Lot 351

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Lot 351
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Einstein Equations, The Matrix & General Relativity, Vectors, "Riemann's Curvature Tensor," & Unified Field Theory
Einstein Equations, The Matrix & General Relativity, Vectors, "Riemann's Curvature Tensor," & Unified Field Theory
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Description:

Einstein Equations, The Matrix & General Relativity, Vectors, "Riemann's Curvature Tensor," & Unified Field Theory

A 1p autograph manuscript in the hand of German Nobel Prize-winning physicist Albert Einstein (1879-1955), comprised of approximately 298 words and no fewer than 18 lines of mathematical equations. N.d., n.p. [but ca. early 1940s, Princeton]. The page of Einstein's notes is numbered "3a" in the upper right corner. A working document, the manuscript was heavily revised by Einstein with multiple cross-outs, one-word substitutions, and even a two-sentence insertion found at the center. Expected wear including even toning and isolated edge darkening mostly confined to the left edge. Partial chips to the creased upper right corner and central left edge mentioned just for accuracy. The lower left corner is also gently creased. Else near fine. Accompanied by an English translation. 8.5" x 11."

In this highly technical scientific manuscript, Einstein delves into the mathematics underlying General Relativity: Riemann’s differential geometry. Einstein’s discovery of Riemannian Geometry enabled the great original breakthrough to General Relativity (in 1912), and he here explores the development of Unified Field Theory along similar lines.

Translated in part:

"The left side from (8) is analogous to Riemann’s curvature tensor, pressed (through the T’) (9) is the attenuation of (8), which is the same as the one we considered in our earlier first section. However, there is still a second part of the attenuation of equations (8), which appears to be of even greater importance for the theory of such fields.

Firstly, some remarks about the equations (8). It refers to the combination of three arbitrary spatial points α, β, and δ; it has the character of a mixed tensor depending on α, and the scalar character depending on β and δ; Since points α, β and δ along with their order are fixed, we are still free to choose any one of the three points as the distinguished point instead of α; It can be shown that the statement of (8) is independent of this choice. Moreover, if we multiply both sides of the equation by (g g g^-1)^-1, where g is an invertible matrix, the resulting equation remains equivalent. However, the order of the points α, β and δ; is important since, apart from their order, (8) on the left-hand side refers to two different matrices…”

The Riemann curvature tensor, also known as the Riemann-Christoffel tensor, was a central mathematical tool in Einstein’s arsenal, expressing the curvature of Riemannian manifolds or tensor fields. "It was Riemannian geometry that enabled Einstein to establish a physical bridge between light, matter and space-time…In the general theory of relativity, Riemannian geometry helped Einstein blend two components – how gravitational forces cause matter to move, and how the presence of matter changes gravitational forces – into one, leading to his most brilliant insight: that gravity in the cosmos is not a force as much as it is the extent to which the space-time continuum is curved.” (Vasudevan Mukunth, "Beyond the Surface of Einstein's Relativity Lay a Chimerical Geometry," The Wire, September 15, 2015.)

In this manuscript, Einstein investigates the application of bivectors to the Riemann tensor. A mathematical object composed of two vectors, a bivector can be used to define an antisymmetric tensor of the second rank; and Einstein here uses bivectors to visualize the curvature of space-time, with one component vector representing the curvature of space and the other the curvature of time. As the above-quoted translation of the manuscript indicates, Einstein is working with bivectors in the context of invertible matrices, typically (for Einstein) a 2 x 2 square matrix so constructed that the matrix’s reciprocal correlate can be generated by inversion (and vice versa). Such invertible matrices were used by Einstein in his field equations to map the metric tensor describing the relationship between space-time and the gravity field; and Einstein also employed them to explain the phenomena of time dilation and gravitational redshift.

Beginning in the early 1940s, Einstein sought to extend General Relativity and to make it the basis for any future Unified Field Theory. Such a “generalized theory of gravitation,” as Einstein himself later called it, constituted Einstein’s final approach to Unified Field Theory. In the course of working with this approach, Einstein investigated the use of new and different types of mathematical objects in his field equations. With the present manuscript we are at the very beginning of Einstein’s exploration of a “generalized theory of gravitation.”

Einstein published two papers on his Bivector explorations: “Bivector Fields” in 1943, and its continuation “Bivector Fields II” in 1944. These articles were published only in English, and the original German text of these articles was extensively transformed being translated into English. While the actual text of the present leaf markedly differs from the published articles, our manuscript bears some correspondence to p. 22 of the latter work. The original German manuscript for these Bivector articles appears to have been lost in the course of time, and the offered leaf is one of the few known surviving portions of its text.

This item comes with a Certificate from John Reznikoff, a premier authenticator for both major 3rd party authentication services, PSA and JSA (James Spence Authentications), as well as numerous auction houses.

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