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

Question

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?

Answer 1

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.