I have recently been thinking a lot about what introductory and background material I want to include in my PhD thesis, as my self-imposed December deadline continues to hurtle towards me at an alarming speed. Concurrent with this thinking, I’ve also recently been enjoying a fantastic book called The Poincaré Conjecture by Donal O’Shea, all about — you guessed it — the Poincaré conjecture.
For those not in the know, the Poincaré conjecture states that
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
To understand this, it’s easier to think about our familiar 2-sphere, examples being the Earth (roughly) or a football. Imagine you have a reel of string, tied to a post. You run in a circle, laying out the string behind you. If you get back to your starting point and pull the string, you will be able to tighten it to a point. An object is simply connected if every loop drawn on it can tighten to a point — an easy to imagine counter example is a torus or doughnut; if you held your string and ran through the hole in the torus, you wouldn’t be able to tighten your loop of string to a point. A closed manifold is one that has no boundary (i.e. you can travel constantly in one direction and return to your starting point; the surface of the Earth is again one such example), and homeomorphic essentially means “is the same as”, modulo some transformation.
Poincaré asserted that this holds true not only for the familiar 2-sphere, but for the more complicated, higher dimensional 3-sphere. The conjecture was stated in 1904, and proved in 2006 by the reclusive Russian mathematician Grigori Perelman. He was offered both the Millennium Prize and the Fields Medal for the proof, but turned them down.
This is all very interesting, but what does it have to do with general relativity, cosmology or my thesis? I’ve been curious about topology for a while, though have never studied it either formally or informally — reading this book by Donal O’Shea is very much my first concerted foray into the subject. However, to reach the statement of the Poincaré conjecture, one first has to understand the concept of curved spaces, and to understand curved spaces, one has to understand flat spaces.
Most of our oldest surviving written mathematics dates to ancient Greece, from roughly the fifth century BCE. One of the greatest legacies of that era is Euclid’s Elements, a comprehensive treatise largely devoted to geometry — the study of the shape of the space we inhabit, and how that space can be mapped and measured in a rigorous and logical way.
The fifth postulate of the Elements runs as follows:
If a line segment intersects two straight lines forming two interior angles on the same side that sum to less that two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.Elements (Book I), Euclid.
A geometry in which this statement holds — a Euclidean geometry — is one in which parallel lines never meet.
Euclidean geometry was king for around two thousand years, until the early days of the nineteenth century saw the emergence of a radical new geometry: the study of spaces in which parallel lines can converge or diverge. For this to happen, the space must be curved in some way — the space is described by a non-Euclidean geometry. The early pioneers in this field were Gauss, Lobachevsky and Bolyai, but the most groundbreaking work was done by a student of Gauss’s: Bernhard Riemann.
In 1854, Riemann delivered a sensational lecture that founded the field that we now call Riemannian geometry, and solidified many of the concepts laid out by Euclid. For example, Euclid offered no definition of what a straight line was; Riemann said that a straight line is one that takes the shortest distance to connect two points: a geodesic. If we know what a straight line is in a space, we can construct triangles, and triangles allow us to determine curvature. If a space is positively curved, the angles in a triangle will sum to more than 180 degrees, and if it’s negatively curved, they will sum to less. Riemann didn’t stop at two dimensions; he extended the geometry to tackle n-dimensional spaces, necessitating the use of tensors — specifically, the Riemann curvature tensor — to measure the curvature in n-dimensions.
We can see the way forward to general relativity now, and in fact, the English mathematician William Clifford also did just that. After reading about Riemann’s lecture, he published a paper, On the Space-Theory of Matter, in which he wrote
I hold in fact: (1) That small portions of space are of a nature analogous to little hills on a surface which is on average flat. (2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave. (3) That this variation of the curvature of space is really what happens in that phenomenon which we call the motion of matter whether ponderable or ethereal.On the Space-Theory of Matter, William Clifford
It is a remarkably prescient statement that predates Einstein by half a century. Unfortunately, Clifford died of tuberculosis not long after writing this paper, and so never developed his ideas. Happily for us cosmologists, Riemannian geometry and the new-found way of computing the curvature of complicated manifolds was here to stay. Einstein united these ideas with the concept of the equivalence principle (acceleration due to gravity is the same as acceleration of any other kind) to give us the most beautiful, profound and accurate description of gravitation ever discovered. From that point on, the field of physical cosmology blossomed.
While O’Shea’s book is mainly about Poincaré and Perelman, I found this exploration of the history of general relativity fascinating. I started to write it up to put somewhere in my thesis, but in the end I think it’s probably too much of a diversion, and not rigorous enough to withstand scrutiny. On the other hand, I felt that it’s too interesting a story not to share. Eventually, this may become a short series of posts about things that fail to make the final cut of my thesis as I write it over the next few months. I also wholeheartedly recommend O’Shea’s book for a much more in-depth and mathematical discussion of topology and curvature than what I have offered here! It’s a highly readable and accessible book.