a geometrical object but not a tensor field and its components change inhomogeneously under local coordinate transformations. The connection is a device introduced for establishing a comparison of vectors in different points of the manifold.

For each vector field and each tangent vector it provides another unique vector field. The expressions and are abbreviated by and X i ,k , respectively, while for a scalar f covariant and partial derivative coincide:. We have adopted the notational convention used by Schouten [ , , ]. Eisenhart and others [ , ] change the order of indices of the components of the connection:. As long as the connection is symmetric, this does not make any difference as. For both kinds of derivatives we have:.

Both derivatives are used in versions of unified field theory by Einstein and others. A manifold provided with only a linear connection L is called affine space. From the point of view of group theory, the affine group linear inhomogeneous coordinate transformations plays a special role: With regard to it the connection transforms as a tensor cf.

Section 2. For a vector density cf. A curve is called an autoparallel if its tangent vector is parallely transported along it at each point :. The equivalence class of autoparallels defined by Equation 18 defines a projective structure on M D [ , ].

From the connection L ij k further connections may be constructed by adding an arbitrary tensor field T to its symmetrised part :. By special choice of T we can regain all connections used in work on unified field theories. We will encounter examples in later sections. The antisymmetric part of the connection, i. The trace of the torsion tensor is called torsion vector ; it connects to the two traces of the affine connection as.

Various subcases of affine spaces will occur, dependent on whether the connection is asymmetric or symmetric , i. In physical applications, a metric always seems to be needed; hence in affine geometry it must be derived solely by help of the connection or, rather, by tensorial objects constructed from it.

This is in stark contrast to Riemannian geometry where, vice versa, the connection is derived from the metric. Such tensorial objects are the two affine curvature tensors defined by. In a geometry with symmetric affine connection both tensors coincide because of. In particular, in Riemannian geometry , both affine curvature tensors reduce to the one and only Riemann curvature tensor. The curvature tensors arise because the covariant derivative is not commutative and obeys the Ricci identity :.

The curvature tensor 22 satisfies two algebraic identities:. From both affine curvature tensors we may form two different tensorial traces each. In the first case , and. V kl is called homothetic curvature , while K jk is the first of the two affine generalisations from and of the Ricci tensor in Riemannian geometry. We get. While V kl is antisymmetric, K jk has both tensorial symmetric and antisymmetric parts:. We use the notation in order to exclude the index k from the symmetrisation bracket.

In order to shorten the presentation of affine geometry, we refrain from listing the corresponding set of equations for the other affine curvature tensor cf. For a symmetric affine connection, the preceding results reduce considerably due to. From Equations 29 , 30 , 32 we obtain the identities:. For the antisymmetric part of the Ricci tensor holds. This equation will be important for the physical interpretation of affine geometry. It may be desirable to derive the metric from a Lagrangian; then the simplest scalar density that could be used as such is given by det K ij.

As a final result in this section, we give the curvature tensor calculated from the connection cf. Equation 20 , expressed by the curvature tensor of and by the tensor :. A manifold carrying both structural elements, i. If the first fundamental form is taken to be asymmetric , i. Equation It depends on the physical interpretation, i.

Thus, in metric-affine and in mixed geometry, two different connections arise in a natural way. With the help of the symmetric affine connection, we may define the tensor of non-metricity by. The inner product of two tangent vectors A i , B k is not conserved under parallel transport of the vectors along X l if the non-metricity tensor does not vanish:. Thomas introduced a combination of the terms appearing in and to define a covariant derivative for the metric [ ], p.

We will have to deal with Equation 47 in Section 6. Connections that are not metric-compatible have been used in unified field theory right from the beginning. In case of such a relationship, the geometry is called semi-metrical [ , ].

We may also abbreviate the last term in the identity 42 by introducing. Then, from Equation 39 , the curvature tensor of a torsionless affine space is given by. Riemann-Cartan geometry is the subcase of a metric-affine geometry in which the metric-compatible connection contains torsion, i. A linear connection whose antisymmetric part has the form. Riemannian geometry is the further subcase with vanishing torsion of a metric-affine geometry with metric-compatible connection.

In this case, the connection is derived from the metric: , where is the usual Christoffel symbol The covariant derivative of A with respect to the Levi-Civita connection is abbreviated by A ; k. The Riemann curvature tensor is denoted. An especially simple case of a Riemanian space is Minkowski space, the curvature of which vanishes:. However, in metric affine and in mixed geometry geodesic and autoparallel curves will have to be distinguished.

As a consequence, the Riemann curvature tensor is also changed; if, however, can be reached by a conformal transformation, then the corresponding spacetime is called conformally flat. Even before Weyl, the question had been asked and answered as to what extent the conformal and the projective structures were determining the geometry: According to Kretschmann and then to Weyl they fix the metric up to a constant factor [ ]; see also [ ], Appendix 1; for a modern approach, cf.

The geometry needed for the pre- and non-relativistic approaches to unified field theory will have to be dealt with separately. There, the metric tensor of space is Euclidean and not of full rank; time is described by help of a linear form Newton-Cartan geometry, cf. In the following we shall deal only with relativistic unified field theories.

The connection to the inhomogeneous coordinates x i is given by homogeneous functions of degree zero, e. Thus, the themselves form the components of a tangent vector. Furthermore, the quadratic form is adopted with being a homogeneous function of degree A tensor field cf.

If we define , with , then transforms like a tangent vector under point transformations of the x i , and as a covariant vector under homogeneous transformations of the. The may be used to relate covariant vectors a i and by.

Thus, the metric tensor in the space of homogeneous coordinates and the metric tensor of M D are related by with. The inverse relationship is given by with. The covariant derivative for tensor fields in the space of homogeneous coordinates is defined as before cf. The covariant derivative of the quantity interconnecting both spaces is given by. Cartan introduces one-forms by. The reciprocal basis in tangent space is given by.

The metric is then given by. We have. The link to the components of the affine connection is given by. In Equation 65 the curvatureform appears, which is given by. Up to here, no definitions of a tensor and a tensor field were given: A tensor T p M D of type r , s at a point p on the manifold M D is a multi-linear function on the Cartesian product of r cotangent- and s tangent spaces in p.

A tensor field is the assignment of a tensor to each point of M D. Usually, this definition is stated as a linear, homogeneous transformation law for the tensor components in local coordinates:. Strictly speaking, tensors are representations of the abstract group at a point on the manifold. Note that. The dual to a 2-form skew-symmetric tensor then is defined by.

In connection with conformal transformations , the concept of the gauge-weight of a tensor is introduced. A tensor is said to be of gauge weight q if it transforms by Equation 56 as. Objects that transform as in Equation 67 but with respect to a subgroup, e. All the subgroups mentioned are Lie -groups, i. Spinors are representations of the Lorentz group only; as such they are related strictly to the tangent space of the space-time manifold.

To see how spinor representations can be obtained, we must use the homomorphism of the group SL 2,C and the proper orthochronous Lorentz group, a subgroup of the full Lorentz group. By picking the special Hermitian matrix. The link between the representation of a Lorentz transformation L ik in space-time and the unimodular matrix A mapping spin space cf. The spinor is called elementary if it transforms under a Lorentz-transformation as. Covariant and covariant dotted 2-spinors correspondingly transform with the inverse matrices,.

Higher-order spinors with dotted and undotted indices transform correspondingly. Next to a spinor, bispinors of the form , etc. Often the quantity is introduced. The reciprocal matrix is defined by. The simplest spinorial equation is the Weyl equation:.

The Dirac equation is in 4-spinor formalism [ 53 , 54 ]:. The group of coordinate transformations acts on the Latin indices. For the example of the Dirac spinor, the adjoint representation of the Lorentz group must be used. In Section 2.

With its help we may formulate the concept of isometries of a manifold, i. If a group G r is prescribed, e. A Riemannian space is called locally stationary if it admits a timelike Killing vector; it is called locally static if this Killing vector is hypersurface orthogonal. In purely affine spaces, another type of symmetry may be defined: ; they are called affine motions [ ]. Within a particular geometry, usually various options for the dynamics of the fields field equations, in particular as following from a Lagrangian exist as well as different possibilities for the identification of physical observables with the mathematical objects of the formalism.

Thus, in general relativity, the field equations are derived from the Lagrangian. This Lagrangian leads to the well-known field equations of general relativity,. In empty space, i. If only an electromagnetic field derived from the 4-vector potential A k is present in the energy-momentum tensor, then the Einstein. Maxwell equations follow:. The components of the metrical tensor are identified with gravitational potentials.

The equations of motion of material particles should follow, in principle, from Equation 92 through the relation. For point particles, due to the singularities appearing, in general this is a tricky task, up to now solved only approximately.

However, the world lines for point particles falling freely in the gravitational field are, by definition, the geodesics of the Riemannian metric. This definition is consistent with the rigourous derivation of the geodesic equation for non-interacting dust particles in a fluid matter description.

It is also consistent with all observations. For most of the unified field theories to be discussed in the following, such identifications were made on internal, structural reasons, as no link-up to empirical data was possible. Due to the inherent wealth of constructive possibilities, unified field theory never would have come off the ground proper as a physical theory even if all the necessary formal requirements could have been satisfied. The latter choice obtains likewise in a purely affine theory in which the metric is a derived secondary concept.

In this case, among the many possible choices for the metric, one may take it proportional to the variational derivative of the Lagrangian with respect to the symmetric part of the Ricci tensor. This does neither guarantee the proper signature of the metric nor its full rank. Several identifications for the electromagnetic 4-potential and the electric current vector density have also been suggested cf.

Complex fields may also be introduced on a real manifold. Such fields have also been used for the construction of unified field theories, although mostly after the period dealt with here cf. Part II, in preparation. In particular, manifolds with a complex fundamental form were studied, e.

Also, geometries based on Hermitian forms were studied [ ]. In later periods, hypercomplex numbers, quaternions, and octonions also were used as basic number fields for gravitational or unified theories cf. Part II, forthcoming. In place of the real numbers, by which the concept of manifold has been defined so far, we could take other number fields and thus arrive, e. In this part of the article we do not need to take into account this generalisation. In most of the cases, the additional dimensions were taken to be spacelike; nevertheless, manifolds with more than one direction of time also have been studied.

In his letter to Einstein of 11 November , he writes [ ], Doc. Perhaps, you are luckier in the search. I am totally convinced that in the end all field quantities will look alike in essence. But it is easier to suspect something than to discover it. Various reasons instilled in me strong reservations: […] your other remarks are interesting in themselves and new to me. Ishiwara, and G. The result is contained in Hilbert , p.

The hints dropped by you on your postcards bring me to expect the greatest. According to him, the deviation from the Minkowski metric is due to the electromagnetic field tensor:. He claims to obtain the same value for the perihelion shift of Mercury as Einstein [ ], p. The meeting was amicable.

In this context, we must also keep in mind that the generalisation of the metric tensor toward asymmetry or complex values was more or less synchronous with the development of Finsler geometry [ ]. Although Finsler himself did not apply his geometry to physics it soon became used in attempts at the unification of gravitation and electromagnetism [ ].

The idea that they keep together the dispersing electrical charges lies close at hand. Thus, the idea of a program for building the extended constituents of matter from the fields the source of which they are, was very much alive around Naturforscherversammlung, 19—25 September [ ] showed that not everybody was a believer in it.

He claimed that in bodies smaller than those carrying the elementary charge electrons , an electric field could not be measured. I wish to see this reason in the fact that it is altogether not permitted to describe the electromagnetic field in the interior of an electron as a continuous space function. The electrical field is defined as the force on a charged test particle, and if no smaller test particles exist than the electron vice versa the nucleus , the concept of electrical field at a certain point in the interior of the electron — with which all continuum theories are working — seems to be an empty fiction, because there are no arbitrarily small measures.

Einstein whether he approves of the opinion that a solution of the problem of matter may be expected only from a modification of our perception of space perhaps also of time and of electricity in the sense of atomism, or whether he thinks that the mentioned reservations are unconvincing and is of the opinion that the fundaments of continuum theory must be upheld.

If, in a certain stage of scientific investigation, it is seen that a concept can no longer be linked with a certain event, there is a choice to let the concept go, or to keep it; in the latter case, we are forced to replace the system of relations among concepts and events by a more complicated one. The same alternative obtains with respect to the concepts of timeand space-distances. In my opinion, an answer can be given only under the aspect of feasibility; the outcome appears dubious to me.

But a more precise reasoning shows that in this way no reasonable world function is obtained. It is to be noted that Weyl, at the end of , already had given up on a possible field theory of matter:. To me, field physics no longer appears as the key to reality; in contrary, the field, the ether, for me simply is the totally powerless transmitter of causations, yet matter is a reality beyond the field and causes its states.

Klein on 28 December , see [ ], p. Yet it retains part of its meaning also with regard to questions concerning the constitution of elementary particles. Because one may try to ascribe to these field concepts […] a physical meaning even if a description of the electrical elementary particles which constitute matter is to be made.

Only success can decide whether such a procedure finds its justification […]. During the twenties Einstein changed his mind and looked for solutions of his field equations which were everywhere regular to represent matter particles:. Let us move into the field chosen by him without too much surprise to see him apparently follow a road opposed to the one successfully walked by the contemporary physicists. After , Einstein first was busy with extracting mathematical and physical consequences from general relativity Hamiltonian, exact solutions, the energy conservation law, cosmology, gravitational waves.

Thus, while lengths of vectors at different points can be compared without a connection, directions cannot. This seemed too special an assumption to Weyl for a genuine infinitesimal geometry:. A metrical relationship from point to point will only then be infused into [the manifold] if a principle for carrying the unit of length from one point to its infinitesimal neighbours is given.

In contrast to this, Riemann made the much stronger assumption that line elements may be compared not only at the same place but also at two arbitrary places at a finite distance. At a point, Equation 98 induces a local recalibration of lengths l while preserving angles, i. If, as Weyl does, the connection is assumed to be symmetric i. With regard to the gauge transformations 98 , remains invariant.

From the 1-form dQ , by exterior derivation a gauge-invariant 2-form with follows. Let us now look at what happens to parallel transport of a length, e. If X is taken to be tangent to C , i. The same holds for the angle between two tangent vectors in a point cf. Yet, also today, the circumstances are such that our trees do not grow into the sky. Due to the additional group of gauge transformations, it is useful to introduce the new concept of gauge-weight within tensor calculus as in Section 2.

Weyl did calculate the curvature tensor formed from his connection but did not get the correct result ; it is given by Schouten [ ], p. His Lagrangian is given by , where the invariants are defined by. Weyl had arranged that the page proofs be sent to Einstein. In communicating this on 1 March , he also stated that. In the most general case, the equations will be of 4th order, though. He then asked whether Einstein would be willing to communicate a paper on this new unified theory to the Berlin Academy [ ], Volume 8B , Document , pp.

Einstein was impressed: In April , he wrote four letters and two postcards to Weyl on his new unified field theory — with a tone varying between praise and criticism. His first response of 6 April on a postcard was enthusiastic:. It is a stroke of genious of first rank. Nevertheless, up to now I was not able to do away with my objection concerning the scale.

However, as long as measurements are made with infinitesimally small rigid rulers and clocks, there is no indeterminacy in the metric as Weyl would have it : Proper time can be measured. As a consequence follows: If in nature length and time would depend on the pre-history of the measuring instrument, then no uniquely defined frequencies of the spectral lines of a chemical element could exist, i.

He concluded with the words. Only for a vanishing electromagnetic field does this objection not hold. Only in a static gravitational field, and in the absence of electromagnetic fields, does this hold:. Einstein saw the problem, then unsolved within his general relativity, that Weyl alluded to, i. Presumably, such a theory would have to include microphysics. But I find: If the ds , as measured by a clock or a ruler , is something independent of pre-history, construction and the material, then this invariant as such must also play a fundamental role in theory.

Yet, if the manner in which nature really behaves would be otherwise, then spectral lines and well-defined chemical elements would not exist. Another famous theoretician who could not side with Weyl was H. However, Weyl still believed in the physical value of his theory. There exists an intensive correspondence between Einstein and Weyl, now completely available in volume 8 of the Collected Papers of Einstein [ ].

We subsume some of the relevant discussions. Weyl did not give in:. Einstein then suggested the affine group as the more general setting for a generalisation of Riemannian geometry [ ], Vol. In particular, it is unimportant which value of the integral is assigned to their world line.

Otherwise, sodium atoms and electrons of all sizes would exist. But if the relative size of rigid bodies does not depend on past history, then a measurable distance between two neighbouring world-points exists. As far as I can see, there is not a single physical reason for it being valid for the gravitational field. The gravitational field equations will be of fourth order, against which speaks all experience until now […].

The quadratic form Rg ik dx i dx k is an absolute invariant, i. If this expression would be taken as the measurable distance in place of ds , then. A very small change of the measuring path would strongly influence the integral of the square root of this quantity. Einstein added:. The last remarks are interesting for the way in which Einstein imagined a successful unified field theory.

In the same way in which Mie glued to his consequential electrodynamics a gravitation which was not organically linked to it, Einstein glued to his consequential gravitation an electrodynamics i. You establish a real unity.

Understandably, no comments about the physics are given [ ], pp. Of course, as he noted, no progress had been made with regard to the explanation of the constituents of matter; on the one hand because the differential equations were too complicated to be solved, on the other because the observed mass difference between the elementary particles with positive and negative electrical charge remained unexplained.

In his general remarks about this problem at the very end of his article, Pauli points to a link of the asymmetry with time-reflection symmetry see [ ], pp. Now as before I believe that one must look for such an overdetermination by differential equations that the solutions no longer have the character of a continuum.

But how? The idea of gauging lengths independently at different events was the central theme. This must also give rise to an identity; and it is found that the new identity expresses the law of conservation of electric charge. Section 4. As he had abandoned the idea of describing matter as a classical field theory since , the linking of the electromagnetic field via the gauge idea could only be done through the matter variables. In October , in the preface for the first American printing of the English translation of the fourth edition of his book Space, Time, Matter from , Weyl clearly expressed that he had given up only the particular idea of a link between the electromagnetic field and the local calibration of length:.

This attempt has failed. Weyl himself continued to develop the dynamics of his theory. As an equivalent Lagrangian Weyl gave, up to a divergence. Due to his constraint, Weyl had navigated around another problem, i. In the paper in , he changed his Lagrangian slightly into. The changes, which Weyl had introduced in the 4th edition of his book [ ], and which, according to him, were of fundamental importance for the understanding of relativity theory, were discussed by him in a further paper [ ].

His colleague in Vienna, Wirtinger , had helped him in this. If J has gauge-weight -1, then Jg ik is such a metric. In order to reduce the new theory to general relativity, in addition only the differential equation. More important, however, for later work was the gauge invariant tensor calculus by a fellow of St. Newman [ ]. In this calculus, tensor equations preserve their form both under a change of coordinates and a change of gauge.

Newman applied his scheme to a variational principle with Lagrangian K 2 and concluded:. We shall discuss these topics in Part II of this article. What is now called Kaluza-Klein theory in the physics community is a mixture of quite different contributions by both scientists. But you understand that, in view of the existing factual concerns, I cannot take sides as planned originally.

Kaluza did not normalize the Killing vector to a constant, i. Of the 15 components of , five had to get a new physical interpretation, i. The component g 55 turned out to be a scalar gravitational potential which, in the static case, satisfies the equation. The Lorentz force appears augmented by an additional term containing g 55 of the order which thus may be neglected. For him, any theory claiming universal validity was endangered by quantum theory, anyway. The remaining covariance group G 5 is given by.

The objects transforming properly under are: the scalar , the vector-potential , and the projected metric. Klein identified the group; however, he did not comment on the fact that now further invariants are available for a Lagrangian, but started right away from the Ricci scalar of M 5 [ ]. The group G 5 is isomorphic to the group H 5 of transformations for five homogeneous coordinates with homogeneous functions of degree 1.

I value your approach more than the one followed by H. If you wish, I will present your paper to the Academy after all. The negative result of his own paper, i. His motivation went beyond the unification of gravitation and electromagnetism:. Clearly, the non-Maxwellian binding forces which hold together an electron. In the first, shorter, part of two, Eddington describes affine geometry; in the second he relates mathematical objects to physical variables.

He starts by calculating both the curvature and Ricci tensors from the symmetric connection according to Equation By this, Eddington claims to guarantee charge conservation:. However, for a tensor density, due to Equation 16 we obtain. Who shall say what is the ordinary gauge inside the electron? Only connections leading to a Lorentz metric can be used if a physical interpretation is wanted. Thus, in general, g kl is not metric-compatible; in order to make it such, we are led to the differential equations for , an equation not considered by Eddington.

This is due to the expression for the inverse of the metric, a function cubic in R kl. Note also that Eddington does not explicitly say how to obtain the contravariant form of the electromagnetic field F ij from F ij ; we must assume that he thought of raising indices with the complicated inverse metric tensor. Now, Eddington was able to identify the energy-momentum tensor T ik of the electromagnetic field by decomposing the Ricci tensor K ij formed from Equation 51 into a metric part R ik and the rest.

His aim was reached in the sense that all three quantities were fixed entirely by the connection; they could no longer be given from the outside. If so, then it must be a purely phenomenological one without any recourse to the nature of the charged elementary particles cf.

At first, Einstein seems to have been reserved cf. To Bohr, Einstein wrote from Singapore on 11 January Eddington has come closer to the truth than Weyl. Like Eddington, Einstein used a symmetric connection and wrote down the equation. By this, the metric was defined as the symmetric part of the Ricci tensor. Due to. Let us note, however, that while transforms inhomogeneously, its transformation law. For a Lagrangian, Einstein used ; he claims that for vanishing electromagnetic field the vacuum field equations of general relativity, with the cosmological term included, hold.

If , then the electric current density j l is defined by. The field equations are obtained from the Lagrangian by variation with regard to the connection and are Einstein worked in space-time. From Equation the connection can be obtained. This equation is an identity if a solution of the field equations is inserted.

From Equation ,. In order that this makes sense, the identifications in Equation are always to be made after the variation of the Lagrangian is performed. For non-vanishing electromagnetic field, due to Equation the Equation now becomes. Einstein concluded:. Except for singular positions, the current density is practically vanishing. Up to the same order,. In general however,. Also, the geometrical theory presented here is energetically closed, i.

His final conclusion was:. Until the end of May , two further publications followed in which Einstein elaborated on the theory. In the second paper, he exchanged the Lagrangian for a new one, i. The resulting equations for the gravitational and electromagnetic fields are the symmetric and skew-symmetric part, respectively, of.

Although the theory offered, for every solution with positive charge, also a solution with negative charge, the masses in the two cases were the same. However, the only known particle with positive charge at the time what is now called the proton had a mass greatly different from the particle with negative charge, the electron. Einstein noted:. The logic of the subsequent derivations in his paper is quite involved.

The first step consisted in the definition of tensor densities. By using both Equation and Equation , Einstein obtained the Einstein. After a field rescaling, he then took a third expression to become his Lagrangian. Nobody can determine empirically an affine connection for vectors at neighbouring points if he has not obtained the line element before. He criticised a theory that keeps only the connection as a fundamental building block for its lack of a guarantee that it would also house the conformal structure light cone structure.

This is needed for special relativity to be incorporated in some sense, and thus must be an independent fundamental input [ ]. From a recent conversation with Einstein I learn that he is of much the same opinion. His outlook on the state of the theory now was rather bleak:. To me, the quantum-problem seems to require something like a special scalar, for the introduction of which I have found a plausible way. But I fail to succeed in giving my pet idea a tangible form: to understand the quantum-structure through an overdetermination by differential equations.

The initial state of an electron moving around a hydrogen nucleus cannot be chosen freely; its choice must correspond to the quantum conditions. In general: not only the evolution in time but also the initial state obey laws. He then ventured the hope that a system of overdetermined differential equations is able to determine.

One of the crucial tests for an acceptable unified field theory for him now was:. In such a way, the un-ambiguity of the initial conditions ought to be understood without leaving field theory. In the introduction to his book, Struik distinguished three directions in the development of the theory of linear connections [ ]:.

The generalisation of parallel transport in the sense of Levi-Civita and Weyl. Schouten is the leading figure in this approach [ ]. Thomas [ ], and T. Thomas [ , ]. Here, only symmetric connections can appear. The idea of mapping a manifold at one point to a manifold at a neighbouring point is central affine, conformal, projective mappings. In his assessment, Eisenhart [ ] adds to this all the geometries whose metric is. Developments of this theory have been made by Finsler, Berwald, Synge, and J.

In this geometry the paths are the shortest lines, and in that sense are a generalisation of geodesics. Affine properties of these spaces are obtained from a natural generalisation of the definition of Levi-Civita for Riemannian spaces. In fact, already in May Jan Arnoldus Schouten in Delft had submitted two papers classifying all possible connections [ , ]. In the first he wrote:.

Weyl, Raum-Zeit-Materie , 2. Section, Leipzig 3. The most general connection is characterised by two fields of third degree, one tensor field of second degree, and a vector field […]. The fields referred to are the torsion tensor S ij k , the tensor of non-metricity Q ij k , the metric g ij , and the tensor C ij k which, in unified field theory, was rarely used.

It arose because Schouten introduced different linear connections for tangent vectors and linear forms. He defined the covariant derivative of a 1-form not by the connection L ij k in Equation 13 , but by. In fact. Furthermore, on p. For such an extension an invariant fixing of the connection is needed, because a physical phenomenon can correspond only to an invariant expression.

According to Schouten. In the following pages will be shown that this difficulty disappears when the more general supposition is made that the original deplacement is not necessarily symmetrical. He then restricted the generality of his approach; in modern parlance, he did allow for vector torsion only:.

On the same topic, Schouten wrote a paper with Friedman in Leningrad [ ]. He relied on the curvature, torsion and homothetic curvature 2-forms [ 32 ], Section III; cf. The spreading of knowledge about properties of differential geometric objects like connection and curvature took time, however, even in Leningrad.

After an uninterrupted search during the past two years I now believe to have found the true solution. After some manipulations, the variation with regard to the metric and to the connection led to the following equations:. However, he cautioned:. 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.

The two covariant derivatives introduced by J. Thomas are and. Thomas then could reformulate Equation in the form. 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 unchanged. 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.

He went on to say:. 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. 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:. 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.

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. The new field equation was picked up by R. In the same spirit as the one of his paper, Einstein said good bye to his theory in a letter to Besso on Christmas in words similar to those in his letter in June:.

Anyway, I now am convinced that, unfortunately, nothing can be made with the complex of ideas by Weyl-Eddington. The equations. I take as the best we have nowadays. But it appears doubtful whether there is room in them for the quanta. It does not allow for electrical masses free from singularities. Moreover, I cannot bring myself to gluing together two items as the l.

Research on affine geometry as a frame for unified field theory was also carried on by mathematicians of the Princeton school. Thus J. We met J. During the period considered here, a few physicists followed the path of Eddington and Einstein. He showed that, in first approximation, he got what is wanted, i. Three months later, Infeld published a note in Comptes Rendus of the Parisian Academy in which he now presented the exact connection as.

Thus, he is back at vector torsion treated before by Schouten [ ]. The Japanese physicist Hattori embarked on a metric-affine geometry derived purely from an asymmetric metrical tensor. He defined an affine connection. The electromagnetic field was not identified with f ik by Hattori, but with the skew-symmetric part of the generalised Ricci tensor formed from. By introducing the tensor , he could write the generalised Ricci tensor as. F ikl is formed from F ik as f ikl from f ik.

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. Infeld could as well have applied this admonishment to his own unified field theory discussed above. Thus, Straneo suggested a unified field theory with only vector torsion as Schouten had done 8 years earlier [ , ] without referring to him.

The field equations Straneo wrote down, i. Straneo wrote further papers on the subject [ , ]. By this, he claimed to have made superfluous the five-vectors of Einstein and Mayer [ ]. 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 [ ].

Einstein, in his papers, did not comment on the missing metric compatibility in his theory and its physical meaning. In this work a generalisation of the equation for metric compatibility, i. The continuation of this research line will be presented in Part II of this article. Lorentz, 16 February On the next day 17 February , 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 — not just in first order as Kaluza had done [ 81 , 82 ]. He came too late: Klein had already shown the same before [ ]. 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:. 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. That Klein had published another important clarifying note in Nature , in which he closed the fifth dimension, seems to have escaped Einstein [ ]. Maxwell equations [ ], p. Mandel of Leningrad was not given credit by Einstein although he also had rediscovered by a different method some of O.

From the geodesics in M 5 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. 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 M 5. A main motivation for Klein was to relate the fifth dimension with quantum physics.

From a postulated five-dimensional wave equation. By this, the reduction of five-dimensional equations as e. Klein had only the lowest term in the series. Beyond incredibly complicated field equations nothing much had been gained [ ]. Even L. Klein and Einstein. He then discussed conformally invariant field equations, and tried to relate them to equations of wave mechanics [ ]. In , N. Presently, the different contributions of Kaluza and O.

An early criticism of this unhistorical attitude has been voiced in [ ]. Indices are raised and lowered with the metrics of V 5 or V 4 , respectively. A consequence then is. Both covariant derivatives are abbreviated by the same symbol A ; k. The covariant derivative of tensors with both indices referring to V 5 and those referring to V 4 , is formed correspondingly. The autoparallels of V 5 lead to the exact equations of motion of a charged particle, not the geodesics of V 4.

From them follows. They also noted that a symmetric tensor F kl could have been interpreted as the second fundamental form, and the formalism would then be the same as local isometric embedding of V 4 into V 5. It is related to the Riemannian curvature of V 4 by. From , by transvection with , the 5-curvature itself appears:.

Dynamic Measurement and Control Trans. Schajer, G. Wood Fiber Sci. Tobias, S. The Institution of Mechan. Yu, R. Download references. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the author s and do not necessarily reflect the views of the National Science Foundation. Reprints and Permissions. Parker, R. Asymmetric tensioning of circular saws. Holz als Roh-und Werkstoff 47, — Download citation. Issue Date : April Search SpringerLink Search.

Abstract The most frequently used tensioning methods are hammering and roll tensioning. Immediate online access to all issues from Subscription will auto renew annually. References Clough, R. Mote Jr. Authors R. Parker View author publications. View author publications. Additional information The authors would like to express their sincere thanks to the National Science Foundation, the University of California Forest Products Laboratory, and to the following corporations: California Cedar Products Co.

Here, only symmetric connections can appear. The idea of mapping a manifold at one point to a manifold at a neighbouring point is central affine, conformal, projective mappings. In his assessment, Eisenhart [ ] adds to this all the geometries whose metric is. Developments of this theory have been made by Finsler, Berwald, Synge, and J. In this geometry the paths are the shortest lines, and in that sense are a generalisation of geodesics.

Affine properties of these spaces are obtained from a natural generalisation of the definition of Levi-Civita for Riemannian spaces. In fact, already in May Jan Arnoldus Schouten in Delft had submitted two papers classifying all possible connections [ , ]. In the first he wrote:. Weyl, Raum-Zeit-Materie , 2. Section, Leipzig 3. The most general connection is characterised by two fields of third degree, one tensor field of second degree, and a vector field […]. The fields referred to are the torsion tensor S ij k , the tensor of non-metricity Q ij k , the metric g ij , and the tensor C ij k which, in unified field theory, was rarely used.

It arose because Schouten introduced different linear connections for tangent vectors and linear forms. He defined the covariant derivative of a 1-form not by the connection L ij k in Equation 13 , but by. In fact. Furthermore, on p. For such an extension an invariant fixing of the connection is needed, because a physical phenomenon can correspond only to an invariant expression. According to Schouten. In the following pages will be shown that this difficulty disappears when the more general supposition is made that the original deplacement is not necessarily symmetrical.

He then restricted the generality of his approach; in modern parlance, he did allow for vector torsion only:. On the same topic, Schouten wrote a paper with Friedman in Leningrad [ ]. He relied on the curvature, torsion and homothetic curvature 2-forms [ 32 ], Section III; cf. The spreading of knowledge about properties of differential geometric objects like connection and curvature took time, however, even in Leningrad.

After an uninterrupted search during the past two years I now believe to have found the true solution. After some manipulations, the variation with regard to the metric and to the connection led to the following equations:. However, he cautioned:. 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.

The two covariant derivatives introduced by J. Thomas are and. Thomas then could reformulate Equation in the form. 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 unchanged.

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. He went on to say:. 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. 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:.

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. 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.

The new field equation was picked up by R. In the same spirit as the one of his paper, Einstein said good bye to his theory in a letter to Besso on Christmas in words similar to those in his letter in June:. Anyway, I now am convinced that, unfortunately, nothing can be made with the complex of ideas by Weyl-Eddington. The equations.

I take as the best we have nowadays. But it appears doubtful whether there is room in them for the quanta. It does not allow for electrical masses free from singularities. Moreover, I cannot bring myself to gluing together two items as the l. Research on affine geometry as a frame for unified field theory was also carried on by mathematicians of the Princeton school.

Thus J. We met J. During the period considered here, a few physicists followed the path of Eddington and Einstein. He showed that, in first approximation, he got what is wanted, i. Three months later, Infeld published a note in Comptes Rendus of the Parisian Academy in which he now presented the exact connection as. Thus, he is back at vector torsion treated before by Schouten [ ].

The Japanese physicist Hattori embarked on a metric-affine geometry derived purely from an asymmetric metrical tensor. He defined an affine connection. The electromagnetic field was not identified with f ik by Hattori, but with the skew-symmetric part of the generalised Ricci tensor formed from. By introducing the tensor , he could write the generalised Ricci tensor as. F ikl is formed from F ik as f ikl from f ik. 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.

Infeld could as well have applied this admonishment to his own unified field theory discussed above. Thus, Straneo suggested a unified field theory with only vector torsion as Schouten had done 8 years earlier [ , ] without referring to him. The field equations Straneo wrote down, i. Straneo wrote further papers on the subject [ , ]. By this, he claimed to have made superfluous the five-vectors of Einstein and Mayer [ ].

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 [ ]. Einstein, in his papers, did not comment on the missing metric compatibility in his theory and its physical meaning.

In this work a generalisation of the equation for metric compatibility, i. The continuation of this research line will be presented in Part II of this article. Lorentz, 16 February On the next day 17 February , 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 — not just in first order as Kaluza had done [ 81 , 82 ]. He came too late: Klein had already shown the same before [ ]. 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:. 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. That Klein had published another important clarifying note in Nature , in which he closed the fifth dimension, seems to have escaped Einstein [ ].

Maxwell equations [ ], p. Mandel of Leningrad was not given credit by Einstein although he also had rediscovered by a different method some of O. From the geodesics in M 5 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. 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 M 5.

A main motivation for Klein was to relate the fifth dimension with quantum physics. From a postulated five-dimensional wave equation. By this, the reduction of five-dimensional equations as e. Klein had only the lowest term in the series. Beyond incredibly complicated field equations nothing much had been gained [ ]. Even L. Klein and Einstein. He then discussed conformally invariant field equations, and tried to relate them to equations of wave mechanics [ ]. In , N. Presently, the different contributions of Kaluza and O.

An early criticism of this unhistorical attitude has been voiced in [ ]. Indices are raised and lowered with the metrics of V 5 or V 4 , respectively. A consequence then is. Both covariant derivatives are abbreviated by the same symbol A ; k. The covariant derivative of tensors with both indices referring to V 5 and those referring to V 4 , is formed correspondingly. The autoparallels of V 5 lead to the exact equations of motion of a charged particle, not the geodesics of V 4.

From them follows. They also noted that a symmetric tensor F kl could have been interpreted as the second fundamental form, and the formalism would then be the same as local isometric embedding of V 4 into V 5. It is related to the Riemannian curvature of V 4 by. From , by transvection with , the 5-curvature itself appears:.

By contraction, and. Two new quantities are introduced:. It turns out that. Also, in a lecture given on 14 October in the Physics Institute of the University of Wien, he still was proud of the 5-vector approach. However, following an idea half of which came from myself and half from my collaborator, Prof.

Mayer, a startlingly simple construction became successful. In this way, we succeeded to recognise the gravitational and electromagnetic fields as a logical unity. 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 in fourdimensional space, which are bound to space by a linear mechanism. Let a s be the 4-vector belonging to ; then such a relation obtains. In the theory equations are meaningful which hold independently of the special relationship generated by. Infinitesimal transport of in fourdimensional space is defined, likewise the corresponding 5-curvature from which spring the field equations. In his report for the Macy-Foundation, which appeared in Science on the very same day in October , Einstein had to be more optimistic:.

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.

However, according to Pauli, Veblen and Hoffmann had spoiled the advantage of projective theory:. The five-dimensional space is just a mathematical device to represent the events points of space-time by these curves. Thus, Veblen and Hoffmann also gained the Klein. Gordon equation in curved space, i. Nevertheless, Hoffman remained optimistic:.

In particular, we do not demand a relationship between electrical charge and a fifth coordinate; our theory is strictly four-dimensional. Of the three basic assumptions of the previous paper, the second had to be given up. The expression in the middle of Equation is replaced by. The field equations were set up according to the method of the first paper; now the 5-curvature scalar was.

It also turned out that with , i. In the last paragraph, the compatibility of the equations was proven, and at the end Cartan was acknowledged:. At about the same time as Einstein and Mayer wrote their second note, van Dantzig continued his work on projective geometry [ , , ]. Together with him, Schouten wrote a series of papers on projective geometry as the basis of a unified field theory [ , , , ] , which, according to Pauli, combine.

In this paper [ ], p. 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. The authors pointed out that. In a sequel to this publication, Pauli and Solomon corrected an error:.

Then we discuss the form of the energy-momentum tensor and of the current vector in the theory of Einstein-Mayer. Michal and his co-author generalised the Einstein-Mayer 5-vector-formalism:. Robertson found a new way of applying distant parallelism: He studied groups of motion admitted by such spaces, e.

Section 6. 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. The contributions from the Levi-Civita connection and from contorsion 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. 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 [ ], p.

However, when Einstein published his contributions in June [ 84 , 83 ], Cartan had to remind him that a paper of his introducing the concept of torsion had. 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 an investigation concerning spaces with simply transitive continuous groups, Eisenhart already in had found the connection for a manifold with distant parallelism given 3 years later by Einstein [ ].

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. 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.

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 — and done so even better cf. Anyhow, in his response Einstein to Cartan on 10 May , [ 50 ], p. After Cartan had sent his historical review to Einstein on 24 May , the latter answered three months later:. The publication should appear in the Mathematische Annalen because, at present, only the mathematical implications are explored and not their applications to physics.

Interestingly, he permitted himself to interpet the physical meaning of geometrical structures :. Cartan that the treatment of continua of the species which is of import here, is not really new. 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. The split, in first approximation, of the tetrad field h ab according to lead to homogeneous wave equations and divergence relations for both the symmetric and the antisymmetric part identified as metric and electromagnetic field tensors, respectively.

Einstein in 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 , in its column News and Views , the respected British science journal Nature reported:. 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. Nature then went on to quote from an interview of Einstein of 26 January in a newspaper, the Daily Chronicle. According to the newspaper, among other statements Einstein made, in his wonderful language, was the following:. 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 [ ], Dokument Nr. 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 — without using that name. His closing sentences are :.

The answer to this question which I have attempted to give in a new paper yields unitary field laws for gravitation and electromagnetism. A few months later in that year, again in Nature , the mathematician H. 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:. It may succeed in predicting some interaction between gravitation and electromagnetism which can be confirmed by observation.

If this were true it would be impossible to calculate the consequences. Indeed, there was a lot of work to do, only in part because Einstein, from one paper to the next, had changed his field equations. As we have seen before, the components of the metric tensor are defined by. 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 , then the connection components turn out to be. Also, the metric is covariantly constant. The Riemannian curvature tensor calculated from Equation turns out to vanish.

As Einstein noted, by g ij from Equation also the usual Riemannian connection may be formed. Moreover, is a tensor that could be used for building invariants. In principle, distant parallelism is a particular bi-connection theory. The connection does not play a role in the following cf. From Equation , obviously the torsion tensor follows cf. He indicated how a Lagrangian could be built and the 16 field equations for the field variables h lj obtained.

In his second note [ 83 ], Einstein departed from the Lagrangian , i. However, as he added in a footnote, pure gravitation could have been characterised by as well. To do so he replaced by and introduced. Einstein concluded that. 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 tensor. He thus arrived at the identity given by Equation 29 , i. By defining , and contracting equation , Einstein obtained another identity :. For the proof, he used the formula for the covariant vector density given in Equation 16 , which, for the divergence, reduces to.

With this first approximation as a hint, Einstein, after some manipulations, postulated the 20 exact field equations:. 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:.

In the meantime, however, he had found a Lagrangian such that the compatibility-problem disappeared. He restricted the many constructive possibilities for 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:. Stodola, Einstein summed up what he had reached.

He presented it as an introduction suited for anyone who knew general relativity. It is here that he first mentioned Equations and Most importantly, he gave a new set of field equations not derived from a variational principle; they are. As Cartan remarked, Equation expresses conservation of torsion under parallel transport:. 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.

From Equation with the help of Equation , , Einstein wrote down two more identities. One of them he had obtained from Cartan:. Here, F k is introduced by. Einstein then showed that. 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, Zaycoff , tried to make the best out of them:. 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. They were published in 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.

From a purely mathematical point of view they were studied previously. Cartan was so amiable as to write a note for the Mathematische Annalen exposing the various phases in the formal development of these concepts.

Later in the paper, he comes closer to the point:. Some of the material in the paper overlaps with results from other publications [ 85 , 90 , 93 ]. Hence 7 identities should exist, four of which Einstein had found previously. 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 [ 92 ], p.

It is reproduced also in [ 90 ]. Then Einstein presented the same field equations as in his paper in Annalen der Mathematik , which he demanded to be. Sixteen field equations were needed which, due to covariance, induced four identities. 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.

In linear approximation, i. 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 [ ]. Then the 4-bein contains three arbitrary functions of one parameter s :. As an exact solution of the field equations , , Einstein and Mayer obtained and.

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. Two days before the paper by Einstein and Mayer became published by the Berlin Academy, Einstein wrote to his friend Solovine:.

Cartan has already worked with it. I myself work with a mathematician S. Mayer from Vienna , a marvelous chap […]. The mentioning of Cartan resulted from the intensive correspondence of both scientists between December and February About a dozen letters were exchanged which, sometimes, contained long calculations [ 50 ] cf. In an address given at the University of Nottingham, England, on 6 June , Einstein also must have commented on the exact solutions found and on his program concerning the elementary particles.

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. In addition to the assumptions 1 , 2 , 3 for allowable field equations given above, further restrictions were made:. After inserting Equation into Equation , 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:. The remaining two types are denoted in the paper by […]. 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 content. Once again, he dropped the subject and moved on to the next.

It seems that this structure has nothing to do with the true character of space […]. However, the correspondence on the subject came to an end in May with a last letter by Cartan. 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.

After having explained the theory and having pointed out the differences to his own affine unified field theory of , he confessed:. 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 […] 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, […] 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.

As usual, Pauli was less than enthusiastic; he expressed his discontent in a letter to Hermann Weyl of 26 August 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]. During the Easter holidays I have visited Einstein in Berlin and found his opinion on modern quantum theory reactionary.

Einstein had sent a further exposition of his new theory to the Mathematische Annalen in August When he received its proof sheets from Einstein, Pauli had no reservations to criticise him directly and bluntly:. 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 […].

It just remains […] to congratulate you or should I rather say condole you? 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.

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. Before he had written to Einstein, Pauli, with lesser reservations, complained vis-a-vis Jordan:.

With such rubbish he may impress only American journalists, not even American physicists, not to speak of European physicists. 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.

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 see letter of Einstein to Pauli on 22 January ; cf. In it, Lanczos cautiously offers some criticism after having made enough bows before Einstein:. 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.

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. He varied g ik and R ik independently [ ]. For Lanczos see J. However, he had kept the metric g ik introduced into the formalism by.

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. An exact, complicated wave equation followed:. In linear approximation, the Einstein vacuum and the vacuum Maxwell equations are obtained, supplemented by the homogeneous wave equation for a vector field [ ]. He also defended Einstein against critical remarks by Eddington [ 62 ] and Schouten [ ], 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.

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. A conference on theoretical physics at the Ukrainian Physical-Technical Institute in Charkow in May , brought together many German and Russian physicists. Unified field theory, quantum mechanics, and the new quantum field theory were all discussed. 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.

Levi-Civita also had sent a paper on distant parallelism to Einstein, who had it appear in the reports of the Berlin Academy [ ]. Levi-Civita introduced a set of four congruences of curves that intersect each other at right angles, called their tangents and used Equation in the form:. They obey. He used the time until the printing was done to give a short preview of his paper in Nature [ ]. 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 […] with those of Einstein.

Einstein had sent him the corrected proof sheets of his fourth paper [ 85 ]. The basic idea was to consider the points of M 4 as equivalent to the ensemble of congruences with tangent vector X 5 i in M 5 with cylindricity condition werden. The space-time interval is defined as the distance of two lines of the congruence on. Mandel did not identify the torsion vector with the electromagnetic 4-potential, but introduced the covariant derivative , where the tensor is skew-symmetric.

We may look at this paper also as a forerunner of some sort to the Einstein. Mayer 5-vector formalism cf. 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. It is shown that the new equations are satisfied to the first order but not exactly.

He then goes on to find a rigourous solution and obtains the metric and the 4-potential [ ]. They found that these field equations did not have a spherically symmetric solution corresponding to a charged point particle at rest. The corresponding solution for the uncharged particle was the same as in general relativity, i. 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 [ , ].

In his paper in July , the physicist Zaycoff had some details:. In this case he has been able to obtain results checking the observed perihelion of mercury. 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 relativity.

In the second of his two brief notes, Salkover succeeded in gaining the most general, spherically symmetric solution [ , ]. This is admitted by the authors in their second paper, in which they present a new calculation. Vallarta also wrote a paper by himself [ ], p.

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 paper [ 88 ]. In Princeton, people did not sleep either. In and , T. Thomas wrote a series of six papers on distant parallelism and unified field theory.

Thomas described the contents of his first paper as follows:. This looks as if he had introduced four vector potentials for the electromagnetic field, and this, in fact, T. The gravitational potentials are still g ik. In his next note, T. Thomas changed his field equations on the grounds that he wanted them to give a conservation law.

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 — provided, of course, that the theory shows itself to be tenable as a theory in agreement with experience.

In his three further installments, T. Thomas moved away from unified field theory to the discussion of mathematical details of the theory he had advanced [ , , ]. Unhindered by constraints from physical experience, mathematicians try to play with possibilities. In the framework of a purely affine theory he obtained a necessary and sufficient condition for this geometry,. The resulting connection is given by. Schouten and van Dantzig also used a geometry built on complex numbers, and on Hermitian forms:.

At first, the possibility of gaining hold on the paths of elementary particles — described as singular worldlines of point particles — was central. 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.

It seems that Einstein, during his visit to Paris in November , had talked to Cartan about his problem of finding the right field equations and proving their compatibility. Starting in December of and extending over the next year, an intensive correspondence on this subject was carried on by both men [ 50 ]. On 3 December , Cartan sent Einstein a letter of five pages with a mathematical note of 12 pages appended. Cartan admitted :.

There are other possibilities giving rise to richer geometrical schemes while remaining deterministic. First, one can take a system of 15 equations […]. 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 […].

As the further correspondence shows, he had difficulties in following Cartan:. I beg you to send me those of your papers from which I can properly study the theory. 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 note.

Now, everything is clear to me. Previously, my assistant Prof. This probably will restrict the free choice of solutions in a region in a far-reaching way — more strongly than the restrictions corresponding to your degrees of determination. In this section, we loosely collect some of these approaches. The mathematicians Struik and Wiener found the task of an amalgamation of relativity and quantum theory wave mechanics attractive:.

A further example for the new program is given by J. Whittaker at the University of Edinburgh [ ] who wished to introduce the wave function via the matter terms:. The object of the present paper is to find these equations […]. Zaycoff, from the point of view of distant parallelism, found the following objection to unified field theory as the only valid one:. By the work of Dirac, wavemechanics has reached an independent status; the only attempt to bring together this new group of phenomena with the other two is J.

Whittaker had expressed himself more clearly:. These are grouped together to form two four-vectors and satisfy wave equations of the second order. Whittaker also had written down a variational principle by which the gravitational and electromagnetic field equations were also gained. However, as the terms for the various fields were just added up in his Lagrangian, the theory would not have qualified as a genuine unified field theory in the spirit of Einstein.

What fancy, if only shortlived, flowers sprang from the mixing of geometry and wave mechanics is shown by the example of H. Although, two years later, Jehle withdrew his claims concerning elementary particles, he continued to apply.

How to combine them with the vectors and tensors appearing in electromagnetic and gravitational theories? As the spinor representation is the simplest representation of the Lorentz group, everything may be played back to spin space.

At the time, this was being done in different ways, in part by the use of number fields with which physicists were unacquainted such as quaternions and sedenions cf. Schouten [ ]. Others, such as Einstein and Mayer, liked vectors better and introduced so-called semi-vectors. Some less experienced, as e. Temple, even claimed that a tensorial theory was necessary to retain it relativistically:. It is contended here that therefore his theory cannot be upheld without abandoning the theory of relativity.

While this story about geometrizing wave mechanics might not be a genuine part of unified field theory at the time, it seems interesting to follow it as a last attempt for binding together classical field theory and quantum physics. Some of the motivation for these papers came from formal considerations, i.

His approach for embedding wave mechanics into a Maxwell-like was continued in further papers, in part in collaboration with J. Fisher; to them it appeared. His conclusion sounds a bit strange:. In this context, another unorthodox suggestion was put forward by A. Anderson who saw matter and radiation as two phases of the same substrate:. Electrons and protons cannot be distinguished from quanta of light, gas pressure not from radiation pressure.

Anderson somehow sensed that charge conservation was in his way; he meddled through by either assuming neutral matter, i. One of the German theorists trying to keep up with wave mechanics was Gustav Mie. In the same year in which Einstein published his theory of distant parallelism, Dirac presented his relativistic, spinorial wave equation for the electron with spin. This event gave new hope to those trying to include the electron field into a unified field theory; it induced a flood of papers in such that this year became the zenith for publications on unified field theory.

Although, as we noted in Section 6. This is a very sketchy outline with a focus on the relationship to unified field theories. An interesting study into the details of the introduction of local spinor structures by Weyl and Fock and of the early history of the general relativistic Dirac equation was given recently by Scholz [ ].

For some time, the new concept of spinorial wave function stayed unfamiliar to many physicists deeply entrenched in the customary tensorial formulation of their equations. For example, J. Some early nomenclature reflects this unfamiliarity with spinors. For the 4-component spinors or Dirac-spinors cf. Ehrenfest, in , still complained :. In , three publications of the mathematician Veblen in Princeton on spinors added to the development. Veblen imbedded spinors into his projective geometry [ ]:.

There are several ways of embodying this invariant theory in a formal calculus. The one which is here employed has its antecedents chiefly in the work ofWeyl, van derWaerden, Fock, and Schouten. It differs from the calculus arrived at by Schouten chiefly in the treatment of gauge invariance, Schouten in collaboration with van Dantzig having preferred to rewrite the projective relativity in a formalism obtainable from the original one by a sort of coordinate transformation, whereas I think the original form fits in better with the classical notations of relativity theory.

These spinors have been recognised by several students Pauli and Solomon, Fock of the subject but their role has probably not been fully understood since it has quite recently been thought necessary to give special proofs of invariance. The transformation law for spinors is the same as before :.

Sections 7. The transformation given above carries the system of lines into the other. After Tetrode and Wigner, whose contributions were mentioned in Section 6. Actually, Weyl had expressed the change in his outlook, so important for the idea of gauge-symmetry in modern physics [ ], pp. We have noted before his refutation of distant parallelism cf. He partly agreed with what Einstein imagined:. For many years, Weyl had given the statistical approach in the formulation of physical laws an important role.

He therefore could adapt easily to the Born-Jordan-Heisenberg statistical interpretation of the quantum state. For Weyl and statistics, cf. Up to now, quantum mechanics has not found its place in this geometrical picture; attempts in this direction Klein, Fock were unsuccessful. Only after Dirac had constructed his equations for the electron, the ground seems to have been prepared for further work in this direction.

Thus is interpreted in the sense of Weyl:. Another note and extended presentations in both a French and a German physics journal by Fock alone followed suit [ , , ]. In the first paper Fock defined an asymmetric matter tensor for the spinor field,.

The covariant derivative then is. The divergence of the complex energy-momentum tensor satisfies. In this point, our theory, developed independently, agrees with the new theory by H. In this regard he found himself in accord with Weyl, whose approach to the Dirac equation he nevertheless criticised:.

Nevertheless, it seems to us that the theory suggested by Weyl for solving this problem is open to grave objections; a criticism of this theory is given in our article. His mass term contained a square root, i. As he remarked, the chances for this were minimal, however [ , ].

In two papers, Zaycoff of Sofia presented a unified field theory of gravitation, electromagnetism and the Dirac field for which he left behind the framework of a theory with distant parallelism used by him in other papers. By varying his Lagrangian with respect to the 4-beins, the electromagnetic potential, the Dirac wave function and its complex-conjugate, he obtained the 20 field equations for gravitation of second order in the 4-bein variables, assuming the role of the gravitational potentials and the electromagnetic field of second order in the 4-potential , and 8 equations of first order in the Dirac wave function and the electromagnetic 4-potential, corresponding to the generalised Dirac equation and its complex conjugate [ , ].

This means that he considered the electron as extended. At this occasion, he fought with himself about the admissibility of the Kaluza-Klein approach:. Mandel, G. Rumer, the author et al. No doubt, there are weighty reasons for such a seemingly paradoxical view. A multi-dimensional causality cannot be understood as long as we are unable to give the extra dimensions an intuitive meaning.

Section 8. In the paper, Zaycoff introduced a six-dimensional manifold with local coordinates x 0 , … , x 5 where x 0 , x 5 belong to the additional dimensions. He wrote two papers, one concerned with the four-dimensional and a second one with the five-dimensional approach [ , ]. For him an important conclusion is that. We reproduce a remark from his publication [ ]:. Most authors introduce an orthogonal frame of axes at every event, and, relative to it, numerically specialised Dirac-matrices.

To be sure, it is much too small by many powers of ten in order to replace, say, the term on the r. Yet it appears important that in the generalised theory a term is encountered at all which is equivalent to the enigmatic mass term. The last, erroneous, sentence must have made Pauli irate.

Everybody should be kept from reading this paper, or from even trying to understand it. Moreover, all articles referred to on p. One of the essential features of quantum mechanics, the non-commutativity of conjugated observables like position and momentum, nowhere entered the approaches aiming at a geometrization of wave mechanics. Einstein was one of those clinging to the picture of the wave function as a real phenomenon in space-time. It should have become a 4-page publication in the Sitzungsberichte.

As he wrote to Max Born:. Will appear soon in Sitz. However, he quickly must have found a flaw in his argumentation: He telephoned to stop the printing after less than a page had been typeset. This did not happen; thus we know of his failed attempt, and we can read how his line of thought began [ ], pp. Thus, in February , Wiener and Vallarta stressed that.

That the micro-mechanical world of the electron is Minkowskian is shown by the theory of Dirac, in which the electron spin appears as a consequence of the fact that the world of the electron is not Euclidean, but Minkowskian. The correction of this misjudgement of Wiener and Vallarta by Fock and Ivanenko began only one month later [ ], and was complete in the summer of [ , , , ].

Gonseth and Juvet, in the first of four consecutive notes submitted in August [ , , , ] stated:. Thus, we will have a frame in which to take the gravitational and electromagnetic laws, and in which it will be possible also for quantum theory to be included. Their further comment is:. Obviously, this artifice will be needed if some phenomenon would force the physicists to believe in a variability of the [electric] charge.

Interestingly, a couple of months later, O. However, as he remarked, his hopes had been shattered [ ]. If the elementary charge of an electron has been measured precisely, then the fifth coordinate is as uncertain as can be. In another paper, Klein suggested the idea that the physical laws in space-time might be implied by equations in five-dimensional space when suitably averaged over the fifth variable.

He tried to produce wavemechanical interference terms from this approach [ ]. At about the same time, W. Klein and noted:. Flint has drawn my attention to a recent paper by O. Klein [ ] in which an extension to five dimensions similar to that given in the present paper is described. It leads also to a wave equation which we can identify as relating to a system containing electrons with opposite spin.

The domain of either electron alone might be rotated in a fifth dimension and we could not observe any difference. He proceeded from the special relativistic homogeneous wave equation in fivedimensional space and, after dimensional reduction, compared it to the Klein-Gordon equation for a charged particle.

Gordon [ ], and by J. The natural frequencies increase monotonically with increased pressure, and the magnitude of the shifts increase with the introduction of additional stressed holes. Hole eccentricity affects which natural frequencies are altered most, so the tensioning can be directed towards optimal increases in certain vibration modes.

The predicted increases in the natural frequencies associated with the potentially unstable modes are limited only by the restrictions of classical plate theory and loading within unstable modes are limited only by the restrictions of classical plate theory and loading within the elastic limit of the saw. This is a preview of subscription content, access via your institution. Rent this article via DeepDyve. Clough, R. Matrix Methods in Structural Mechanics. Google Scholar. Dym, C. Leyden, The Netherlands: Norrhoff Internat.

Gulick, F. Jeffrey, G. Royal Soc. London, Series A. Mote, C. Service Report No. Mote, J. Stability control analysis of rotating plates by finite element: Emphasis on slots and holes. Dynamic Measurement and Control Trans. Schajer, G. Wood Fiber Sci. Tobias, S. The Institution of Mechan. Yu, R.

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