Topology of (Anti) de Sitter space

[Slereah]

Almost every book that I can think of derives the de Sitter spacetime as a hyperboloid in a (4+1) D spacetime, with the topology [itex]ℝ \times S^3[/itex], but I'm having trouble finding something justifying it. "Exact solutions of Einstein's Field Equations" gives what seems like an [itex]ℝ^{4}[/itex] version :

[itex]ds^2 = \frac{dt^2 - dx^2 - dy^2 - dz^2}{1+\frac{K}{4}(t^2 - x^2 - y^2 - z^2}[/itex]

Which makes sense, since this is basically the Lorentzian version of the stereographic projection. My guess is that this suffers from geodesic incompleteness for the same reason that the stereographic projection does.

The most natural choice would be [itex]S^{4}[/itex] of course, but beyond the obvious CTC problems, it doesn't even admit a Lorentzian metric.

So why [itex]ℝ \times S^3[/itex] in particular? Are there any other nicely behaved topologies for that metric, other than whatever quotient manifolds of it, I guess?

 

[Ambitwistor]

it is important to always question and seek justification for theories and concepts. In this case, the de Sitter spacetime is commonly derived as a hyperboloid in a (4+1) D spacetime with the topology ℝ \times S^3. However, you are having trouble finding something that justifies this choice and are questioning if there are other possibilities.

First of all, it is worth noting that the choice of ℝ \times S^3 as the topology for de Sitter spacetime is not arbitrary. It is based on the symmetry and properties of the spacetime and is supported by mathematical and physical evidence.

One way to justify this choice is by looking at the symmetries of de Sitter spacetime. It is a maximally symmetric space, meaning that it has the same symmetries at every point. In this case, the symmetries are given by the de Sitter group, which is isomorphic to SO(4,1) (the group of rotations and boosts in a (4+1) D spacetime). This group naturally acts on ℝ \times S^3, making it a suitable choice for the topology of de Sitter spacetime.

Another way to justify this choice is by looking at the physical properties of de Sitter spacetime. It is a vacuum solution of Einstein's field equations, meaning that it describes a universe with no matter or energy. In this case, the Ricci scalar (a measure of the curvature of spacetime) is constant and positive, which is consistent with the topology of ℝ \times S^3.

It is also worth mentioning that the choice of ℝ \times S^3 is not the only option for the topology of de Sitter spacetime. As you mentioned, the most natural choice would be S^4, but this leads to causality violation (CTCs) and does not admit a Lorentzian metric. Other possibilities include quotients of ℝ \times S^3, such as S^3/\Gamma, where \Gamma is a discrete group of isometries. These quotients maintain the symmetries and properties of de Sitter spacetime, but have a different topology.

In conclusion, the choice of ℝ \times S^3 as the topology for de Sitter spacetime is supported by its symmetries and physical properties. While there are other possibilities, they either violate causality or do not maintain the desired properties. As