Aber leichter ist ahnen als finden.
On the History of Unified Field Theories
Einstein's next letter of 17 January is skeptical: Various reasons instilled in me strong reservations: Satzes erscheinen die elektrody. Einstein's answer to Hilbert on 15 November shows that he had also been busy along such lines: 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: The meeting was amicable. Although Finsler himself did not apply his geometry to physics it soon became used in attempts at the unification of gravitation and electromagnetism [ ].
Bach wrote also about Weyl's theory Bach [ 2 ]. The idea that they keep together the dispersing electrical charges lies close at hand. Der Gedanke liegt nahe, dass diese es sind, die die auseinanderstrebenden elektrischen Ladungen zusammenhalten. However, Pauli's remark after Weyl's lecture in Bad Nauheim 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. Therefore, I'd like to ask Mr.
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.
Einstein's answer is tentative and evasive: We just don't know yet 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. 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. 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 [ Only success can decide whether such a procedure finds its justification [ Denn man kann versuchen, denjenigen Feldbegriffen [ 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.
Stachel [ ]. Next came Kaluza's five-dimensional unification of gravitation and electromagnetism, and Eddington's affine geometry. 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.
If, as Weyl does, the connection is assumed to be symmetric i. Let us now look at what happens to parallel transport of a length, e. The same holds for the angle between two tangent vectors in a point cf. Perhaps, having in mind Mie's ideas of an electromagnetic world view and Hilbert's approach to unification, in the first edition of his book, Weyl remained reserved: Yet, also today, the circumstances are such that our trees do not grow into the sky. He went through the entire mathematics of Weyl's theory, curvature tensor, quadratic Lagrangian field equations and all; he even discussed exact solutions.
Studied mathematics at the University of Vienna where he obtained his doctoral degree in Became a professional officer during the First World War. He obtained professorships in Graz and Vienna and, in , at the University of Amsterdam. He specialised in the theory of invariants cf. Einstein, being the first expert who could keep an eye on Weyl's theory, immediately objected, as we infer from his correspondence with Weyl. Weyl had arranged that the page proofs be sent to Einstein. There is a fully determined action principle, which, in the case of vanishing electricity, leads to your gravitational equations while, without gravity, it coincides with Maxwell's equations in first order.
In the most general case, the equations will be of 4th order, though. Im allgemeinsten Fall werden die Gleichungen allerdings 4. 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.
Es ist ein Genie-Streich ersten Ranges. Allerdings war ich nicht imstande, meinen Massstab-Einwand zu erledigen. There, Einstein argued that if light rays would be the only available means for the determination of metrical relations near a point, then Weyl's gauge would make sense. 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. 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: Presumably, such a theory would have to include microphysics. Yet, if the manner in which nature really behaves would be otherwise, then spectral lines and well-defined chemical elements would not exist.
Wenn das mit einer Uhr bzw. Lorentz; in a paper on the measurement of lengths and time intervals in general relativity and its generalisations, he contradicted Weyl's statement that the world-lines of light-signals would suffice to determine the gravitational potentials [ ]. However, Weyl still believed in the physical value of his theory.
Einstein and Weyl 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. Even before Weyl's note was published by the Berlin Academy on 6 June , many exchanges had taken place between him and Einstein. On a postcard to Weyl on 8 April , Einstein reaffirmed his admiration for Weyl's theory, but remained firm in denying its applicability to nature.
Weyl had given an argument for dimension 4 of space-time that Einstein liked: Weyl did not give in: Und daran muss ich als Mathematiker durchaus festhalten: The first was that Weyl's theory preserves the similarity of geometric figures under parallel transport, and that this would not be the most general situation cf.
He repeated this argument in a letter to his friend Michele Besso from his vacations at the Baltic Sea on 20 August , in which he summed up his position with regard to Weyl's theory: 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.
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Then, Weyl's fundamental hypothesis is incorrect on the molecular level, anyway. 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 [ Writing from his vacations on 18 September , Weyl presented a new argument in order to circumvent Einstein's objections. A very small change of the measuring path would strongly influence the integral of the square root of this quantity.
Schouten, Pauli, Eddington, and others Sommerfeld seems to have been convinced by Weyl's theory, as his letter to Weyl on 3 June shows: 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. So wie Mie seiner konsequenten Elektrodynamik eine Gravitation angeklebt hatte, die nicht organisch mit jener zusammenhing, ebenso hat Einstein seiner konsequenten Gravitation eine Elektrodynamik d. Sie stellen eine wirkliche Einheit her. As compared to his criticism with respect to Eddington's and Einstein's later unified field theories, he is speaking softly, here. 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.
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. 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. For Eddington, Weyl's theory of gauge-transformation was a hybrid: 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. Further research Pauli, still a student, and with his article for the Encyclopedia in front of him, pragmatically looked into the gravitational effects in the planetary system, which, as a consequence of Einstein's field equations, had helped Einstein to his fame.
He showed that Weyl's theory had, for the static case, as a possible solution a constant Ricci scalar; thus it also admitted the Schwarzschild solution and could reproduce all desired effects [ , ]. Weyl himself continued to develop the dynamics of his theory. Due to his constraint, Weyl had navigated around another problem, i. Now he had arrived at second order field equations. Eddington criticised Weyl's choice of a Lagrangian as speculative: Einstein's conclusion was that, by writing down a metric with gauge-weight 0, it was possible to form a theory depending only on the quotient of the metrical components.
Eisenhart wished to partially reinterpret Weyl's theory: He demanded that the connection remain metric-compatible from which, trivially, Weyl's gauge-vector must vanish. Dienes applied the same argument to Eddington's generalisation of Weyl's theory [ 49 ]. Other mathematicians took Weyl's theory at its face value and drew consequences; thus M.
More important, however, for later work was the gauge invariant tensor calculus by a fellow of St. John's College in Cambridge, M. Newman [ ]. In this calculus, tensor equations preserve their form both under a change of coordinates and a change of gauge. The more important development, however, was the extension to non-Abelian gauge-groups and the combination with Kaluza's idea.
We shall discuss these topics in Part II of this article. Born in Ybbs, Austria. Studied mathematics at the University of Vienna.
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From a full professor at the University of Vienna, but accepted professorship at University of Innsbruck, returning to Vienna only in Wrote an important paper on the general theta function and had an exceptional range in mathematics function theory, algebra, number theory, plane geometry, theory of invariants. Born in Tokaj, Hungary. Kaluza's idea of looking at four spatial and one time dimension originated in or before ; by then he had communicated it to Einstein: In a subsequent letter to Kaluza of 5 May Einstein still was impressed: 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. From the equations of motion, charge conservation also followed in Kaluza's linear approximation. For him, any theory claiming universal validity was endangered by quantum theory, anyway. From the cylinder condition, a grave objection toward Kaluza's approach results: Here, contact is made to the projective formulation of Kaluza's theory cf.
While towards the end of May Einstein had not yet fully supported the publication of Kaluza's manuscript, on 14 October he thought differently: I value your approach more than the one followed by H. If you wish, I will present your paper to the Academy after all. Wenn Sie wollen, lege ich Ihre Arbeit doch der Akademie vor [ This led to a joint publication which was submitted just one month after Einstein had finally presented a rewritten manuscript of Kaluza's to the Berlin Academy [ ].
The negative result of his own paper, i. In any case, apart from an encouraging letter to Kaluza in in which he called Kaluza's idea the only serious attempt at unified field theory besides the Weyl—Eddington approach, Einstein kept silent on the five-dimensional theory until Born near Brest, then in Russia.
First a Talmud student with a keen interest in mathematics. Worked with Einstein for at least a decade — as his calculational assistant. He held a university position in Minsk from on and later became a member of the Belorussian Academy of Sciences. From his youth he was inflicted with elephantiasis. His motivation went beyond the unification of gravitation and electromagnetism: Clearly, the non-Maxwellian binding forces which hold together an electron.
By this, Eddington claims to guarantee charge conservation: 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. This is due to the expression for the inverse of the metric, a function cubic in R k l.
Eddington's affine theory thus can also be seen as a bi-connection theory. Eddington's main goal in this paper was to include matter as an inherent geometrical structure: 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.
Lorentz did not like the large number of variables in Eddington's theory; there were 4 components of the electromagnetic potential, 10 components of the metric and 40 components of the connection: His notation representing the covariant derivative by a lower index is highly ambiguous, though, and will be avoided. 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.
By this, the metric was defined as the symmetric part of the Ricci tensor. Except for singular positions, the current density is practically vanishing. Also, the geometrical theory presented here is energetically closed, i. His final conclusion was: 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.
In the third paper as well, Einstein's desire to create a unified field theory satisfying all his criteria still was not fulfilled: His equations, again, did not give a singularity-free electron. Nobody can determine empirically an affine connection for vectors at neighbouring points if he has not obtained the line element before. Therefore, unlike you and Einstein, I deem the mathematician's discovery of the possibility to found a geometry on an affine connection without a metric as meaningless for physics, in the first place.
Niemand kann empirisch einen affinen Zusammenhang zwischen Vektoren in benachbarten Punkten feststellen, wenn er nicht vorher bereits das Linienelement ermittelt hat. He criticised a theory that keeps only the connection as a fundamental building block for its lack of a guarantee that it would also house the conformal structure light cone structure.
This is needed for special relativity to be incorporated in some sense, and thus must be an independent fundamental input [ ]. Likewise, Eddington himself did not appreciate much Einstein's followership. 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: From our knowledge about quantum states, in particular as it developed in the wake of Bohr's theory during the past decade, this characteristic feature of theory does not correspond to reality. The initial state of an electron moving around a hydrogen nucleus cannot be chosen freely; its choice must correspond to the quantum conditions. One of the crucial tests for an acceptable unified field theory for him now was: Besso from 5 January In such a way, the un-ambiguity of the initial conditions ought to be understood without leaving field theory.
Schouten is the leading figure in this approach [ ]. Here, only symmetric connections can appear. 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.
V 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 [ It arose because Schouten introduced different linear connections for tangent vectors and linear forms. For such an extension an invariant fixing of the connection is needed, because a physical phenomenon can correspond only to an invariant expression.
Thus, while Einstein and Weyl influenced Eddington, Schouten apparently did his research without knowing of Eddington's idea. 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 [ ]. A similar, but less detailed, classification of connections than Schouten's has also been given by Cartan.
Other mathematicians were also stimulated by Einstein's use of differential geometry in his general relativity and, particularly, by the idea of unified field theory. Perhaps he considered his papers on the geometry of unified field theory as a sin of his youth: In Poggendorff , among the 33 papers listed, all are from his later main interest. He modified Eddington's approach to the extent that he now took both a non-symmetric connection and a non-symmetric metric, i.
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: But this must be due to either a calculational error, or to a printer's typo because in the paper of J. 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. After having shown that his new theory contains the vacuum field equations of general relativity for vanishing electromagnetic field, Einstein then proved that, in a first-order approximation, Maxwell's field equations result cum grano salis: 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.
On the History of Unified Field Theories
Im Makroskopischen zweifle ich nicht an ihrer Richtigkeit. 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.
Hierin unterscheidet sich die Elektrodynamik von der Gravitation; deshalb erscheint mir auch das Bestreben, die Elektrodynamik mit dem Gravitationsgesetz zu einer Einheit zu verschmelzen, nicht mehr gerechtfertigt. 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.
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. New calculations seem to show that these equations yield the motion of the electrons. But it appears doubtful whether there is room in them for the quanta. Aus den neuen Rechnungen scheint sich zu ergeben, dass diese Gleichungen die Bewegung der Elektronen liefern. Aber es erscheint zweifelhaft, ob die Quanten darin Platz haben.
It does not allow for electrical masses free from singularities. Moreover, I cannot bring myself to gluing together two items as the l. Ferner kann ich mich nicht dazu entschliessen, zwei Sachen zusammenzuleimen wie die rechte und die linke Seite einer Gleichung , die logisch-mathematisch nichts miteinander zu schaffen haben.
To me, this seems to be a case of whiggish historical hindsight. According to Bargmann, Einstein's lasting result is that he pointed out the importance of the discrete symmetry operations [ 6 ]. Einstein's paper practically excludes that they form the antisymmetric part of an asymmetric metric tensor.
Thomas, after having given a review of Weyl's, Einstein's, and Schouten's approaches, said about his own work: 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. Thus, he is back at vector torsion treated before by Schouten [ ]. By introducing the tensor f i j k: As in Hattori's theory two connections are used, Infeld criticised that Hattori had not explained what his fundamental geometry should be: He then gave another example for a theory allowing the identification of the electromagnetic field tensor with the antisymmetric part of the Ricci tensor: At first, one does not see how a choice can be made among the various non-Riemannian geometries providing us with the gravitational and Maxwell's equations.
The proper world geometry which ought to lead to a unified theory of gravitation and electricity can only be found by an investigation of its physical content. The field equations Straneo wrote down, i. Straneo kept the energy-momentum tensor of matter as an extraneous object including the electromagnetic field as well as the electric current vector. Straneo wrote further papers on the subject [ , ]. 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. Born in Cracow, Poland. Studied at the University of Cracow and received his doctorate in Of Indian origin; born in Goa he moved to America in with his learned father and graduated from Harvard University in in mathematics, history and languages. Mathematician, historian, and Sanskrit scholar. In he became Einstein's assistant with the explicit understanding that he work with him on distant parallelism.
It seems that Mayer was appreciated much by Einstein and, despite being in his forties, did accept this role as a collaborator of Einstein. After coming to Princeton with Einstein in , he got a position at the Mathematical Institute of Princeton University and became an associate of the Institute for Advanced Study. Wrote a joint paper with T. Act I Einstein became interested in Kaluza's theory again due to O. Einstein wrote to his friend and colleague Paul Ehrenfest on 23 August 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 [ 79 , 80 ].
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. Der ganze Inhalt findet sich in der Arbeit von O. Thus, the three of them had no chance to find out that Kaluza had made a mistake: Klein's results [ ].
A main motivation for Klein was to relate the fifth dimension with quantum physics. It was Klein's papers and the magical lure of a link between classical field theory and quantum theory that raised interest in Kaluza's idea — seven years after Kaluza had sent his manuscript to Einstein. Klein acknowledged Mandel's contribution in his second paper received on 22 October , where he also gave further references on work done in the meantime, but remained silent about Einstein's papers [ ].
By this, the reduction of five-dimensional equations as e. Klein had only the lowest term in the series. The 5th dimension is assumed to be a circle, topologically, and thus gets a finite linear scale: Beyond incredibly complicated field equations nothing much had been gained [ ]. He also suggested that one should not accept the cylinder condition, a suggestion looked into by Darrieus who introduced an electrical 5-potential and 5-current, and deduced Maxwell's equations from the five-dimensional homogeneous wave equation and the five-dimensional equation of continuity [ 41 ].
He then discussed conformally invariant field equations, and tried to relate them to equations of wave mechanics [ ]. Klein's lure lasted for some years. Presently, the different contributions of Kaluza and O. An early criticism of this unhistorical attitude has been voiced in [ ]. Petersburg renamed later Petrograd and Leningrad. Studied at Petrograd University and spent his whole carrier at this University. Fundamental contributions to quantum theory Fock space, Hartree—Fock method ; also worked in and defended general relativity. From lecturer at the University of Leningrad, and from research work at the Physics Institute of this university.
Perhaps, he had since absorbed Mandel's ideas which included a projection formalism from the five-dimensional space to space-time [ , , , ]. Another motivation is also put forward: After a listing of all the shortcomings of Kaluza's theory, the new approach is introduced: Both covariant derivatives are abbreviated by the same symbol A ; k.
Einstein and Mayer made three basic assumptions: Two new quantities are introduced: Thus, by another formalism, Einstein and Mayer rederived what Klein had obtained in his first paper on Kaluza's theory [ ]. The authors' conclusion is: 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. Auf diese Weise gelang es, das Gravitations- und das elektromagnetische Feld als logische Einheit zu erfassen.
Hence, no physical progress is made, [if at all] at most only in the sense that one can see that Maxwell's equations are not just first approximations but appear on as good a rational foundation as the gravitational equations of empty space. Electrical and mass-density are non-existent; here, splendour ends; perhaps this already belongs to the quantum problem, which up to now is unattainable from the point of view of field [theory] in the same way as relativity is from the point of view of quantum mechanics. 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. Veblen had already worked on projective geometry and projective connections for a couple of years [ , , ]. However, according to Pauli, Veblen and Hoffmann had spoiled the advantage of projective theory: There is a one-to-one correspondence between the points of space-time and a certain congruence of curves in a five-dimensional space for which the fifth coordinate is the curves' parameter, while the coordinates of space-time are fixed.
The five-dimensional space is just a mathematical device to represent the events points of space-time by these curves. Geometrically, the theory of Veblen and Hoffmann is more transparent and also more general than Einstein and Mayer's: It can house the additional scalar field inherent in Kaluza's original approach. Thus, Veblen and Hoffmann also gained the Klein—Gordon equation in curved space, i. In his note, Hoffmann generalised the formalism such as to include Dirac's equations without gravitation , although some technical difficulties remained.
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. In a second note, Einstein and Mayer extended the 5-vector-formalism to include Maxwell's equations with a non-vanishing current density [ ]. Of the three basic assumptions of the previous paper, the second had to be given up.
In the last paragraph, the compatibility of the equations was proven, and at the end Cartan was acknowledged: 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. In a note added after proofreading, the authors showed that they had noted Schouten and Dantzig's papers [ , ].
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. This has made it necessary to introduce a new expression for the energy-momentum tensor and [ 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.
After a break during the First World War, he received his Ph. After teaching at the medical school, he became professor of geometry at the Univerity of Cagliari in From there he moved on to the same position at the University of Florence in Despite his premature death, Bortolotti published about a hundred papers, notably in differential geometry. Born in Richmond, England. Studied mathematics and theoretical physics at Oxford University and received his doctorate in Became an assistant at Princeton University and worked there with Einstein in — From professor at Queens College in New York.
His scientific interests were in relativity theory, tensor analysis, and quantum theory. 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. I even remember having tried, at Hadamard's place, to give you the most simple example of a Riemannian space with Fernparallelismus by taking a sphere and by treating as parallels two vectors forming the same angle with the meridians going through their two origins: Einstein had believed to have found the idea of distant parallelism by himself.
In this regard, Pais may be correct. Every researcher knows how an idea, heard or read someplace, can subconsciously work for years and then surface all of a sudden as his or her own new idea without the slightest remembrance as to where it came from. It seems that this happened also to Einstein. It is quite understandable that he did not remember what had happened six years earlier; perhaps, he had not even fully followed then what Cartan wanted to explain to him.
In any case, Einstein's motivation came from the wish to generalise Riemannian geometry such that the electromagnetic field could be geometrized: 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.
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. Also, he took Einstein's treatment of Fernparallelism as a special case of his more general considerations.
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. I also could show that the field equations, in first approximation, lead to equations that correspond to the Newton—Poisson theory of gravitation and to Maxwell's theory. Nevertheless, I still am far from being able to claim that the derived equations have a physical meaning. The reason is that I could not derive the equations of motion for the corpuscles.
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.
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.
This article then became reprinted in March by the British astronomy journal The Observatory [ 84 ]. 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. 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. Even the French-Belgian writer and poet Maurice Maeterlinck had heard of Einstein's latest achievement in the area of unified field theory. 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 Of course, in space-time with a Lorentz metric, the 4-bein-transformations do form the proper Lorentz group. Moreover, Y l m k: In principle, distant parallelism is a particular bi-connection theory. At the end of the note Einstein compared his new approach to Weyl's and Riemann's: Comparison at a distance of both lengths and directions.
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 [ 87 ]. He left aside the Hamiltonian principle and started from identities for the torsion tensor, following from the vanishing of the curvature tensor 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. His Lagrangian is a particular linear combination of the three possible scalar densities, as follows: Stodola, Einstein summed up what he had reached Einstein's next publication was the note preceding Cartan's paper in Mathematische Annalen [ 88 ]. He presented it as an introduction suited for anyone who knew general relativity.
Most importantly, he gave a new set of field equations not derived from a variational principle; they are 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. One of them he had obtained from Cartan: The changes in his approach Einstein continuously made, must have been hard on those who tried to follow him in their scientific work.
They were published in as the first article in the new journal of this institute [ 91 ]. On 23 pages he clearly and leisurely outlined his theory of distant parallelism and the progress he had made. As to references given, first Cartan's name is mentioned in the text: 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 [ 83 , 89 , 92 ]. The counting of independent variables, field equations, and identities is repeated from Einstein's paper in Mathematische Annalen [ 88 ]. Hence 7 identities should exist, four of which Einstein had found previously.
He now presented a derivation of the remaining three identities by a calculation of two pages' length. It is reproduced also in [ 89 ]. Interestingly, right after Einstein's article in the institute's journal, a paper of C. Proca, who had attended Einstein's lectures, gave an exposition of them in a journal of his native Romania.
He was quite enthusiastic about Einstein's new theory: Then Einstein presented the same field equations as in his paper in Annalen der Mathematik , which he demanded to be 1 covariant, 2 of second order, and 3 linear in the second derivatives of the field variable h i k. While these demands had been sufficient to uniquely lead to the gravitational field equations with cosmological constant of general relativity, in the teleparallelism theory a great deal of ambiguity remained.
Sixteen field equations were needed which, due to covariance, induced four identities. 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. Einstein's next note of one and a half pages contained a mathematical result within teleparallelism theory: From any tensor with an antisymmetric pair of indices a vector with vanishing divergence can be derived [ 92 ].
Jetzt nur noch schnell die Welt erklart, dann geh ich mit dem Hund. Wie kann das Universum unendlich sein und sich gleichzeitig ausdehnen? Fur Laien ist Physik voller Widerspruche. Ganz nebenbei erklart dieses Buch warum das so ist, und macht Physik verstandlich. Physik kann vieles, aber eine Antwort auf die Frage, was die Welt zusammenhalt, vermag sie nicht zu geben. Physik sucht nach Fakten, nicht nach Wahrheiten.
So beschloss ich eine alternative Theorie auf der Basis wissenschaftlicher Logik zu entwickeln. Nach mehrjahriger Arbeit war das Ergebnis eine uberraschend kurze und einfache Erklarung, die ich hiermit vor- und zur Diskussion stellen mochte. Read more Read less. Special offers and product promotions Rs cashback on Rs or more for purchases made through Amazon Assistant. Offer period 1st September to 30th September. Cashback will be credited as Amazon Pay balance within 15 days from purchase.
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