Einstein

1915 · réf. 5863 — Berlin: Königlichen Akademie der Wissenschaften 1915. First edition. First editions very rare offprint of the first two of the papers published in November 1915 that document his final version of the general theory of relativity. "In the half century and more of Einstein's work in science one discovery stands above all as his greatest achievement. It is his general theory of relativity" Norton. . EINSTEIN'S COMPLETION OF THE GENERAL THEORY OF RELATIVITY. First editions very rare offprint of the first two of the papers published in November 1915 that document Einstein's final version of the general theory of relativity. "In the half century and more of Einstein's work in science one discovery stands above all as his greatest achievement. It is his general theory of relativity" Norton. "There was difficulty reconciling the Newtonian theory of gravitation with its instantaneous propagation of forces with the requirements of special relativity; and Einstein working on this difficulty was led to a generalization of relativity - which was probably the greatest scientific discovery that was ever made" Dirac quoted in Chandrasekhar p. 3. Einstein's special theory of relativity 1905 showed that the laws of physics are the same in all inertial i.e. non-accelerating frames of reference. It was then natural to ask whether it was possible to extend this principle of relativity to the more general case of frames of reference in arbitrary states of motion. This problem became linked to a theory of gravitation with Einstein's 'equivalence principle' of 1907 according to which the effects of gravity are locally equivalent to those of accelerated motion. Einstein's first steps towards a geometrical theory of gravitation were taken in August 1912 when his friend Marcel Grossmann provided the necessary mathematical tools. "Some time between August 10 and August 16 it became clear to Einstein that Riemannian geometry is the correct mathematical tool for what we now call general relativity theory. The impact of this abrupt realization was to change his outlook on physics and physical theory for the rest of his life" Pais p. 210. The resulting 'Entwurf' theory 1913 had much in common with the final theory of 1915 but based on a fallacious argument Einstein abandoned the requirement that the theory should be 'generally-covariant' i.e. that arbitrary frames of reference should be allowed. "After three years of fruitless peregrinations the revelation came to Einstein that he had been constantly on the wrong track although in 1913 he had been so near to the right solution" Lanczos p. 211. On November 4 1915 he presented to a plenary session of the Prussian Academy a new version of general relativity 'Zur allgemeinen Relativitätstheorie' "based on the postulate of covariance with respect to transformations with determinant 1" and stated that he had "completely lost confidence" in the 'Entwurf' equations. On November 18 he published his calculation of the precession of the perihelion of Mercury based on the new theory: its agreement with observation confirmed that the theory was correct the Entwurf theory predicted half the observed value of the precession. "In June 1905 while still a patent examiner in Bern Einstein submitted his famous work on the electrodynamics of moving bodies to the Annalen der Physik. This work contained his special theory of relativity in which he asserted the equivalence of all inertial frames of reference as a fundamental postulate of physics. The question which then naturally arose was whether it was possible to extend this principle of relativity to the more general case of frames of reference in arbitrary states of motion. But he could find no workable basis for such an extension until he tried to incorporate gravitation into his new special theory of relativity for a review article in 1907 'Uber das Relativitätsprinzip und die ausdemselben gezogenen Folgerungen' Jahrbuch der Radioaktivitat und Elektronik 4 1907 411-62. The difficulties of this task led him to a new principle later to be called the 'principle of equivalence.' "On the basis of the fact that all bodies fall alike in a gravitational field Einstein postulated the complete physical equivalence of a homogeneous gravitational field and a uniform acceleration of the frame of reference. This extended the principle of relativity to the case of uniform acceleration. It also foreshadowed the problem whose complete solution would lead him to his general theory of relativity: the construction of a relativistically acceptable theory of gravitation based on the principle of equivalence" Norton p. 258. One application of the equivalence principle proved crucial to the subsequent development of his ideas on general relativity. Einstein considered an observer standing on a rotating disc - a non-inertial accelerating reference frame. According to special relativity measuring rods aligned with the circumference of the disc will contract due to their motion whereas measuring rods positioned along the radius of the disc will not. Hence the ratio of the circumference of the disc to its diameter will be less than π. "The spatial geometry for the rotating observer is therefore non-euclidean. Invoking the equivalence principle Einstein concluded that this will be true for an observer in a gravitational field as well. This then is what first suggested to Einstein that gravity should be represented by curved space-time. "To describe curved space-time Einstein turned to Gauss's theory of curved surfaces a subject he vaguely remembered from his student days at the ETH in Zürich. He had learned it from the notes of his classmate Marcel Grossmann. Upon his return to his alma mater as a full professor of physics in 1912 Einstein learned from Grossmann now a colleague in the mathematics department of the ETH about the extension of Gauss's theory to spaces of higher dimension by Riemann and others. Riemann's theory provided Einstein with the mathematical object with which he could unify the effects of gravity and acceleration: the metric field" Janssen p. 65. The first product of this collaboration was the Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation published before the end of June 1913 which contained many of the essential features of the final general theory of relativity; most importantly it introduced the 'metric' of space-time. In Minkowski's formulation of special relativity 1908 the most important quantity is the 'world function' of two events which determines the metric and causal structure of space-time. If these events have coordinates x y z t and x' y' z' t' in some inertial reference frame the world function is: c2t' - t2 - x' - x2 - y' - y2 - z' - z2 where c is the speed of light. Its crucial property is that it depends only on the two events and not on the choice of inertial reference frame - in other words it is unchanged 'invariant' when x y z t and x' y' z' t' are both subjected to any Lorentz transformation. Einstein and Grossmann began with the world function in differential form: ds2 = c2dt2 - dx2 - dy2 - dz2 If we now subject x y z t to an arbitrary coordinate transformation not necessarily a Lorentz transformation this takes the general form ds2 = g11dx12 g12dx1dx2 . ; the collection of quantities gμν which in general depend on the coordinates x1 x2 x3 x4 is called the metric. Based on analogy with Newton's theory Einstein expected that the gravitational equations should be of the form Gμν = Tμν where Gμν is a purely geometric quantity constructed solely from the metric gμν and its derivatives up to the second order and the 'stress-energy tensor' Tμν contains the information about the matter that is producing the gravitational field including energy density momentum fluxes and stresses. The question was: what exactly should Gμνbe Einstein and Grossmann found that generally covariant equations did not seem to be compatible with energy-momentum conservation or reduce to the equations of Newtonian gravitational theory for weak static fields both essential requirements of the correct theory. Einstein therefore decided to settle in the 'Entwurf' for equations with very limited covariance - instead of arbitrary changes in coordinates only linear ones were allowed. The restricted covariance of the 'Entwurf' field equations continued to bother him until in late August 1913 he convinced himself that such restrictions are unavoidable by means of the infamous "hole argument" first published as an addendum to the reprint of the 'Entwurf' article in Zeitschrift für Physik in January 1914. This ingenious argument showed correctly that if the gravitational equations were generally covariant the metric gμν would not be uniquely determined by the matter distribution i.e. by Tμν. He concluded incorrectly that this implied that general covariance must be ruled out the hole argument does not work if only linear coordinate transformations are allowed. The appropriate analogy is with electromagnetism: the metric is analogous to the scalar and vector potentials of electromagnetism and it was well known certainly to Einstein that these potentials are not uniquely determined by the charges and currents producing the electromagnetic field. That the 'Entwurf' theory was incorrect was made clear by Einstein's attempt in collaboration with Michele Besso another former classmate to explain the motion of the perihelion of Mercury. In 1859 Urbain Jean Joseph Le Verrier had observed the 'precession' of Mercury's orbit: this orbit is an ellipse but the ellipse is not fixed in space but slowly rotates. From early on in his search for a new relativistic theory of gravitation Einstein had been interested in the problem of Mercury's perihelion. In a letter to his friend Conrad Habicht in 1907 Einstein had already expressed his hope that such a theory would explain the anomalous advance of Mercury's perihelion. Besso visited Einstein in Zürich in June 1913 and the two men calculated the precession expected on the basis of the 'Entwurf' theory. Disappointingly it was only about half the observed anomaly. Einstein left Zürich in March 1914 to take up a professorship in Berlin which was to be his home until December 1932. He made no further progress on the gravitational equations until the summer of 1915 although a detailed exposition of the 'Entwurf' theory was published in October 1914 in which Einstein maintained the need for restricted covariance and even claimed that this determined the gravitational Lagrangian uniquely. "Einstein still believed in the 'old' theory as late as July 1915 between July and October he found objections to that theory and his final version was conceived and worked out between late October and November 25 . What made Einstein change his mind between July and October Letters to Sommerfeld and Lorentz show that he had found at least three objections against the old theory: 1 its restricted covariance did not include uniform rotations 2 the precession of the perihelion of Mercury came out too small by a factor of about 2 and 3 his proof of October 1914 of the uniqueness of the gravitational Lagrangian was incorrect. Einstein got rid of all these shortcomings in a series of four brief articles offered here . "On November 4 Einstein presented to the plenary session of the Prussian Academy a new version of general relativity 'based on the postulate of covariance with respect to transformations with determinant 1'. He began this paper by stating that he had 'completely lost confidence' in the equations proposed in October 1914. At that time he had given a proof of the uniqueness of the gravitational Lagrangian. He had realized meanwhile that this proof 'rested on misconception' and so he continued 'I was led back to a more general covariance of the field equations a requirement which I had abandoned only with a heavy heart in the course of my collaboration with my friend Grossmann three years earlier' . "Einstein and Grossmann had concluded that the gravitational equations could be invariant under linear transformations only and Einstein's justification for this restriction was based on the belief that the gravitational equations ought to determine the gμν uniquely a point he continued to stress in October 1914. In his new paper he finally liberated himself from this three-year-old prejudice. That is the main advance on November 4. His answers were still not entirely right. There was still one flaw a much smaller one which he eliminated three weeks later. But the road lay open. He was lyrical. 'No one who has really grasped it can escape the magic of this new theory.' "The remaining flaw was of course Einstein's unnecessary restriction to unimodular transformations. The reasons which led him to introduce this constraint were not deep I believe. He simply noted that this restricted class of transformations permits simplifications of the tensor calculus . The new equations are a vast improvement over the Einstein-Grossmann equations and cure one of the ailments he had diagnosed only recently: unimodular transformations do include rotations with arbitrarily varying angular velocities. In addition he proved that the new equations can be derived from a variational principle and that the conservation laws are satisfied" Pais pp. 250-252. On November 11 he submitted a 'Nachtrag' to his paper of a week earlier. "Einstein proposes a scheme that is even tighter than the one of a week earlier. Not only shall the theory be invariant with respect to unimodular transformations . but more strongly it shall satisfy the condition that the determinant of the matrix gμν is equal to minus one . During the next two weeks Einstein believed that this new condition had brought him closer to general covariance . One week later he remarked that 'no objections of principle' can be raised against it" ibid. pp. 252-253. Norton p. 309 points out that Einstein had in fact made a significant advance in this paper: namely he had finally found generally covariant field equations that reduced to the Newtonian equations in the weak field limit" ibid. p. 253. On November 18 still retaining the restrictions of his paper of a week earlier Einstein presented in 'Erklarung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie'"two of his greatest discoveries. Each of these changed his life. The first result was that his theory explains . quantitatively . the secular rotation of the orbit of Mercury discovered by Le Verrier . without the need of any special hypothesis. This discovery was I believe by far the strongest emotional experience in Einstein's scientific life perhaps in all his life. Nature had spoken to him. He had to be right. 'For a few days I was beside myself with joyous excitement'. Later he told Fokker that his discovery had given him palpitations of the heart. What he told de Haas is even more profoundly significant: when he saw that his calculations agreed with the unexplained astronomical observations he had the feeling that something actually snapped in him . "Einstein's discovery resolved a difficulty that was known for more than sixty years. Urbain Jean Joseph Le Verrier had been the first to find evidence for an anomaly in the orbit of Mercury and also the first to attempt to explain this effect . In 1859 he found that the perihelion of Mercury advances by thirty-eight seconds per century due to 'some as yet unknown action on which no light has been thrown . a grave difficulty worthy of attention by astronomers'" ibid. pp. 253-254. A more accurate measurement of 43 seconds was made by Simon Newcomb in 1882 and this was precisely the value predicted by the new theory. The prediction of the bending of light in a gravitational field was treated only briefly in 'Erklarung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie' probably because no accurate measurement of it had been made so this prediction could not be confirmed at the time. Einstein had realised in 1907 based on the equivalence principle that some bending of light should occur but he believed that the effect was too small to be observed. In 1911 he realized that the effect could be detected for starlight grazing the sun during a total eclipse and found that the amount of bending in that case is 0''.87 - this value could in fact have been computed by Newton from his law of gravitation and his corpuscular theory of light. In 3 Einstein discovered that general relativity implies a bending of light by the sun equal to 1".74 twice the Newtonian value. This factor of 2 set the stage for a confrontation between Newton and Einstein. "It was not until May 1919 that two British expeditions obtained the first useful photographs and not until November 1919 that their results were formally announced . In March 1917 the Astronomer Royal Sir Frank Watson Dyson drew attention to the excellence of the star configuration on May 29 1919 an eclipse date for measuring the alleged deflection . Two expeditions were mounted one to Sobral in Brazil led by Andrew Crommelin from the Greenwich Observatory and one to Principe Island off the coast of Spanish Guinea led by Eddington. Before departing Eddington wrote 'The present eclipse expeditions may for the first time demonstrate the weight of light i.e. the Newton value; or they may confirm Einstein's weird theory of non-Euclidean space; or they may lead to a result of yet more far-reaching consequences - no deflection' . The expeditions returned. Data analysis began. According to a preliminary report by Eddington to the meeting of the British Association held in Bournemouth on September 9-13 the bending of light lay between 0''.87 and double that value. Word reached Lorentz. Lorentz cabled Einstein . Then came November 6 1919 the day on which Einstein was canonized" Pais 304-305. At a joint meeting of the Royal Society and the Royal Astronomical Society on that date Dyson concluded his remarks with the statement "'After a careful study of the plates I am prepared to say that they confirm Einstein's prediction. A very definite result has been obtained that light is deflected in accordance with Einstein's law of gravitation'" ibid. p. 305. Three remarks may be made on the speed with which after eight years of struggle Einstein completed these final papers on his theory. The first is that Einstein had come very close to the correct gravitational equations in the second half of 1912 - they are recorded in his 'Zurich notebook' - but he discarded them because of his arguments against general covariance as we have seen. Once he no longer believed in these arguments he could return to the work carried out in the Zurich notebook and complete it. The second is that the detailed calculations in 3 relating to Mercury's perihelion were in fact very similar to those he had carried out with Besso in 1913 and so required relatively little extra effort. The final point is that Einstein was in competition with the great Göttingen mathematician David Hilbert. Weil 75 76 77; Chandrasekhra 'The general theory of relativity: Why "It is probably the most beautiful of all existing theories" Journal of Astrophysics 5 1984 pp. 3-11; Eisenstaedt The Curious History of Relativity 2006; Janssen 'Of pots and holes: Einstein's bumpy road to general relativity' Annalen der Physik 14 Supplement 2005 pp. 58-85; Lanczos Einstein Decade: 1905-1915 1974; Norton 'How Einstein found his field equations: 1912-1915' Historical Studies in the Physical Sciences 14 1984 pp. 253-316; Pais Subtle is the Lord 1982. Large 8vo 253 x 180 mm pp. 778-786 & 799-801. Original printed wrappers. Königlichen Akademie der Wissenschaften unknown