EINSTEIN, Albert.

Über das Relativitätprinzip und die aus demselben gezogenen Folgerungen [On the relativity principle and the conclusions drawn from it]. Offprint from: Jahrbuch der Radioaktivität und Elektronik, Band 4, Heft 16.

[Leipzig: S. Hirzel, 1907].

First edition, extremely rare author’s presentation offprint (with ‘Überreicht vom Verfasser’ (Presented by the Author) stamped on front wrapper), from the library of the great German physicist Arnold Sommerfeld, of this crucially important transitional paper, in which Einstein introduced the equivalence principle, that uniform acceleration and gravitation are equivalent in their physical effects, which launched him on his path to general relativity. “Einstein's efforts to incorporate gravitation into the theory of relativity led him in 1907 to formulate a new formal principle, later named the principle of equivalence. He stressed that, when gravitational effects are taken into account, it is impossible to maintain the privileged role that inertial frames of reference still have in the original relativity theory. He concluded that, if gravitation is to be included, it is necessary to extend the relativity principle. The search for a group of transformations, wider than the Lorentz group, under which the laws of physics remain invariant when gravitation is included, lasted from 1907 until the end of 1915, leading finally to what Einstein considered his greatest achievement, the general theory of relativity” (Collected Papers 2, p. xxix). “On p. 443 are probably the first explicit statements both of the equivalence of inertial and gravitational mass and of the equation for mass in terms of energy [E = mc²] now regarded as the theoretical basis for the release of atomic energy” (Weil). In 1905, “Einstein said that all energy of whatever sort has mass. It took even him two years more to come to the stupendous realization that the reverse must also hold: that all mass, of whatever sort, must have energy. ... With mass and energy thus wholly equivalent, Einstein was able in 1907, in a long and mainly expository paper published in the Jahrbuch der Radioactivität [the offered paper], to write his famous equation E = mc² ... In presenting his equation in 1907 Einstein spoke of it as the most important consequence of his theory of relativity” (Hoffmann, Albert Einstein, p. 81). “Of greatest importance is the last part of the paper which generalizes the principle of relativity from uniformly moving systems to uniformly ‘accelerated’ systems. ... He introduces the principle of equivalence which claims that the problem of a uniform and stationary gravitational field on the one hand, and the system moving with a constant acceleration without any gravitation on the other hand, are physically indistinguishable situations. This principle put him in a position to find out what effect gravitation has on an arbitrary physical phenomenon, because all he had to do was to observe that phenomenon from an accelerated reference system. He thus obtains the speeding up of clocks in a field of increased gravitational potential, which must lead to a universal red shift of the spectral lines coming from the Sun, and likewise to a bending of light rays near to the limb of the Sun. Furthermore, this hypothesis at once makes it clear why inertial mass and gravitational mass must be, under all circumstances, strictly proportional to one another. ... Hence the principle of the energy value of inertial mass must be extended to the gravitational mass” (Lanczos, The Einstein Decade, p. 153). Later Einstein wrote that when he was working on this paper, “There occurred to me the happiest thought of my life, in the following form. The gravitational field has only a relative existence in a way similar to the electric field generated by magnetoelectric induction. Because for an observer falling freely from the roof of a house there exists – at least in his immediate surroundings – no gravitational field” [Einstein’s emphasis] (Pais, Subtle is the Lord, p. 178). Although Einstein submitted the paper on 4 December 1907; it was published in the January 22 issue of the Jahrbuch. This is one of Einstein’s rarest major papers in offprint form. RBH lists three copies: Plotnick (Christie’s 2002); Einstein’s own collection of his offprints (Christie’s 2008); and Richard Green (Christie’s 2008). OCLC lists 6 copies worldwide (Morgan; Princeton; Stanford; Trinity College, Cambridge; Queen’s University, Kingston ON; Thomas Fisher). This copy was presented by Einstein to one of the leading physicists of the time, surely hoping to make himself known in the scientific world when he was still a technical expert in the Swiss Patent Office.

Provenance: Arnold Sommerfeld (1868-1951) (his signature and characteristic numbering in red pencil (‘11’) on front cover). “The son of a physician, Sommerfeld was educated at the University of Königsberg. After teaching briefly at the universities of Göttingen, Clausthal, and Aachen he was appointed professor of physics at the University of Münich in 1906. Sommerfeld should have retired in 1936 in favour of his pupil, Werner Heisenberg. Opposition from the Nazi party to Heisenberg’s appointment prolonged Sommerfeld’s tenure and it was not in fact until late 1939 that he finally retired, to be succeeded not by Heisenberg but by Wilhelm Müller, a Nazi aerodynamicist without a single publication in physics to his credit. Although Sommerfeld and Heisenberg were not Jewish, they were regarded by the Nazis as Jewish sympathizers. Sommerfeld, however, survived the war and returned to his Münich chair in 1945, continuing to work at physics until he died in a car accident in 1951” (Oxford Reference). “Arnold Sommerfeld was one of the most distinguished representatives of the transition period between classical and modern theoretical physics. The work of his youth was still firmly anchored in the conceptions of the nineteenth century; but when in the first decennium of the century the flood of new discoveries, experimental and theoretical, broke the dams of tradition, he became a leader of the new movement, and in combining the two ways of thinking he exerted a powerful influence on the younger generation. This combination of a classical mind, to whom clarity of conception and mathematical rigour are essential, with the adventurous spirit of a pioneer, are the roots of his scientific success, while his exceptional gift of communicating his ideas by spoken and written word made him a great teacher” (Max Born, p. 275).

“His first important paper on relativity theory after 1905 is the 1907 review. This article was written at the request of {Johannes] Stark, the editor of the Jahrbuch. On September 25, 1907, Einstein had accepted this invitation. On November 1, Einstein further wrote to Stark: ‘I am now ready with the first part of the work for your Jahrbuch. I am working zealously on the second [part] in my unfortunately scarce spare time.’ Since this second part contains the remarks on gravitation, it seems probable that Einstein’s ‘happiest thought’ came to him sometime in November 1907. We certainly know where he was when he had this idea. In his Kyoto lecture he told the story: ‘I was sitting in a chair in the patent office at Bern when all of a sudden a thought occurred to me. ‘If a person falls freely he will not feel his own weight!’ I was startled. This simple thought made a deep impression on me. It impelled me toward a theory of gravitation’ …

“Three main issues are raised in Section V of the Jahrbuch article.

The Equivalence Principle. ‘Is it conceivable that the principle of relativity also holds for systems which are accelerated relative to each other?’ That is Einstein’s starting question. Then he gives the standard argument. A reference frame Σ₁ is accelerated in the x direction with a constant acceleration γ. A second frame Σ₂ is at rest in a homogeneous gravitational field which imparts an acceleration –γ in the x direction to all objects. ‘In the present state of experience, we have no reason to assume that … Σ₁ and Σ₂ are distinct in any respect, and in what follows we shall therefore assume the complete physical equivalence of a gravitational field and the corresponding acceleration of the reference frame. This assumption extends the principle of relativity to the case of uniformly accelerated motion of the reference frame … he began by applying his new postulate to the Maxwell equations, always for uniform acceleration. He did not raise the question of the further extension to nonuniform acceleration until 1912, the year he first referred to his hypothesis as the ‘equivalence principle.’

The Gravitational Red Shift. Many textbooks on relativity ascribe to Einstein the method of calculating the red shift by means of the Doppler effect of light falling from the top to the bottom of an upwardly accelerating elevator. That is indeed the derivation he gave in 1911. However, he was already aware of the red shift in 1907. The derivation he gave at that time is less general, more tortured, and yet, oddly, more sophisticated. It deserves particular mention because it contains the germ of two ideas that were to become cornerstones of his final theory: the existence of local Lorenz frames and the constancy of the velocity of light for infinitesimally small paths …

Maxwell’s Equations; Bending of Light; Gravitational Energy = mc². Indomitably Einstein goes on. He tackles the Maxwell equations next. [He concludes that Maxwell’s equations have the same form in a uniformly accelerated reference frame as in a non-accelerated frame but with a modified velocity of light.] ‘It follows that the light rays … are bent by the gravitational field.’ Second, he examines the energy conservation law in [the accelerated frame] and finds ‘a very notable result … In a gravitational field, one must associate with every energy E an additional position-dependent energy which equals the position-dependent energy of a ‘ponderable’ mass of magnitude E/mc². The law [E = mc²] therefore holds not only for inertial but also for gravitational mass’ …

“This review does not have the perfection of the 1905 paper on special relativity. The approximations are clumsy and mask the generality of the conclusions. Einstein was the first to say so, in 1911. The conclusion about the bending of light is qualitatively correct, quantitatively wrong – though, in 1907, not yet logically wrong. Einstein was the first to realize this, in 1915. Despite all this I admire this article at least as much as the perfect relativity paper on 1905, not as much for its details as for its courage” (Pais, pp. 179-182).

“In 1920, Einstein recalled how he first arrived at the ideas behind the equivalence principle:

‘While I was occupied (in 1907) with a comprehensive survey of the special theory for the ‘Yearbook for Radioactivity and Electronics,’ I also had to attempt to modify Newton’s theory of gravitation in such a way that its laws fitted into the theory. Attempts along these lines showed the feasibility of this enterprise, but did not satisfy me, because they had to be based on physical hypotheses that were not well-founded. Then there came to me the most fortunate thought of my life in the following form:

‘Like the electric field generated by electromagnetic induction, ... the gravitational field only has a relative existence. Because, for an observer freely falling from the roof of a house, during his fall there exists—at least in his immediate neighborhood—no gravitation field. Indeed, if the observer lets go of any objects, relative to him they remain in a state of rest or uniform motion, independently of their particular chemical or physical composition [note by AE: air resistance is naturally ignored in this argument]. The observer is thus justified in interpreting his state as being at rest.

‘Through these considerations, the unusually extraordinary experimental law, that all bodies fall with equal acceleration in the same gravitational field, immediately obtains a deep physical significance. For if there were just one single thing that fell differently from the others in the gravitational field, then with its help the observer could recognize that he was falling in a gravitational field. If such a thing does not exist—which experiment has shown with great precision—then there is no objective basis for the observer to regard himself as falling in a gravitational field. Rather, he has the right to regard his state as one of rest and, with respect to a gravitational field, his neighborhood as field free. The experimental fact of the material-independence of the acceleration due to gravity is thus a powerful argument for the extension of the relativity postulate to coordinate systems in non-uniform relative motion with respect to each other .... The generalization of the relativity principle thus indicates a speculative path towards the investigation of the properties of the gravitational field.’

“Einstein alludes here to his initial attempts to set up a special-relativistic theory of gravitation, but gives no details. In 1933 he gave the fullest account of how he ‘arrived at the equivalence principle by a detour [Umweg]’ through such attempts. After mentioning his doubts after 1905 about the privileged dynamical role of inertial systems, and his early fascination by Mach’s idea that the acceleration of a body is not absolute, but relative to the rest of the bodies in the universe, he turns to the events of 1907:

‘I first came a step closer to the solution of the problem when I attempted to treat the law of gravitation within the framework of special relativity. Like most authors at the time, I attempted to establish a field law for gravitation, since the introduction of an unmediated action at a distance was no longer possible, at least in any sort of natural way, on account of the abolition of the concept of absolute simultaneity.

‘The simplest thing naturally was to preserve the Laplacian scalar gravitational potential and to supplement Poisson’s equation in the obvious way by a term involving time derivatives, so that the special theory of relativity was satisfactorily taken into account. The equation of motion of a particle also had to be modified to accord with the special theory. The way to do so was less uniquely prescribed, since the inertial mass of a body might well depend on its gravitational potential. This was even to be expected on the basis of the law of the inertia of energy.

‘However, such investigations led to a result that made me highly suspicious. For according to classical mechanics, the vertical acceleration of a body in a vertical gravitational field is independent of the horizontal component of its velocity. This is connected with the fact that the vertical acceleration of a mechanical system, or rather of its center of mass, in such a gravitational field turns out to be independent of its internal kinetic energy. According to the theory I was pursuing, however, such an independence of the gravitational acceleration from the horizontal velocity, or from the internal energy of a system, did not occur.

‘This did not accord with an old fact of experience, that all bodies experience the same acceleration in a gravitational field. This law, which can also be formulated as the law of equality of inertial and gravitational mass, now appeared to me in its deep significance. I was most highly amazed by it and guessed that in it must lie the key to the deeper under- standing of inertia and gravitation.’

“Turning from later reminiscences, let us see how Einstein presented his approach to gravitation in 1907:

‘Up to now we have only applied the principle of relativity, i.e., the presupposition that the laws of nature are independent of the state of motion of the reference system, to acceleration-free reference systems. Is it conceivable that the principle of relativity also holds for systems that are accelerated relative to each other?

‘This is not the place for an exhaustive treatment of this question. Since, however, it is bound to occur to anyone who has followed the previous applications of the relativity principle, I shall not avoid taking a position on the question here. Consider two systems in motion Σ₁ and Σ₂. Let Σ₁ be accelerated in the direction of its X -axis, and let γ be the magnitude (constant in time) of this acceleration. Let Σ₂ be at rest, but in a homogeneous gravitational field that imparts an acceleration –γ in the direction of the X -axis to all objects. As far as we know, the laws of physics with respect to Σ₁do not differ from those with respect to Σ₂; this is due to the circumstance that all bodies in a gravitational field are equally accelerated. So we have no basis in the current state of our experience for the assumption that the systems Σ₁and Σ₂differ from each other in any respect; and therefore in what follows shall assume the complete physical equivalence of a gravitational field and the corresponding acceleration of a reference system.

‘This assumption extends the principle of relativity to the case of uniformly-accelerated translational motion of the reference system. The heuristic value of this assumption lies in the circumstance that it allows the replacement of a homogeneous gravitational field by a uniformly accelerated reference system, which to a certain extent is amenable to theoretical treatment.’

“Some further comments on this equivalence in his next paper on gravitation in 1911 are illuminating. He notes that in both systems, objects subject to no other forces fall with constant acceleration:

‘For the accelerated system K′ [corresponding to the 1907 Σ₁], this follows directly from the Galileian principle [of inertia]; for the system K at rest in a homogeneous gravitational field [corresponding to the 1907 Σ₂], however, it follows from the experimental fact that in such a field all bodies are equally strongly uniformly accelerated. This experience of the equal falling of all bodies in a gravitational field is the most universal with which the observation of nature has provided us; in spite of that, this law has not found any place in the foundations of our physical picture of the world … From this standpoint one can as little speak of the absolute acceleration of a reference system, as one can of the absolute velocity of a system according to the usual [special] theory of relativity. [Naturally, one cannot replace an arbitrary gravitational field by a state of motion of the system without a gravitational field; just as little as one can trans- form all points of an arbitrarily moving medium to rest by a relativity transformation.] From this standpoint the equal falling of all bodies in a gravitational field is obvious.

‘As long as we confine ourselves to purely mechanical processes within the realm of validity of Newtonian mechanics, we are certain of the equivalence of the systems K and K′. Our point of view will only have a deeper significance, however, if the systems K and K′ are equivalent with respect to all physical processes, i.e., if the laws of nature with respect to K agree completely with those with respect to K′. By assuming this, we obtain a principle that, if it really is correct, possesses a great heuristic significance. For by means of theoretical consideration of processes that take place relative to a uniformly accelerated reference system, we obtain conclusions about the course of processes in a homogeneous gravitational field.’

“With hindsight, one can see that Einstein’s attempt to find the best way to implement mathematically the physical insights about gravitation incorporated in the equivalence principle was hampered significantly by the absence of the appropriate mathematical concepts. His insight, as he put is a few years later, that gravitation and inertia are “essentially the same” [wesensgleich], cries out for implementation by their incorporation into a single inertio-gravitational field, represented mathematically by a non-flat affine connection on a four-dimensional manifold. But the concept of such a connection was only developed after, and largely in response to, the formulation of the general theory. So Einstein had to make do with what was available: Riemannian geometry and the tensor calculus as developed by the turn of the century, i.e., based on the concept of the metric tensor, without a geometrical interpretation of the covariant derivative” (Stachel, pp. 83-86).

BRL 20; Stanitz 94; Weil *21. Born, ‘Arnold Johannes Wilhelm Sommerfeld 1868-1951,’ Obituary Notices of Fellows of the Royal Society 8 (1952), pp. 275-296. Stachel, ‘The first two acts,’ pp. 81-112 in: Gravitation in the Twilight of Classical Physics. The Promise of Mathematics (Renn & Schemmel, eds.), 2007.

8vo (231 x 157 mm), pp. 411-462. Original printed wrappers (upper cover a bit soiled, lower part of spine worn, light vertical crease for posting), faint ink stain to page 418, spine strip with wear and tear.

Item #6410

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