Manhattan Rare Book Company

“It is a wonderful thing that the explanation for the Mercury anomaly emerges so convincingly from such an abstract idea.” — Karl Schwarzschild

RARE FIRST EDITION IN ORIGINAL WRAPPERS OF KARL SCHWARSCHILD’S DISCOVERY OF THE FIRST EXACT SOLUTION TO THE FIELD EQUATIONS OF EINSTEIN’S GENERAL THEORY OF RELATIVITY — A SOLUTION THAT PROVIDED THE KEY TO PREDICTING THE ORBITS OF PLANETS AND SATELLITES.

By the end of 1915, Albert Einstein had published the final version of the field equation of general relativity (the “Einstein Field Equation”, or EFE), which relates distributions of matter and energy to the geometry of the spacetime surrounding those distributions. A key remaining task was developing solutions to the equation — i.e., descriptions of specific spacetime geometries that, when associated with specific distributions of matter and energy, satisfy the EFE. This was no trivial task. The EFE actually represents, in a highly compressed notation, ten separate second-order differential equations. “As differential equations, [the EFEs] are extremely complicated; the Ricci scalar and tensor are contractions of the Riemann tensor, which involves derivatives and products of the Christoffel symbols, which in turn involve the inverse metric and derivatives of the metric. … The equations are also nonlinear …. It is therefore very difficult to solve [the EFE] in any sort of generality ….” (Sean M. Carroll, Spacetime and Geometry: An Introduction to General Relativity, 2004).

A solution to the EFE takes the form of a “metric tensor” (or “metric” for short) — a mathematical description of a particular spacetime geometry. The metric can be expressed in the form of a “line element” that can be used for calculation of path lengths through spacetime. The metric is, in fact, the key to doing physics in the curved spacetimes predicted by general relativity. Among other things, “(1) the metric supplies a notion of ‘past’ and ‘future’; (2) the metric allows the computation of path lengths and proper time [“proper time” is the flow of time as measured by an observer moving through the spacetime]; (3) the metric determines the ‘shortest distance’ between two points, and therefore the motion of test particles [“test particles” are idealizations of planets, satellites, and other comparatively small objects moving through the spacetime in issue]; [and] (4) the metric replaces the Newtonian gravitational field ….” (Carroll, op cit.).

The first published solution to the EFE — presented in the paper offered here — was Karl Schwarzschild’s solution for the spacetime geometry around a static, spherically symmetric mass distribution. Expressed in the form of the associated line element, the Schwarzschild metric is:

ds² = – (1 – (2GM/r)) dt² + (1 – (2GM/r))^-1 dr² + r²dΩ²

Schwarzschild’s solution was important for a number of reasons. First, to an excellent approximation, stars, and in particular the sun, are static and spherically symmetric, so that Schwarzschild’s solution provides an entrée to the problem of determining planetary orbits, which are geodesics (roughly, paths of “shortest length”) in the spacetime geometry described by the Schwarzschild metric.

Second, the solution had important links to a problem that had been at the heart of the historical development of general relativity. One of the triumphs of Einstein’s theory had been the natural way in which it accounted for the anomalous perihelion precession of the planet Mercury — that is, the fact that the point in its orbit at which Mercury makes its closest approach to the Sun shifts in regular fashion from one orbit to the next, instead of remaining fixed, as Newton’s theory of gravity would predict. The existence of the anomalous perihelion precession had been known to astronomers since LeVerrier’s observations in the mid-nineteenth century, but it had defied previous attempts at a convincing explanation. Einstein realized that his new theory of gravity (which, in essence, is what general relativity was) would give him a good shot at explaining this phenomenon in a way that emerged naturally from a broad theoretical framework rather than relying on unsupported empirical assumptions (such as the hypothesis that there was an unknown planet inside the orbit of Mercury (see Richard Baum & William Sheehan, In Search of Planet Vulcan: The Ghost in Newton’s Clockwork Universe, 1997), or on unmotivated ad hoc tweaks to Newton’s theory. Einstein’s early attempts to solve the perihelion problem in the context of the emerging theory of general relativity yielded a disappointingly inaccurate estimate of the magnitude of the precession. (See Hanoch Guttfreund & Jürgen Renn, The Road to Relativity, 2015). Shortly before he presented the final version of the EFE, Einstein published a revised analysis of the perihelion problem. Although the revised analysis accurately accounted for the phenomenon, and was firmly grounded in Einstein’s new theory, it was derived through the use of approximations. Schwarzschild’s metric, on the other hand, provided the basis for an exact calculation of the paths of planets moving through the spacetime geometry induced by the Sun’s spherically symmetric matter distribution.

Finally, the Schwarzschild metric — particularly when extended to an “interior” solution describing the metric inside a star (an extension that Schwarzschild accomplished in a subsequent paper) — provided key insights into stellar evolution and the existence and nature of “black holes,” as well as more speculative phenomena such as wormholes. “The Schwarzschild solution without question remains one of the most important known exact solutions of Einstein’s equation.” (Robert M. Wald, General Relativity, 1984).

Schwarzschild discovered this solution to the EFE while fighting for Germany on the Russian front during World War I. After volunteering for service, he had been “commissioned as a lieutenant and attached to the headquarters staff of an artillery unit, serving first in France and later on the Eastern front. His assignment was to calculate trajectories for long-range missiles ….” (DSB). His artillery work “does not seem to have taken up all his time, for as soon as he had received Einstein’s reports to the Prussian Academy [i.e., the tetralogy of papers in which Einstein had estimated the perihelion precession and then restated the EFE in its final form] he managed to provide the first complete solution of the field equations in the case of a large mass with a spherical gravitational field. On behalf of the absent Schwarzschild, Einstein submitted his calculations to the [Prussian Royal] academy on January 13, 1916. Schwarzschild’s precise calculations made no difference to the result [for Mercury’s perihelion precession] found by Einstein’s approximations.” (Albrecht Fölsing, Albert Einstein, 1997.)

Schwarzschild transmitted the solution to Einstein in a letter dated December 22, 1915. Addressing his correspondent as “Verehrter Herr Einstein!” (“Esteemed Mr. Einstein!”), he said that “[i]n order to become versed in your gravitational theory, I have been occupying myself more closely with the problem you posed in the paper on Mercury’s perihelion and solved to the 1st-order approximation. … Thereupon, I took my chances and made an attempt at a complete solution. A not-overly lengthy calculation yielded the following result [which he proceeded to describe]. … It is a wonderful thing that the explanation for the Mercury anomaly emerges so convincingly from such an abstract idea. As you see, the war is kindly disposed toward me, allowing me, despite fierce gunfire at decidedly terrestrial distance, to take this walk into your land of ideas.” (Einstein Papers, vol. 8A, Doc. 169). Einstein received this letter — sent from the front, during a war, in the middle of a holiday season — in time to respond to it by December 29 (a fact that forces one to reflect that the postal system, at least, hasn’t improved in the last century). Addressing Schwarzschild as “Hoch geehrter Herr Kollege!” (“Highly esteemed Colleague!”), Einstein said that Schwarzschild’s calculation was “extremely interesting. I hope you publish it soon! I would not have thought that the strict treatment of the [mass-]point problem was so simple.” He also noted that he was “very satisfied with the theory [of general relativity]. It is not self-evident that it already yields Newton’s approximation; it is all the more gratifying that it also provides the perihelion motion and line shift, although it is not yet sufficiently secure. Now the question of light deflection is of most importance.” (Id., Document 176). Schwarzschild sent Einstein his complete paper, and Einstein responded at length on January 9, stating that “[t]he mathematical treatment of the subject appeals to me exceedingly. Next Thursday I am going to deliver the paper before the Academy with a few words of explanation.” (Id., Document 181). This last letter is also interesting since in it, Einstein steps back from his theory to consider its implications for the ontology of space, which he describes in Machian terms, stating that “[u]ltimately, according to my theory, inertia is simply an interaction between masses, not an effect in which ‘space’ of itself were involved, separate from the observed mass. The essence of my theory is precisely that no independent properties are attributed to space on its own.”

“While serving on the front in Russia, Schwarzschild developed symptoms of a rare, painful, and at the time incurable, malady called pemphigus [now generally controllable through medication] … Schwarzchild was invalided home in March 1916 and spent the last two months of his life in a hospital.” DSB. He died on May 11.

This issue of Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften which is offered here (1916; No. VII) also includes a paper by Einstein: “Eine neue formale Deutung der Maxwellschen Feldgleichungen der Eletrodynamik,” in which he “reformulates Maxwell’s equations [of electromagnetism] in the language of tensor calculus and in light of his recent discovery of the field equations of general relativity.” (Alice Calaprice, Daniel Kennefick & Robert Schulmann, An Einstein Encyclopedia, 2015).

Offered with: a copy of Issue No. XXXIV in original wrappers of the Sitzungsberichte for the same year containing Einstein’s obituary for Schwarzschild.

IN: Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, Vol. VII (7) and Vol. XXXIV (34). Berlin: Königlichen Akademie der Wissenschaften, 1916. Octavo, original wrappers. Both issues housed together in a custom box. A little soiling to wrappers; text fine and largely unopened. VERY RARE IN ORIGINAL WRAPPERS.

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