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Lot 36: Albert Einstein rare 1922 autograph manuscript incorporating a variant of The General Theory of Relativity’s principal formula

Est: $60,000 USD - $80,000 USDSold:
Lion Heart AutographsJune 15, 2016New York, NY, US

Item Overview


Rare Einstein Manuscript Incorporating “(Rik – ½gikR) - λgik = 0,” a Variant of His Central Formula for General Relativity, “The Most Elegant Theory in the History of Science,” to Account for Matter and Cosmological Structure – Einstein’s response to Erich Trefftz’s Paper on the “Einstein Equations” ********* EINSTEIN, ALBERT. (1879-1955). German-American physicist and Nobel Prize winner. AMsS. (Signed in the title,“Einstein’schen”). 2pp. 4to. N.p., N.d. The original autograph manuscript for Einstein’s paper “Bemerkung zu der Abhandlung von E. Trefftz: ‘Das statische Gravitationsfeld zweier Massenpunkte’ in der Einsteinschen Theorie’” (“Observation on the work of E. Trefftz: ‘Static Gravitational Field of Two Point Masses in the Einstein Theory’”), published in the Proceedings of the Berlin Academy in late 1922; the first paper published by Einstein after winning the Nobel Prize for physics. In our German manuscript Einstein states his interest in “the importance for the problem of cosmological questions, i.e.: for the question of the geometric structure of the universe,” refers to “Euclidean-Galilean isotropic and homogeneous space,” mentions the “well-known Schwarzschild solution,” and talks about singularity and differential equations. ********* In his annus mirabilis of 1905, Einstein famously published five papers “that changed the face of physics,” (Einstein’s Miraculous Year: Five Papers That Changed the Face of Physics, ed. John Stachel). One of these ground-breaking papers introduced the special theory of relativity. It presented a deep revision of our common notions of space and time, and another paper published later that year as an addendum contained the famous equivalence between energy, E, and m, mass, and c, the speed of light, expressed as: E=mc2 ********* Ten years later, recognizing the need to account for the effect of gravity on space and time, Einstein completed an even more significant achievement, the general theory of relativity, essentially a field theory of gravity. At the heart of this theory were the generally covariant field equations of gravitation, the so-called Einstein equations, which are written in the form: Rik – ½gikR = - kTik ********* where gik denotes the metric tensor, the fundamental object of Einstein’s relativistic theory describing the properties of curved space-time, Rik denotes the Ricci tensor, formed from the metric, R its contraction, the Riemann curvature scalar, and Tik the energy-momentum tensor representing the energy and matter content in space-time. k is a constant, proportional to the gravitational constant. ********* These equations were realized by Einstein in November 1915 in a dramatic breakthrough during intense competition with the world’s leading mathematician David Hilbert. Einstein’s general theory of relativity was published in March 1916 in Annalen der Physik and his “revolutionary hypothesis has withstood the test of time, despite numerous expert attempts to find flaws,” (“After 100 years, Einstein’s theory stands test of time,” Phys.org, Santini). The Einstein equations of 1915 form the foundation of today’s large-scale astrophysical and cosmological physics. Though their practical applications are minor when compared to their historic scientific contributions, modern GPS navigation systems, for example, would not function as precisely as they do without taking into account tiny effects that are encoded in the Einstein equation. Almost exactly one hundred years after their publication in 1915, a huge international research enterprise announced the long-sought for experimental confirmation of one of their central predictions: gravitational waves. ********* Einstein’s equations of 1915 were not his last word on the subject. Just two years later, in 1917, he applied his equations to the problem of explaining the structure of the cosmos on a large scale and found that he would need to modify his equations by adding another term, containing a constant, which he denoted λ and called “cosmological.” Although, he is reported to have regretted this modification as the “greatest blunder” of his life, that cosmological constant “has refused to die,” (J. Earman, “Lambda: The Constant That Refuses to Die.” Archive for History of Exact Sciences 55 [2001]: 189-220). In fact, it has gained new prominence in recent years with the discovery of the accelerated expansion of the universe, a discovery that earned S. Perlmutter, B.P. Schmidt and A.G. Riess the 2011 Nobel Prize in physics. ********* With the famous gravitational constant and for the special case of a vacuum, where the energy-momentum tensor Tik vanishes, Einstein’s gravitational field equations read: (Rik – ½gikR) - λgik = 0 ********* which is the equation cited as “(1a)” in our manuscript. By a mathematical operation called contraction, equation (1a) implies that λ = - R/4 in the case of a vacuum. Substituting this expression for λ into equation (1a), one obtains the equation: Rik – ¼ gikR = 0 ********* The latter equation, written out by Einstein, is given as equation “(1)” in the present manuscript. It was advanced by Einstein in a paper of 1919 as a candidate for a slightly modified field equation to account both for the structure of matter and for cosmological structure. ********* After their publication, Einstein’s field equations and his discussion of their relevance for cosmological speculation immediately sparked great interest among mathematicians and physicists who began to investigate the consequences of Einstein’s new theory, both for its mathematical and physical properties. Our Einstein manuscript must be seen in this larger context. ********* Reacting to Einstein’s field equations with its cosmological term several authors soon found solutions for special cases. Among them were the famous mathematician Hermann Weyl (1884-1955), the lesser known Friedrich Kottler (1886-1956), Rudolf Bach alias Rudolf Förster (1885-1941), and the engineer Erich Trefftz (1888-1937). All these authors published a particular solution to the vacuum cosmological field equations, now known as the KTW (“Kottler-Trefftz-Weyl”) solution, (Geyer, K.H. “Geometrie der Raum-Zeit der Massbestimmung von Kottler, Weyl und Trefftz.” Astronomische Nachrichten 301 [1980], 135-149). It is cited and discussed in Einstein’s paper as equations “(2)” and “(3)”, resulting in: ds2 = (1 + A + Bw2) dt2 - ____dw2____ - w2 (dυ2 + sin2 dϕ2) w 1 + A/w + Bw2 ********* This equation contains two other well-known solutions as special cases (Cf. Laue, Max von. “Die Lösungen der Feldgleichungen der Schwere von Schwarzschild, Einstein und Trefftz und ihre Vereinigung.” Preussische Akademie der Wissenschaften [Berlin] Physikalisch-mathematische Klasse. Sitzungsberichte 1923, 26-30). One special case is the famous Schwarzschild line element, the basis for the physics of black holes. The other special case is de Sitter’s solution, relevant in the cosmological context. ********* Erich Trefftz was an applied mathematician and engineer. He was Professor of Mathematics at the Technical University of Aachen at the time he published his 1922 paper (the one about which Einstein is writing), “Das statische Gravitationsfeld zweier Massenpunkte in der Einsteinschen Theorie” in Mathematische Annalen on the solution of Einstein’s equations, but moved to the Technische Hochschule in Dresden at just about the time when Einstein’s response was published. Trefftz had a sound mathematical training, having studied for some time in Goettingen with such luminaries as David Hilbert, Carl Runge, and Ludwig Prandtl. His paper on Einstein’s equations is somewhat unusual among his scientific work, which largely dealt with technical problems and questions of applied mathematics, (Stein, Erwin. “An Appreciation of Erich Trefftz.” Computer Assisted Mechanics and Engineering Sciences 4 [1997], 301-304). ********* In his criticism of Trefftz’ paper, Einstein does not deny that Trefftz offers a valid solution to the field equation. Rather, he questions Trefftz’ interpretation of the solution. Trefftz had argued that it was describing the static, i.e., time-independent gravitational field of two massive spherical bodies, an important model case for the physical interpretation of the general theory of relativity. Einstein questions the claim that in the interpretation as a field of two gravitating bodies, the solution truly was time-independent. ********* In a letter to Nobel Prize-winning German physicist Max von Laue (1879-1960), Trefftz rebutted a technical point which was part of Einstein’s argument, (Albert Einstein Archives, Call-No. 23-050). We do not know whether Einstein ever responded to Trefftz’ counter-argument, but Laue, a colleague and good friend of Einstein, and himself a winner of the 1914 Nobel Prize in physics, was also interested in this subject. He published a paper of his own, in which he gave a detailed discussion of the general KTW-solution and its physical interpretation (see Laue, op. cit.). For a modern discussion of the KTW-solution see Geyer, op.cit. ********* In fact, it was Laue who presented and discussed Einstein’s article to the Prussian Academy for publication in its session of November 23, 1922. The paper was then published on December 21, 1922. Shortly after, Laue presented his own paper to the Academy in their session of February 15, 1923. His discussion remains silent about Einstein’s comment on Trefftz. Laue, who according to a written note of provenance that accompanies our manuscript once owned our original, must have kept Einstein’s manuscript and sent it to Trefftz for perusal. All this happened when Einstein was traveling. He had left Berlin in early October 1922 for a half-year long trip that would take him to Japan and, on his way back, to Palestine and Spain. During that time, in November 1922, he was awarded the 1921 Nobel Prize for Physics. He only returned to Berlin in late March 1923. Trefftz responded to Laue and returned the manuscript belatedly with a letter, dated June 5, 1923, (Albert Einstein Archives, Call-No. 23-050). ********* Einstein’s two-page manuscript represents the original for Einstein’s published paper with the same title “Bemerkung zu der Abhandlung von E. Trefftz: ‘Das statische Gravitationsfeld zweier Massenpunkte in der Einsteinschen Theorie.’” A comparison shows that the manuscript is, in all probability, the handwritten version used for typesetting the published paper. It contains additional handwritten corrections in pencil that were executed in the published version. The manuscript is 98% complete, breaking off in mid-sentence at the bottom of the second page. The printed version contains an additional 3½ concluding lines. World renowned Einstein scholar, Walter Isaacson, has described Einstein’s formula for General Relativity as, “The Most Elegant Theory in the History of Science.” ********* The published paper was reprinted in Volume 13 of the Collected Papers of Albert Einstein, Princeton University Press, 2012 (2, Doc. 387, pp.594-597), with extensive commentary. ********* Heavily folded with some separation along the folds of the first page. The second page is repaired on the verso along a horizontal fold tear. A paperclip mark in the upper left corner and some scattered staining. Light pencil notations throughout. In good condition. Accompanied by a sheet on which a note in German details the provenance: “Einstein-Manuscript, given by Herr v. Laue 1948 in Göttingen / Alex. Dingas / For Miss G. Schrupf / to be used in any way deemed suitable, possibly even for sale / Alexander Dingas / Berlin 12/ April 1964.” ********* Our thanks to Einstein scholar, author and professor of the history of mathematics, Dr. Tilman Sauer of the University of Mainz, for his invaluable assistance in describing our manuscript.

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