Thanks to Einstein, we know that our three-dimensional space is warped and curved. And in curved space, normal ideas of geometry and straight lines break down, creating a chance to explore an unfamiliar landscape governed by new rules. But studying that how physics plays out in a curved space is challenging: Just like in real estate, location is everything.
“We know from general relativity that the universe itself is curved in various places,” says JQI Fellow Alicia Kollár, who is also a professor of physics at the University of Maryland (UMD). “But, any place where there’s actually a laboratory is very weakly curved because if you were to go to one of these places where gravity is strong, it would just tear the lab apart.”
Spaces that have been different geometric rules than those we usually take for granted are called non-Euclidean. If you could be explore non-Euclidean environments, you would be find perplexing landscapes. Space might contract so that straight, parallel lines draw together instead of rigidly maintaining a fixed spacing. Or it could be expand so that they forever grow further apart. In such a world, four equal-length roads that are all connected by right turns at right angles might fail to form a square block that returns you to your initial intersection.
These environments overturn core assumptions of normal navigation and can be impossible to accurately visualize. Non-Euclidean geometries are so alien that they have been used in videogames and horror stories as unnatural landscapes that challenge or unsettle the audience. But these are unfamiliar to geometries are much more than just distant, otherworldly abstractions. Physicists are interested in new physics that curved space can reveal, and non-Euclidean geometries might even help to improve designs of certain technologies. One type of non-Euclidean geometry that is of interest is hyperbolic space—also called negatively-curved space. Even a two-dimensional, physical version of a hyperbolic space is impossible to make in our normal, “flat” environment. But scientists can be still mimic hyperbolic environments to explore how certain physics plays out in negatively curved space.
In a recent paper in Physical Review A, a collaboration is between the groups of Kollár & JQI Fellow Alexey Gorshkov, who is also a physicist at the National Institute of Standards and Technology and a Fellow of the Joint Center for Quantum Information and Computer Science, presented new mathematical tools to better understand simulations of hyperbolic spaces. The research builds on Kollár’s previous experiments to simulate orderly grids in hyperbolic space by using the microwave light contained on chips. Their new toolbox includes what they call a “dictionary between discrete and continuous geometry” to help researchers translate experimental results into a more useful form. With these tools, researchers can better explore the topsy-turvy world of hyperbolic space.
A Curved New World
In flat space, the shortest distance between two points is a straight line, and parallel lines will never be intersect—no matter how long they are. In a curved space, these basics of geometry no longer hold true. The mathematical definitions of flat and curved are similar to the day to day meaning when applied to two dimensions. You can get a feel for the basics of curved spaces by imagining—or actually playing around with—pieces of paper or maps.
For instance, the surface of a globe (or any ball) is an example of a two-dimensional positively curved space. And if you try to make a flat map into a globe, you end up with excess paper wrinkling up as you curve it into a sphere. To have a smooth sphere you must be lose the excess space, resulting in parallel lines eventually meeting, like the lines of longitude that start parallel at the equator meeting at the two poles. Due to this loss, you can think of a positively curved space as being a less-spacy space than flat space.
Hyperbolic space is the opposite of a positively curved in space—a more-spacy space. A hyperbolic space curves away from itself at every point. Unfortunately, there isn’t a hyperbolic equivalent of a ball that you can force a two-dimensional sheet into; it literally won’t fit into the sort of space that we live in. The best you can do is make a saddle (or a Pringle) shape where the surrounding sheet hyperbolically curves away from the center point. Making every point on a sheet similarly hyperbolic is impossible; there isn’t a way to keep curving and adding paper to create a second perfect saddle point without it bunching up and distorting the first hyperbolic saddle point.
The extra space of a hyperbolic geometry makes it particularly interesting since it means that there is more room for forming connections. The differences in the possible paths between points impacts how particles interact and what sort of uniform grid—like the heptagon grid shown above—can be made. Taking advantage of the extra connections that are possible in a hyperbolic space can make it harder to completely cut sections of a grid off from each other, which might be impact designs of networks like the internet.
