Fifty years before Albert Einstein proposed his revolutionary theory of relativity, the German mathematician and physicist Bernhard Riemann dreamed up the concept of curved space, which played a key role in Einstein's formulation. "Riemann is responsible for most of the fundamental ideas underlying modern cosmology, as well as some other parts of physics, yet he has been largely overlooked," said Robert Osserman, professor emeritus of mathematics at Stanford.
Osserman described Riemann's major contributions in a presentation Feb. 16 at the annual meeting the American Association for the Advancement of Science in Philadelphia. His presentation was part of a session that he organized titled "From Riemann to Strings: The Ongoing Romance between Geometry and Physics." Riemann, who was born in 1826, died at the age of 39 from tuberculosis. In his short career he published only a dozen papers. Nevertheless, his ideas have had a major impact on the course of both modern mathematics and physics. "He was the first person to fundamentally rethink geometry since Greek times," Osserman said.
Osserman described two key ideas from Riemann's vision: the concept of spaces that can be curved in all dimensions; and "Riemann surfaces," an abstract concept of a surface that engineers have applied to the study of aerodynamics and hydrodynamics, and theoretical physicists have more recently drawn upon heavily in their formulations of string theory.
In his general theory of relativity, Einstein used Riemann's concept of curved space as the basis for his elegant explanation of gravitation. Massive objects put a dimple in space in their vicinity. So when other physical objects, including photons, which don't have any mass, wander into the object's vicinity, they encounter this curved space. This curvature determines the path the objects follow, in a way that was formerly attributed to the force of attraction that we call gravity.
In much the same way that Riemann conceived of curving and twisting space in innovative ways, he also described a set of abstract surfaces that were created by cutting and pasting together normal surfaces in ways that cannot be done with real surfaces, but can be done abstractly.
"You can do a lot of mathematics on those abstract surfaces. So this has been an amazingly important idea for many parts of mathematics, and now for physics," Osserman said.
Another session speaker, Edward Witten from the Institute for Advanced Study at Princeton, who is one of the main architects of string theory, discussed some of the relations between physics and math in the 20th and 21st centuries. String theory suggests replacing pointlike particles with infinitesimal vibrating strings as the basic units of the physical world.
The other speakers in the session were Jerry Kazdan from the University of Pennsylvania, who talked about "Local and Global Aspects of Geometry," and David Morrison of Duke University, who discussed "The Small-Scale Structure of Spacetime."