Seeking the Shape of the Universe: Confronting the Hyperbolic World, From Henri Poincaré to the Cosmic Microwave Background

Abstract

This dissertation engages the intellectual adventure to empirically determine the size and shape of the universe – to imagine, measure, and map the cosmos in the long 20th century. This journey developed in three stages, which in broad strokes can be seen as imagining and theorizing, modeling, and testing the shape of space; my account complicates this narrative, demonstrating the role that mathematical imagination played at the core of each stage. Late-19th-century mathematical advances regarding non-Euclidean geometry and intrinsic curvature encouraged growing numbers of mathematicians and astronomers to pursue the possibility of astronomically measuring the curvature of space. Part I historicizes the revolutionary mathematical world in which the French mathematician, celestial mechanist, and physicist Henri Poincaré established the “hyperbolic crystal” at the heart of mathematics. Part II explores the remarkable puzzle posed by Poincaré in 1892 about fictional beings in a “Hyperbolic World” creating their geometry and physics. Poincaré saw the need for a “new geometry,” analysis situs (algebraic topology), to distinguish between spaces which admit the same metric (are empirically indistinguishable via metric geometry) and yet have topologically-distinct “shapes.” He set out single-handedly to create it.

Part III features the co-revolutionizing worlds of of mathematics and theoretical physics in Poincaré’s mathematical corpus, Hermann Minkowski’s development of spacetime theory, Einstein’s development of general relativity, and emerging relativistic cosmological models of the universe. I show how Poincaré’s mathematical creations were intensely studied and absorbed by Göttingen mathematicians and scientists, thereby shaping the practice of 20th-century mathematics and physics. The profound topological concepts and tools that originated with Poincaré’s analysis situs served, together with increasingly sophisticated tools of observational cosmology, as a means to interrogate the cosmos and potentially to establish its shape. The Cosmic Microwave Background (CMB) radiation – the “first light” of the universe, remnant heat from the big bang – became after its discovery in 1964 a primary observational means to explore the universe at its largest scales. I assess how in the 1990s cosmologists and mathematicians exchanged ideas and practices in assessing data from NASA’s satellite mission Cosmic Microwave Background Explorer (COBE) and its implications for the shape of space.