# AdS$_4$ and $\mathbb{H}^4$: What is the difference between them?

This figure (source) shows the embedding of 4D hyperbolic space $$\mathbb{H}^4$$ and 4D de Sitter space dS$$_4$$ in 5D Minkowski space $$\mathbb{M}^5$$. $$\mathbb{H}^4$$ is a hyperboloid of two sheets and dS$$_4$$ is a hyperboloid of one sheet. However, I also understand that 4D anti-de Sitter space AdS$$_4$$ can be embedded in $$\mathbb{M}_5$$, and that it is also hyperbolic but simply connected everywhere. I want to know why the author calls the figure on the right de Sitter space but he calls the figure on the left hyperbolic space rather than anti-de Sitter space.

Is AdS$$_4$$ just one of the $$\mathbb{H}^4$$ hyperboloids? Do the two possible hyperbolic embeddings correspond to the $$\{\mp\pm\pm\pm\}$$ metric signature freedom? Does $$\mathbb{H}^4$$ have a Lorentzian signature or is it just the Euclidean version of AdS$$_4$$? If so, how can I reconcile the simply connected property of AdS$$_4$$ with the disconnected property of $$\mathbb{H}^4$$?

$$\text{AdS}_n$$ is a sphere of timelike radius in a space of two timelike and $$n-1$$ spacelike dimensions.* $$\text{AdS}_n$$ itself has one timelike dimension.

For comparison:

$$\mathbb H^n$$ is a sphere of timelike radius in a space of one timelike and $$n$$ spacelike dimensions,** and has zero timelike dimensions itself;

$$\text{dS}_n$$ is a sphere of spacelike radius in a space of one timelike and $$n$$ spacelike dimensions, and has one timelike dimension itself.

* Actually, it's usually taken to be the universal cover of that sphere, since otherwise it's periodic in time, i.e., has closed causal loops.

** Usually with opposite points identified, so that there's only one sheet.

• Thank you. Could you please make a similar statement about de Sitter space: "dS$_n$ is a sphere of ???? radius in a space of ????" Thanks! Mar 19 at 5:17
• @hodopsmith I already did... Mar 19 at 5:31
• I somehow looked over that when I skipped to your asterisks. Oops! Thanks! This is actually a great description for what I'm working on and I am going to paraphrase you in my paper. Mar 19 at 5:39
• @benrg, I am just curious. What are Ricci scalar curvatures for this both spacetimes? Mar 19 at 9:37
• @JanGogolin The Ricci scalar is some small constant times the Gaussian curvature. In terms of the cosmological constant the Gaussian curvature is $Λ/3$ if I'm remembering right. It might also be taken to be $\pm1$ (i.e. a sphere of radius $1$). Mar 20 at 7:16