# Is the limit from $d\mathcal{S}_4$ to Minkowski spacetime smooth?

For a static observer, the boundary of observed $$dS^4$$ spacetime locates on $$r=\ell$$, $$\ell$$ is the de Sitter radius which is reversely proportional to spacetime curvature or say cosmological constant $$\Lambda^{\frac{1}{2}}$$ up to a constant factor.

There is seemingly a physical and smooth limit $$\ell\rightarrow\inf$$ back to flat spacetime, and thus the cosmological boundary becomes the null-infinity.

For a finite $$\ell$$, the bifurcation 2-sphere $$B$$ is located on $$U=V=0$$ without any singularity or ambiguity. One should notice that this 2-sphere is a common boundary of two horizons $$\mathcal{H}^-,\mathcal{H}^+$$.

If one takes the flat limit, seemingly the $$\mathcal{H}^-,\mathcal{H}^+$$ deform to null-infinity $$\mathcal{I}^-,\mathcal{I}^+$$ defined on Minkowski spacetime, and $$B$$ becomes the spatial infinity $$i^0$$. But, as a famous result, $$i^0$$ in the description of Penrose's diagram is singular and is not a common boundary of $$\mathcal{I}^-,\mathcal{I}^+$$. Or exactly, $$\mathcal{I}^-,\mathcal{I}^+$$ have no a common boundary.

Is there something wrong with this limit? or say is this limit smooth?

It depends on what you call a limit of something, and how do you judge convergence (and thus smoothness). Take for example the function $$f(x) = C + 1/(x-a)$$. Now take the limit $$a \to \infty$$. Is $$\tilde{f}(x) = C$$ a "smooth" limit of $$f(x)$$ as $$a \to \infty$$?
In a weak sense yes, for any finite $$x_0$$, $$f(x_0) - \tilde{f}(x_0)$$ goes to zero eventually. Or said differently, we can choose arbitrary finite domains where the two functions converge to each other perfectly well. However, for any finite $$a$$, there will be neighborhoods in $$\mathbb{R}$$ where $$f - \tilde{f}$$ is arbitrarily large. In that sense the two functions never converge to one another on the entire domain $$\mathbb{R}$$.
With de-Sitter space-time the case is similar. Its global structure will never be anything like Minkowski space-time, for any value of $$\Lambda$$. In other words, one has to change causal topology between the two space-times, which will always involve some non-smoothness. However, one can, similarly to the example with the function $$f$$, define the limit by focusing on certain local domains. (In terms of physics, this corresponds e.g. to observers, particles etc. that live for a finite time and probe a finite amount of space.)
This is similar to, say, Schwarzschild space-time as $$M\to0$$. For every finite $$M$$, there will be a black hole horizon, a curvature singularity, the second domain behind the Einstein-Rosen bridge etc. Global convergence of space-times into one another is a very strong condition that you may have a hard time satisfying in most cases.