Historical Background Lecture 2

Bernhard Riemann’s Habilitation Lecture, his Debt to Carl Friedrich Gauss and his Gift to Albert Einstein

Abstract

We review Carl Friedrich Gauss’ contributions to classical differential geometry, astronomy and physics and his influence on his “student” Bernhard Riemann. Riemann’s public lecture, his “Habilitation”, is discussed as an extension of Gauss’ most important contribution to differential geometry, the fact that a surface’s curvature is an intrinsic property, to n-dimensional manifolds. Riemann and Gauss foresaw that geometry of space time belonged to the arena of physics and experiment, and their works laid the foundation for Einstein’s formulation of general relativity.

This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8). The term “textbook” in these Supplemental Lectures will refer to that work.

Keywords: differential geometry, Gaussian curvature, Riemannian geometry, curvature tensor. Curved space time, gravitation

Gauss, Geometry and Physics

Carl Friedrich Gauss (1777-1855) was one of the greatest and most influential mathematicians of all time. After being recognized as a prodigy in Brunswick, Germany, Gauss studied at the University of G ttingen from 1795 to 1798 where he did seminal work on

constructive geometry and number𝑜𝑜̈ theory. After his stint at G ttingen, he returned to Brunswick and continued his studies, assisted by support from the Duke of𝑜𝑜̈ Brunswick. In 1801 Gauss’ life changed decisively. In 1801 the dwarf “planet” Ceres had been observed briefly and several points on its trajectory, which headed toward the sun, were recorded by the Italian astronomer Giuseppe Piazzi. Gauss, working with Franz Xaver von Zach, an astronomer Gauss had known in G ttingen, analyzed the scant data and worked out the dwarf planet’s

trajectory around the sun𝑜𝑜̈ to impressive accuracy. In order to accomplish this, Gauss developed analysis methods based on conic sections, differential equations, fast Fourier transforms and he even estimated the errors in his predictions by developing and applying the curve fitting method of least squares. He predicted the orbit of Ceres around the sun and calculated when and where it would be found in the sky after passing behind the sun. Ceres was rediscovered by Zach on December 7 1801 almost exactly where Gauss had predicted, contrary to the calculations of most other experienced astronomers! It turns out that Gauss had invented several new methods of calculation and data analysis in the process. Much of his work predated that of others (for example, Joseph Fourier’s first paper on the subject occurred in 1807).

After this success Gauss secured the position of Professor of Astronomy and Director of the Observatory in G ttingen in 1807. Gauss published several groundbreaking works on astronomy

and applied mathematics𝑜𝑜̈ in the following years. Gauss’ interests turned to differential geometry in 1818 when his prodigious calculational skills led to his appointment to the team making a geodetic survey of the Kingdom of Hannover. Gauss invented the heliotrope to aid in measuring positions for the survey. In this period, which lasted a decade, Gauss also worked on more theoretical mathematics. These included the possibility of non-Euclidean geometries, a very controversial subject at the time. One of his longtime associates was Farkas Wolfgang Bolyai, whose son, Janos, discovered non-Euclidean geometry in 1829. (This development is discussed in the textbook where the Riemann disc approach to spaces of constant negative curvature is presented in a problem set.) Farkas sent Gauss the work of his son in 1832 and Gauss made the famous remark that “there was nothing in it that he hadn’t known for many years”, insulting Bolyai and losing his friendship. Gauss never published in this subject because he feared it would be too controversial and might reflect poorly on his reputation.

Underlying his interest in non-Euclidean geometries was his deep research in differential geometry and topology. These efforts led to his work on the intrinsic properties of surfaces, the invention of Gaussian curvature and the great “Theorema Egregium”, the proof that the curvature of a surface is an intrinsic, bending invariant property of any surface. (This topic is developed in detail in the textbook.) Gauss’ work established that curvature does not depend on how a surface is parametrized or how it is embedded in three dimensional Euclidean space: the curvature can be calculated from the surface’s “metric”, length and angle measurements on the surface itself. Gauss’ work in differential geometry led to Riemann’s development of Riemannian geometry, the generalization of intrinsic differential geometry to manifolds in higher dimensions and the eventual development of general relativity by Einstein.

In 1830 Gauss began a long term collaboration with the physicist Wilhem Weber at the University of G ttingen which led to his famous “Gauss’ Law”, the differential statement that charges are the local𝑜𝑜̈ sources of electric fields, studies in magnetism, the discovery of Kirchhoff’s rules of electrical circuits, and the construction of the first electromechanical telegraph in 1833. In addition, Gauss developed more accurate methods of mapping the earth’s magnetic field and engaged in this project for several years.

Wilhelm Weber’s fame attracted Bernhard Riemann to the University of G ttingen in the

1850s. We discuss this era next. 𝑜𝑜̈

Bernhard Riemann, the Invention of Modern Differential Geometry and its Impact on Physics

Bernhard Riemann (1826-1866) enrolled in the University of G ttingen in 1846, but not in the field of mathematics, rather in the Theology department at the behest𝑜𝑜̈ of his father, a Lutheran minister. Riemann, however, took courses from Gauss and Moritz Stern, who recognized Riemann’s extraordinary depth. Riemann turned his studies exclusively to mathematics and decided to move to Berlin University in 1847 which had a larger mathematics department. Once there Riemann was drawn to Dirichlet who, like Riemann, preferred intuitive thinking to long rigorous arguments and calculations. (This preference guided Riemann throughout his career, but it did cause him some problems later in life when his dependence on the Dirichlet Principle was criticized by Weierstrass. This problem was later resolved by David Hilbert in 1901 when he formulated Dirichlet’s Principle with more precision and vindicated Riemann.)

In 1849 Riemann returned to the University of G ttingen where he acted as W. Weber’s assistant while Gauss acted as Riemann’s Ph.D. thesis𝑜𝑜̈ advisor. Riemann developed a strong background in theoretical physics under Weber’s guidance. Riemann’s thesis concerned the theory of complex variables. He invented Riemann surfaces, which introduced topological concepts into complex variables for the first time. Gauss was very impressed with Riemann’s originality. This led to Gauss recommending Riemann for a post at the University of G ttingen

and the opportunity for him to work toward his “Habilitation”, a degree which would permit𝑜𝑜̈ him to become a university lecturer. Riemann spent almost 2 ½ years on this project.

The Habilitation consisted of several parts. In one part Riemann analyzed the properties of functions which could be represented by trigonometric series. In another he invented the notion of Riemann integrability. The third requirement was the presentation of a public lecture on a topic chosen by his advisor, Gauss. Gauss chose the topic of geometry (to Riemann’s surprise, which he expressed in letters to his father). Since Riemann was very shy at this point in his life, giving a lecture on a subject he had not concentrated on was a daunting challenge. In addition, the lecture was required to be conceptual and could not be too technical for a public audience. Riemann devoted himself to this chore and produced a hallmark in mathematics, the conceptual basis for differential geometry in n-dimensional spaces. This work built upon Gauss’ work on the intrinsic properties of two dimensional surfaces and effectively generalized it to higher dimensions.

An English translation of Riemann’s Habilitation lecture (“ ber die Hypothesen welche der

Geometrie zu Grunde liegen”) can be found here: 𝑈𝑈̈

http://lymcanada.org/wp-content/uploads/sites/2/2015/01/Riemann_Habilitation-Dissertation.pdf In the first part of the lecture Riemann defines the coordinates and metric of an n- dimensional manifold, a Riemannian space. He introduces the idea of shortest lines, geodesics, and notes that the space is locally Euclidean. He then indicates how curvature would be defined and measured in this space and effectively introduced the Riemannian curvature tensor and counts the number of independent components the tensor possesses, through a famous and simple original argument. In the second part of the lecture Riemann discusses the relation of geometry to the space time of the physical world. He ends with the remark:

“This path leads out into the domain of another science, into the realm of physics…”

Gauss was very impressed with the lecture and he spoke to Weber with unusual enthusiasm. For once, Gauss did not claim that he had “done it first many years earlier”, although it clearly was a generalization of Gauss’ perspective on differential geometry, both in theory (Theorema Egrigium) and practice (the geodetic survey of Hannover). Riemann was subsequently granted a lectureship at the university. In 1859 Riemann became the chairman of the mathematics department.

From the perspective of the textbook, the fruition of Riemann’s public lecture was Einstein’s theory of general relativity. That formulation relies on the tensor methods of Riemann to capture the covariance of the theory’s geometric properties and physical laws in the setting of dynamically curved space time. It also fulfills Riemann’s dream that the geometry of space time should be determined experimentally. There was a 61 year gap between these two milestones in mathematics and physics.

REFERENCES

1. Carl Friedrich Gauss: Titan of Science by C.W. Dunningtom, The Mathematical Association of America, Washington, DC, 2004. 2. Bernhard Riemann 1826-1866 Turning points in the Conception of Mathematics, D. Laugwitz (translated by A. Shenitzer), Springer Science Publishers, New York, 1999.