This post is the first half of a discussion about Anti de Sitter space / Conformal Field Theory correspondence. You can read more backstory about why we want to know about AdS / CFT here.
The first side of our search covers spaces. No, not the cosmic emptiness outside the bounds of our atmosphere. We're talking about the geometry of the natural world. It's difficult to generalize because we're really only used to thinking about two-dimensional geometry (flat shapes on a flat surface) but different, more varied geometries can be useful in physics as well. You'll see more clearly once we get started.
In the buildup to fancy terms that involve new concepts we'll inevitably run into fancy terms that describe old concepts, i.e. jargon for something we already know. Euclidean Space fills that gap because we already know the rules for Euclidean spaces, only no one told us that it fits under the Euclidean umbrella. Conveniently, 'the Euclidean umbrella' is a metaphor and not mathematical jargon.
If we recall from our days of grade school geometry, Euclid was concerned with the basics. Lines between points, angles in shapes, congruency and whatnot. Euclidean space covers anything that follows Euclid's five postulates, which state a bunch of stuff that boils down to this: draw whatever we want on a flat piece of paper and we'll be drawing in Euclidean space. Angles of triangles add up to 180, lines extend infinitely in any direction and the parallel postulate is upheld (needless to say, the parallel postulate is a more complex topic than we require for this exercise - trust me that we won't violate this postulate drawing on an infinitely large piece of paper with no creases, folds or curved surfaces).
There. We already know how to define Euclidean space.
Elliptical space is also familiar, although the rules for Elliptical spaces probably aren't something we learned in grade school. Elliptical space is the first space we'll talk about that is part of a group of spaces often referred to as non-Euclidean spaces. Any space that isn't specifically Euclidean space is, of course, non-Euclidean. Put simply, Elliptical space describes the surfaces of a sphere. The easiest way to think about this is to imagine the geometry of the Earth (a slight oblate spheroid, but close enough). Some of the rules are different but we're comfortable enough with spheres that there won't be any surprises.
Firstly, there are no such things are parallel lines in Elliptical spaces. Think about longitudinal lines; they may appear parallel at the equator but all longitudinal lines converge at the north and south poles. We might think that latitudinal lines violate this rule, but they too intersect - with themselves! Additionally, triangles on the surface of a sphere always have angles which add up to greater than 180??. This leads us to our first real curiosity: the Earth is so huge that a good approximation for a description of space on the scale of a human is Euclidean! That is to say, drawing a triangle on in the dirt in your backyard (i.e. on the surface of a very large sphere), it will appear that all three angles add up to 180??, but they're actually slightly more than 180?? with accurate enough measurements. We'll run into this feature again, so keep it in mind.
The usefulness of Elliptical space is, of course, navigating the globe. This is why the path of an airplane traversing a large section of the globe appears curved, like flights from Emirates on FlightAware. The Euclidean representation of an airplane path looks like it's not taking the shortest path between two points because the airplane seems to go out of its way to reach the intended destination. Unexpected plane diversions aside, the airplane is actually traveling in a straight line along the surface of a sphere. That path is, in fact, the shortest distance between the two points.
That wasn't so hard. Now when we take a long plane trip and plot it on a map, we can abuse the ear of our row-mate by explaining that our travels are best described by Elliptical space rather than Euclidean space!
If we can understand that Euclidean and Elliptical space are geometries on a plane and a sphere, respectively, than it seems natural that we can construct geometries from other basic shapes. Though most people without a couple calculus courses under their belt will suggest cubic- and pyramidal-geometries, the mathematically logical shape to define is the hyperbolic paraboloid. I imagine we're not as familiar with the hyperbolic paraboloid, a saddle-shaped surface which haunts many a freshman calculus student, so we won't spend much time on it, except for the important points:
Conceptually, hyperbolic space offers a foil to the features of the Elliptical space. Whereas a triangle on the surface of a sphere will have angles adding up to greater than 180??, the hyperbolic space offers triangles which always sum to less than 180??. It seems like the only purpose to the study of hyperbolic geometry is to expand your own personal understanding of non-Euclidean geometries, although I'd be curious to hear if anyone knows of other uses. I've heard of some investigation into the idea that the universe is "shaped" like a hyperbolic paraboloid, but I'm fairly sure there's no generally agreed upon shape to the universe currently. You can read further into ideas of how the universe might be shaped.
Minkowski, de Sitter and Anti de Sitter Spaces
Similarly to how the surface of the Earth looks flat on the human-scale but is clearly a sphere on larger scale, each of these three space descriptions (Euclidean, Elliptical and Hyperbolic) is a really great description of space at small speeds. So why do we need any other description of space? Well, thanks to Einstein's special relativity, we know that as we push tiny particles to higher and higher speeds, they approach the speed of light but never reach it. Considering that calculating features of very, very tiny particles pushed to very, very fast speeds are one of the main uses for space geometries, incorporating special relativity into these space descriptions is a necessity to getting accurate calculations.
Our last three spaces are lumped together because we've already covered their likeness in this document. Minkowski space, quickly conceived by Hermann Minkowski after Einstein's early-1900s discoveries, is simply Euclidean space that obeys the tenants of special relativity. Likewise, de Sitter space is, in physics parlance, the Lorentzian analogue of Elliptical space. However, physicists are not always known for their creative naming schemes (with some exceptions), and thus Anti de Sitter space, Hyperbolic space that behaves as special relativity says it should, was unfortunately coined. We noted the curious "oppositeness" between Elliptical and Hyperbolic space before, so the "Anti" makes logical sense if you're familiar with the functions they perform.
I think this gives us a good idea of what Anti de Sitter space is, plus lots of knowledge about spaces in general and how we can use spaces in physics. We are only halfway to our goal, however. Next week we'll attempt to discover what "Conformal Field Theory" is and finish up our discussion of how it ties back into Anti de Sitter space as AdS/CFT Correspondence. As always, keep in touch via the comments below if there are questions or just... comments.
Other questions we didn't answer:
- Are there any other uses to hyperbolic geometry?
- For all the amazing documentation on hyperbolic space, there is little information on toroidal space. Why?
- What is the shape of the universe?
Ready? Move on to the next step: What is a Conformal Field Theory?
The specifics of special relativity are beyond the scope of this particular exercise, but there are excellent descriptions for the beginner across the web.
Lorentz mathematically described the effects of special relativity before Einstein put all the pieces together, but this could just as well be called the "relativistic analog" of Elliptical space.