When Einstein Proposed a Limit to the Universe

The idea of a finite universe isn't a very comforting thought. For something to be countable—finite—it must have an end, which is where there are no longer any new things to be counted. And the only real end we deal with is death, where life stops counting. Everything else keeps going on and on and on because time is counting, and counting is the same right here now as it is in the farthest, darkest corner of the universe.

Einstein proposed the idea of a measurable universe in 1921, via an address titled "Geometry and Expansion" delivered at the Prussian Academy of Sciences in Berlin. It consists of a lot of musing, but a rigorous, formal kind of musing about the end of everything.

Einstein was wondering about mathematics and reality and when, if ever, they should properly meet. The problem is that math deals with idealized forms—perfect circles, undeviating straight lines, points. That's fine because everything seems to be made of such stuff (straight lines and neat angles), but not quite. These are formal constructions, "as void of all content of intuition or experience," Einstein declared. The world is messier.

It's the formalism, the consistency of these concepts (or axioms) that separates math from the other sciences, e.g. why mathematics is built on proofs and the rest of science is built on theories. Mathematical logic can guarantee the truth of something through inductive deductive reasoning, while science is limited to the real world and evidence-based assurances. Science is deductive inductive.

Einstein had smashed head-on into this distinction in the process of formulating his theory of relativity. This collision has to do with Euclidean geometry, which is the geometry of straight lines demarcating finite, real spaces in two and three dimensions. If you were to geometrically (on paper) represent whatever is currently in front of you, the shapes and angles between those shapes, the result would be an example of Euclidean geometry. It's fine for most things, and the fourth dimension is hard to draw.

To be a bit more specific, Euclidean geometry is geometry that follows a few simple but firm rules: A straight line segment can be drawn between two points; a straight line segment can be extended indefinitely as a line; a circle can be drawn such that the line segment is its radius. Simple, obvious stuff. It's how we build things.

But it's not the geometry of our universe. In reality, it's the sort of formal, idealized math we were talking about before, though it usually does just find as an approximation. The difference has to do with Einstein's relativity, which demonstrated that space-time itself can be deformed by gravity and bodies in motion. And, as space-time warps, no longer can we properly draw a straight line between points. We think we have, sure, but deep down there's no longer any such thing as straightness. All is warped.

Well, almost. There is some straightness after all, in a sense. We have the thing that forces space-time to warp in the first place, which is the constant speed of light propagating through empty space. The c in E = mc^2.

"If two ideal clocks are going at the same rate at any time and at any place, being then in immediate proximity to each other, they will always go at the same rate, no matter where and when they are again compared with each other at one place," Einstein offered. "If this law were not valid for real clocks, the proper frequencies for the separate atoms of the same chemical element would not be in such exact agreement as experience demonstrates."

So all hope is not lost. And the apparent failure of Euclidean geometry to describe a relativistic universe doesn't mean that no geometry is capable of describing it. For this we have Riemannian geometry, which enabled Einstein to formulate relativity in the first place. Simply, it replaces the neat straight lines and points of Euclid with a smooth surface, a manifold that can be warped and deformed but still described mathematically, albeit not nearly with so much ease.

Finally, this is where we get to the finite, measurable universe. Riemannian geometry—the geometry of an elastic, relativistic space-time—should transform into Euclidean geometry as things get smaller and smaller, Einstein suggested. The result would be a complete "practical geometry."

"The question whether the universe is spatially finite or not seems to me decidedly a pregnant question in the sense of practical geometry," he said.

In Euclidean geometry, we imagine an infinite universe as an unbounded one. That is, if we take some inexhaustible supply of small wooden cubes and keep stacking them and stacking them, there is never an end to the space in which we might stack. Our tower could grow forever and ever.

This isn't very satisfying, however, and Einstein offered something much better: a finite, yet unbounded universe. This view or interpretation is made possible by a spherical, three-dimensional geometry, the space-time curve of all mass together. The finite, unbounded universe as suggested by Einstein is a sphere, which is a horrible thing to do to a brain.

"Now this is the place where the reader's imagination boggles," Einstein declared. "'Nobody can imagine this thing,' he cries indignantly. 'It can be said, but cannot be thought. I can represent to myself a spherical surface well enough, but nothing analogous to it in three dimensions.'" The reader has a point.

Here is Einstein's example:

In the adjoining figure let K be the spherical surface, touched at S by a plane, E, which, for facility of presentation, is shown in the drawing as a bounded surface. Let L be a disc on the spherical surface. Now let us imagine that at the point N of the spherical surface, diametrically opposite to S, there is a luminous point, throwing a shadow L' of the disc L upon the plane E. Every point on the sphere has its shadow on the plane. If the disc on the sphere K is moved, its shadow L' on the plane E also moves. When the disc L is at S, it almost exactly coincides with its shadow. If it moves on the spherical surface away from S upwards, the disc shadow L' on the plane also moves away from S on the plane outwards, growing bigger and bigger. As the disc L approaches the luminous point N, the shadow moves off to infinity, and becomes infinitely great.

Can you sort of see it? How the disc as it moves up further and further on the sphere is forced to project farther and farther and larger and larger on the plane until, eventually, its projection is just shooting off toward infinity. So maybe just imagine our universe as that plane, which some parts stretching out forever in the intuitively satisfying sense of "forever," but what's being illustrated is that forever-ness can originate from a finite space.

So: a finite universe means that we can keep counting and counting forever, but we'll run out of things to count. Increasing space, increasing nothingness. In Einstein's words: "The smaller that mean density, the greater is the volume of universal space." Have fun with that.