Think about 2D surfaces. Which ones are the most symmetric? A flat plane is a very nice space: every point is as good as any other (there’s nothing intrinsic to any point to distinguish any point from any other, except arbitrarily), and no direction is particularly special either. A space like that has a lot of symmetries. A sphere also has a lot of symmetries, it also has no directions or points which are distinguished until you declare, “this is my North Pole” arbitrarily. (The earth isn’t a perfect sphere of course and we can use the imperfections as the way we define north and south.) The last type of symmetric space looks like a saddle (like on a horse). It bends one way in one direction and bends the other way in the other direction. An idealized saddle also has no distinguished directions or points.
The analogs of these things in higher dimensions, and where one of the directions is time, are important in general relativity. The analog of the plane is called “flat space” or “Minkowski space”. The analog of the sphere is “de Sitter space”. Finally, the analog of the saddle is “anti-de Sitter space” (usually abbreviated AdS, with a lowercase d). It’s a bit of an odd space in a lot of ways. When you look at what space looks like at any given time, it’s a bit like M.C. Escher’s “Angels and Devils”.
Surprisingly, Anti-de Sitter space is the easiest space to understand quantum aspects of gravity in. That’s because anti-de Sitter space is curved in such a way that the complicated stuff can be neatly separated from the easy stuff. You can start from something you understand well and turn on the complexity piece by piece. Roughly speaking it’s because the gravitational stuff becomes less important as you go farther and farther away from any matter you’re considering, in a way which is even faster than this happens in flat space or de Sitter space. It turns out that we can exactly understand everything in this gravitational theory by mapping the physics one-to-one to a nongravitational model which we understand really well. There’s a lot of evidence that the map works perfectly. This is called the AdS/CFT correspondence. A lot of work goes into testing the correspondence and attempting to prove it, and this is a big research area.
de Sitter space doesn’t have the same desirable properties. Nevertheless there has been great progress in understanding quantum properties of de Sitter in the last year [0]. These results would not have been possible without understanding AdS first.
Flat space quantum gravity remains challenging, although again some progress has been made recently too [1].
[0] arxiv:2110.14670
[1] arxiv:1905.09809 and many others
The analogs of these things in higher dimensions, and where one of the directions is time, are important in general relativity. The analog of the plane is called “flat space” or “Minkowski space”. The analog of the sphere is “de Sitter space”. Finally, the analog of the saddle is “anti-de Sitter space” (usually abbreviated AdS, with a lowercase d). It’s a bit of an odd space in a lot of ways. When you look at what space looks like at any given time, it’s a bit like M.C. Escher’s “Angels and Devils”.
Surprisingly, Anti-de Sitter space is the easiest space to understand quantum aspects of gravity in. That’s because anti-de Sitter space is curved in such a way that the complicated stuff can be neatly separated from the easy stuff. You can start from something you understand well and turn on the complexity piece by piece. Roughly speaking it’s because the gravitational stuff becomes less important as you go farther and farther away from any matter you’re considering, in a way which is even faster than this happens in flat space or de Sitter space. It turns out that we can exactly understand everything in this gravitational theory by mapping the physics one-to-one to a nongravitational model which we understand really well. There’s a lot of evidence that the map works perfectly. This is called the AdS/CFT correspondence. A lot of work goes into testing the correspondence and attempting to prove it, and this is a big research area.
de Sitter space doesn’t have the same desirable properties. Nevertheless there has been great progress in understanding quantum properties of de Sitter in the last year [0]. These results would not have been possible without understanding AdS first.
Flat space quantum gravity remains challenging, although again some progress has been made recently too [1].
[0] arxiv:2110.14670
[1] arxiv:1905.09809 and many others