The Irish Press caught the bait: In the text supplied by Einstein which became also widely distributed, he said:. Even from this point of view I can see no special advantages over the theoretical possibilities known before, rather the opposite. Einstein replied coolly and curtly:. I would have liked to only respond to it after your first enthusiasm about the new field equations will have given place to a more sober judgment perhaps the letter is written still too early. Of course, progress is made by your decision to take a specific Lagrangian; also, the mathematical side of your thoughts to me seems extraordinarily clear.
Nevertheless, my reservations with regard to a non-irreducible object as a basis continue unabatedly. He then expressed in more detail, why for him, only irreducible tensors are the variables to be used. To a friend he had written:. The result is fascinatingly beautiful. I could not sleep a fortnight without dreaming of it.
The report of L. He distinguished between the three cases: The further classification depended on additional symmetry conditions on the basic variable s. It was perhaps this remark which induced J. He derived such transformations changes from the weakened condition for auto-parallels cf. If a symmetric connection is used, they are given by The impression prevails that the basic geometrical concepts have nothing to do with physics.
Einstein did express it like this: In a paper of , he showed a pragmatic attitude: He set out to solve approximately the field equations with asymmetric metric and asymmetric connection He also offered three alternatives for an energy pseudo- tensor which all vanish for a single plane wave. Thus, the paper contained no new fundamental insights. A favorable reaction came from a young Harvard mathematician R. He assumed h ij to represent gravitation but found the direct link of the skew-symmetric part of g ij with the electromagnetic field as incorrect.
By an elementary calculation presented a year later also by M. This equation then is rewritten as one of the usual Maxwell equations in a space with metric h ij with a complicated r. Likewise, an additional condition was laid on the curvature tensor cf. Fortunately, this naive strategy of imposing additional conditions with the aim to obtain interpretable field equations, did not have many followers. Papapetrou [ ]. His ansatz contained five unknown functions of the radial coordinate.
An assumption used was that asymptotically, i. At this point, this seems no serious objection to the theory, because other static spherically symmetric SSS solution might exist. Thus, the radial electromagnetic field is constant. Moreover, this constant electric field does not influence the gravitational field. Papapetrou also discussed approximate solutions and concluded for them that k ij does not describe the electromagnetic field but the electromagnetic potential.
This would rule out an interpretation of the solution in terms of an electric field.
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All these solutions would not have been acceptable to Einstein in the sense of describing sources of electricity , because they were not free of singularities. They are, in suitable cases, first approximations for weak fields. It is now my belief that, for a serious and rigorous field theory, one must insist that the field be free of singularities everywhere. We shall come back to the demand that exact solutions ought to be free of singularities in Sections 9. Note that this result depends on the identification of the symmetric part of the metric with the gravitational field potential.
A year later, a different proof was given by E. Straus for the weak field equations. In the same paper, Straus concluded: And then, immediately, the time-dependence of the free functions was dropped. The main conclusion drawn by Takeno et al. This condition is consistent with derived by M. In his paper of discussed above in Section 8. I do not believe this, mainly because I do not believe the ultimate particles to be identifiable individuals that could be described in this fashion. Moreover, in the symmetric theory i. In view of the research done since, e. In the meantime, Einstein had gone on struggling with his field equations and, in a letter to M.
Solovine of 25 November , had become less optimistic [ ], p. I will not be able to finish it [the work]; it will be forgotten and at a later time arguably must be re-discovered. It happened this way with so many problems. In his correspondence with Max Born during the second half of the s, Einstein clung to his refusal of the statistical interpretation of quantum mechanics. According to him, physics was to present reality in space and time without, as it appeared to him, ghostly interactions at a distance.
In a letter of 3 March , he related this to UFT:. Yet the calculatory difficulties are so great that I shall bite the dust until I myself have found an assured opinion of it. In spite of such reservations, Einstein carried on unflagging with his research. In his next publication on UFT [ ], he again took a complex asymmetric metric field. By adding four such terms, the Hermitian metric tensor can be obtained. In order to again be able to gain the weak field equations, an additional assumption was made: The motivation behind this trick is to obtain the compatibility equation 30 from , indirectly.
The skew-symmetric part of is formed and a trace taken in order to arrive at. With the help of this equation, finally reduces to As in [ ], Einstein did not include homothetic curvature into the building of his Lagrangian with the same unconvincing argument: Einstein summed up the paper for Pauli on one page or so and concluded: The few things we were able to calculate strengthened my confidence in this theory.
To me it deems that, even if such solutions do exist in a suitably chosen field theory, it would not be possible to relate them with the atomic facts in physics in the way you wish, namely in a way that avoids the statistical interpretation, in principle. As will be seen in Section 9.
Did God tell this into your ear? But the way of proceeding is: These identities are the means to find the equations. Six weeks later, on 30 September , Einstein had changed his mind: But the compatibility for this stronger system is problematic; i. At first, Einstein seems to have followed a strategy of directly counting equations, variables, and identities. However, early in he seems to have had a new idea: In this extended group, the old gravitational equations are no longer covariant […].
We will come back to his final decision in Section 9. Weyl, and the astronomers G. UFT was left aside [ ]. I shall not succumb to that temptation. For 25 years Einstein had devoted his main scientific work to the problem of the structure of matter and radiation. He tried to gain an insight:. In this he is following the heroic method that proved so successful […] in the theory of relativity. Unfortunately there are many possible approaches, and since each requires a year or more of intensive computation, progress has been heartbreakingly slow.
Straus wrote an article about UFT: The others, big names and lesser known contributors except for the mathematician J. Schouten, shunned this topic. He described his intentions in going beyond general relativity and essentially presented the content of his paper with E. The proof of their physical usefulness is a tremendously difficult task, inasmuch as mere approximations will not suffice. His first paper of opened with a discussion, mostly from the point of view of mathematics, concerning the possibilities for the construction of UFT with a non-symmetric fundamental tensor.
As the fundamental tensor is no longer considered symmetrical, the symmetry of the connection as in Riemannian geometry must also be weakened. By help of the conjugate quantities of Section 2. In fact, for the conjugate: Now, he wanted to test the field equations by help of some sort of Bianchi-identity such as cf. After a lengthy calculation, he arrived at: The anti-Hermitian part of is given by. Equation then can be written as. Therefore, Einstein demanded that the contribution of to the Eq. We will meet again in Section A split of into symmetric and skew symmetric parts inside the bracket would give the equation:.
At best, as a sufficient condition, the vanishing of the symmetric and skew-symmetric parts separately could take place.
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Besides the additional equation would hold:. As we will see, by a further choice cf. Einstein first reformulated into:. Thus, in [ ], with a new approach via identities for the curvature tensor and additional assumptions, Einstein had reached the weak field equations of his previous paper [ ]. No physical interpretation of the mathematical objects appearing was given by him. The book was announced with fanfare in the Scientific American [ ]:.
Solovine, on 25 January , was:. A few weeks ago, it has caused a loud rustling noise in the newspaper sheets although nobody except the translator had really seen the thing. In the book, the translator is identified to have been Sonja Bargmann, the wife of Valentine, who also had translated other essays by Einstein. In Appendix II, with the assumption that. Einstein arrived at the field equations 30 , , and In place of the equivalent equations , or with can be used.
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Moreover, if in addition is taken into account, then also. This is due to a relation to be met again below [ cf. He then set up such a variational principle The following identity holds: In all three systems, equations , , are to be used for expressing the components of the asymmetric connection by the components of the asymmetric metric. The metric then is determined by the remaining equations for the Ricci tensor. The only remarks concerning a relationship between mathematical objects and physical observables made by Einstein at the very end of Appendix II are:.
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There are at present few physicists who share this view. McCrea reflected his own modesty. Although more cautious, he was very clear: After all, a human being is only a poor wretch! Pauli commented on this 3rd edition. Starting from the Cauchy problem, i. This discussion calls back into memory the intensive correspondence Einstein had carried on between and with the French mathematician E. Cartan on an equivalent problem within the theory of teleparallelism, cf.
At the time, he had asked whether his partial differential equations PDEs had a large enough set of solutions. It had taken Cartan a considerable effort of convincing Einstein of the meaningfulness of his calculations also for physics [ ], pp. Already in the 3rd Princeton edition of The Meaning of Relativity , in Appendix II [ ], Einstein tried to get to a conclusion concerning the compatibility of his equations by counting the independent degrees of freedom but made a mistake.
As mentioned above, W. Pauli had noticed this and combined it with another statement of his rejection of the theory. Compatibility was shown later by A. Lichnerowicz [ ] cf. In the meantime, Mme. Choquet-Bruhat, during her stay at Princeton in and , had discussed the Cauch-problem with Einstein such that he might have received a new impulse from her.
Let us postpone the details and just list his results. Similarly, he found for the Maxwell vacuum equations,. Einstein noted that by introducing the vector potential A i , and taking into account the Lorentz gauge, i. He ascribed the increase in the number of freely selectable coefficients loss of strength to the gauge freedom for the vector potential. For the Einstein vacuum equations, he obtained. He then gave a name to the number of free coefficients calculated before: Such a formula was derived for involutive, quasi-linear systems of PDEs by a group of relativists around F.
Hehl at the University of Cologne at the end of the s [ ], Eqs. Following an invitation by the editors of Scientific American to report on his recent research, Einstein made it clear that he would not give. That should be done only with theories which have been adequately confirmed by experiment.
Two questions were very important, though not yet fully answered: However, it was not a compatible system as were the other two. But this does not prove that it corresponds to nature. Experience alone can decide the truth. Yet we have achieved something if we have succeeded in formulating a meaningful and precise question. Painstaking efforts and probably new mathematics would be required before the theory could be confronted with experiment.
There were not only skeptics but people like Dr. He pointed out that the theory permits a class of similarity solutions, i. Nevertheless, Einstein remained optimistic; in the same letter to Max Born, in which he had admitted his shortcoming vis-a-vis the complexity of his theory, he wrote:. Therefore, the game is over, and the geometric model of the macrocosm has been constructed. However, at the end of his article, Finzi pointed out that it might be difficult to experimentally verify the theory, and thought it necessary to warn that even if such an empirical base had been established, this theory would have to be abandoned after new effects not covered by it were observed.
Einstein still grappled with the problem of how to set up a convincing system of field equations. Without specification of the Lagrangian , from the field equations follow — without use of the multiplier-term — to be:. A paragraph then was devoted to the choice of the proper Lagrangian. Einstein started from and removed a divergence term in.
The road to the weak field equations — followed here still did not satisfy Einstein, because in it the skew-symmetric parts of both the metric and the connection could also be taken to be purely imaginary. The prize payed is the exclusion of a physical interpretation of the torsion tensor. A new principle applying to physical theories in general is put forward: The relation of geometrical objects to physical observables remained unchanged when compared to the 3rd edition [ ]. This was due not only to the arbitrariness in picking a particular field equation from the many possibilities, but also to the failure of the theory to include a description of concepts forming an alternative to quantum theory.
At the very end of his Meaning of Relativity he explained himself in this way:. Nevertheless, I consider it unjustified to assert, a priori, that such a theory is unable to cope with the atomic character of energy. An indirect answer to this opinion was given by F. Dirac, do not yet have any workable alternative to put in its place.
Such kind of sober judgment did not bother The New York Times which carried an almost predictable headline: Privately, in a letter to M. Solovine of 28 May , Einstein seemed less assured. Because I do not know, whether there is physical truth in it. From the viewpoint of a deductive theory, it may be perfect economy of independent concepts and hypotheses. In the Festschrift for Louis de Broglie on the occasion of his 60th birthday George, Einstein again summarized his approach to UFT, now in an article with his assistant Bruria Kaufman [ ].
In a separate note, as kind of a preface he presented his views on quantum theory, i. A list of objections to the majority interpretation of quantum theory was given. At the end of the note, a link to UFT was provided:. A reasonable general relativistic field theory could perhaps provide the key to a more perfect quantum theory. This is a modest hope, but in no way a creed. As in the 4th edition of his book [ ], the geometrical basics were laid out, and one more among the many derivations of the weak field equations of UFT given before was presented.
In order to make the variational principle invariant, due to. The further derivation of the field equations led to the known form of the weak field equations:. In addition, a detailed argument was advanced for ruling out the strong field equations. It rests partially on their failure to guarantee the possibility to superpose weak fields. He then discussed the occurring geometric objects as representations of this larger group and concluded: The next paper with Bruria Kaufman may be described as applied mathematics [ ].
Einstein returned to the problem, already attacked in the paper with E. Straus, of solving 30 for the connection in terms of the metric and its derivatives. The authors first addressed the question: The situation was complicated by the existence of the algebraic invariants of the non-symmetric g ik as well as by the difficulty to solve for the connection as a functional of the metric tensor. Bose [ 52 ]. Tonnelat published earlier and presenting a solution were not referred to at all [ , , , ]. In the sequel, g ik is given a Lorentz signature. It is shown in the paper that among the three algebraic invariants to be built from only two are independent:.
Einstein agreed to let his 74th birthday be celebrated with a fund-raising event for the establishing of the Albert Einstein College of Medicine of Yeshiva University, New York. Roughly two weeks later, according to A. It announced the appearance of the 4th Princeton edition of The Meaning of Relativity with its Appendix II, and reported Einstein as having stated that the previous version of of the theory had still contained one important difficulty.
In a letter to Carl Seelig of 14 September , Einstein tried to explain the differences between the 3rd and the 4th edition of The Meaning of Relativity:. This development is closed now in the sense that the form of the field laws is completely fixed. Yet, the question about its physical foundation still is completely unsettled. This follows from the fact that comparison with experience is bound to the discovery of exact solutions of the field equations which seems impossible at the time being. This result was confirmed by Callaway in Callaway identified the skew part of the fundamental tensor with the electromagnetic field and applied a quasistatic approximation built after the methods of Einstein and Infeld for deriving equations of motion for point singularities.
In fact, as Bonnor then showed in the lowest approximation linear in the gravitational, quadratic in the electromagnetic field , the static spherically symmetric solution contains only two arbitrary constants e, m besides p 2 which can be identified with elementary charge and mass; they are separately selectable [ 33 ]. However, in place of the charge appearing in the solutions of the Einstein-Maxwell theory, now for e 2 the expression e 2 p 2 , and for occurred in the same solution.
The definition of mass seemed to be open, now. For vanishing electromagnetic field, the solution reduced to the solution for the gravitational field of general relativity. In a discussion concerning the relation of matter and geometry, viz. Callaway tried to mediate between the point of view of A. Einstein with his unified field theory already incorporating matter, geometrically, and the standpoint of J. Wheeler who hoped for additional relations between matter and space-time fixing the matter tensor as in the case of the Einstein-Maxwell theory [ 70 ], p.
The last paper with B. In it, the authors followed yet two other methods for deriving the field equations of UFT. The following relation resulted:. Although the wanted transposition-invariant field equations did come out, the authors were unhappy about the trick introduced. The question arises naturally whether we cannot find a form of the variational function which will itself be transposition-invariant, […].
From we see that does not transform like a connection. As a function of the Ricci tensor:. A short calculation shows that is invariant under As a Lagrangian, now was taken. Variation with respect to the variables. Although the authors do not say it, Eqs.
Modulo the field equations,. Kaufman — in simplifying the derivations as well as the form of the field equations. From a letter to his friend Solovine in Paris of 27 February , we note that Einstein was glad: It amounted to assign to the vanishing magnetic current density and to the electric current density. We hold to the principle that the stronger system has to be preferred to any weaker system, as long as there are no special reasons to the contrary.
In her answer of 28 February , B. Hence g ik can just as well be this multiplier. The field equations we would get from this Lagrangian would be identical with the equations in our paper, except that they would be expressed in terms of g ik. In order to do that one would have to survey all possible additional tensors which could be used in the Lagrangian. Until , more than a dozen people had joined the research on UFT and had published papers. Nevertheless, apart from a mentioning of H. Kaufman was well aware of this and would try to mend this lacuna in the same year, after Einstein had passed away.
Einstein [ ], B. After she presented essential parts of the joint paper with Einstein, Kaufman discussed its physical interpretation and some of the consequences of the theory. With , i. Both in terms of the number of papers published until the beginning of , and of researchers in UFT worldwide, she did poorly. Tonnelat as well as one or two of their collaborators, and some work done in Canada and India. The many publications coming from Italian groups were neglected by her as well as contributions from Japan, the United States and elsewhere which she could have cited. Unfortunately, his proof, within general relativity, that static, regular solutions behaving asymptotically like a Schwarzschild point particle with positive mass are locally Euclidean, could not be carried over to UFT.
This was due to the complications caused by the field equations , While could be solved, in principle, for the , as functions of , its subsequent substitution into led to equations too complicated to be solved — except in very special cases. The search for solutions of the weak field equations had begun already with exact spherically symmetric, static solution derived by a number of authors cf. In a final section, B. His field equations replacing , were cf. At first, in the affine theory, g ik is defined by the l.
This is a serious desideratum. On the other hand we ought not to be disheartened by proofs, offered recently by L. Ikeda and others, to the effect, that this theory cannot possibly account for the known facts about electrodynamic interaction. Some of these attempts are ingenious, but none of them is really conclusive. In the strong field equations, led to the vanishing of the electric current density. In 2nd order, the charge-current tensor was defined by , and the wave equation then.
Since , Cornelius Lanczos had come to Dublin, first as a visiting, then as a senior professor, and, ultimately, as director at the Dublin Institute for Advanced Studies. In any case, there is no further published research on UFT by him. Kaufman [ ] and solved for the connection. The solution to , is given as:. Mishra then linearized the metric and showed the result to be equivalent to the linearized Einstein-Straus equations cf. In a joint paper with M. This was due to some unknown terms in the 2nd and 3rd order of the approximation.
After a modification of the field equations according to the method of Bonnor [cf. A hope for overcoming the difficulty of relating mathematical objects from UFT to physical observables was put into the extraction of exact solutions. In simple cases, these might allow a physical interpretation by which the relevant physical quantities then could be singled out. One most simple case with high symmetry is the static spherically symmetric sss field. He built another expression:. If a ij is chosen as a metric, then the unique solution of general relativity in this sss case, e.
Wyman also questioned the boundary condition used at spacelike infinity: Apparently, this left no great impression; the search for sss solutions continued. In the case of sss fields, only one additional field equation, e. The solutions describe spread out charges while the masses are banned into singularities. Again, the exact solutions described spread out charges of both signs with an infinity of singular surfaces. They were unphysical because they contained no parameter for the mass of the sources [ 32 ].
In her book, M. Later in Italy, F. Vanstone mistakenly believed he had found time-dependent spherically symmetric solutions, but the time dependency can be easily removed by a coordinate transformation [ ]. Rao had calculated some, but not all components of the connection for the case of a time-dependent spherically symmetric field but had failed to find a time-dependent solution [ ]. Unfortunately, all this work did not bring further insight into the physical nature of the sss solutions.
He also proved the following theorem: Still worse, in a sobering contribution from the Canadian mathematician Max Wyman and his German colleague Hans Zassenhaus cast doubt on any hope for a better understanding of the physical contents of UFT by a study of exact solutions. Strong words, indeed, but not without reason:. The only hope to extract from this maze the proper mathematical expressions to use for physical quantities would thus have to be physical in nature.
This is due to the approximated Newtonian equations of motion following from the geodesic equation for h ij.
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Hence, for this solution, mass and charge are unrelated. This result casts into doubt much of the work on exact solutions independently of any specific assignment of mathematical objects to physical variables. Nevertheless, the work of assembling a treasure of exact solutions continued. Yet the bit we have been able to calculate has strengthened my trust in this theory.
How would he have dealt with the fact, unearthed in , that such non-singular solutions not always offered a convincing physical interpretation, or even were unphysical? Plane wave solutions of the weak and strong field equations of the form. Already up to here, diverse assignments of geometrical objects to physical quantities observables were encountered. We now assemble the most common selections. From the fact that the exact, statical, spherically symmetric solution of the weak field equations derived by A.
Perhaps the metric chosen to describe the gravitational potential ought not to be identified with h ik! Let the inverse of g ik be given by. From a study of the initial value problem, A. A related suggestion made by several doctoral students of M. Lenoir was to use a ij with and as metric [ , , ], [ ], p. The torsion tensor appeared in the definition of this metric.
The same ambiguity arose for the description of the electromagnetic field: In her discussion of two possibilities, St. The first choice was supported by Pham Tan Hoang [ , ]. Tonnelat, in her books, also discussed in detail how to relate observables as the gravitational and electromagnetic fields, the electric current density, or the energy-momentum tensor of matter to the geometric objects available in the theory [ ], Chapter VI; [ , ]; cf.
For the electromagnetic field tensor, four possibilities were claimed by her to be preferable: Tonnelat opted for m ik , and also for the electric current density vector. The field induction is defined via: On the other hand, Eq. We learned above in of Section 9. Ikeda used yet another definition of the electromagnetic field tensor. If electrical currents are to be included, the following choices for the current density were considered by Einstein, by Straus, cf. The second choice would either violate the weak field equations or forbid any non-zero current density.
These alternatives are bound to the choice for the induction. Late in his life, Einstein gave the interpretation of magnetic current density [ ]. Another object lending itself to identification with the electromagnetic field would be homoth-etic curvature encountered in Section 2. This choice has been made by Sciama, but with a complex curvature tensor [ ]. In this case The vector potential thus is identified with the torsion vector.
In a paper falling outside of the period of this review, H. Treder suggested to also geometrize spinorial degrees of freedom by including them in the asymmetric metric; it took the form [ , 67 ]:. It is obvious that the assumed mapping of geometrical objects to physical variables had to remain highly ambiguous because the only arguments available were the consistency of the interpretation within unified field theory and the limit to the previous theories Einstein-Maxwell theory, general relativity , thought to be necessarily encased in UFT.
As we have seen, the hope of an eventual help from exact solutions had to be abandoned. Again a precise attribution to geometrical objects could not be found. The symmetries Einstein had introduced, i. There are only a few papers with these symmetries as their topic. Kaufman to the theorems by Emmy Noether [ , ]. As a necessary condition for the field equations following from a variational principle to be J-invariant, she derived. According to the 2nd theorem by E.
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