It has been said that the Big Bang started from a singularity. Think about a balloon radially growing over time. Fix a time $t_0, t_1 > 0$, and let $M_0, M_1$ be two balloons at time $t_0, t_1$ respectively. I can find a two-parameter diffeomorphism $\phi(t_0, t_1): M_0 \rightarrow M_1$.
However, I cannot find a diffeomorphism if I let $t_0 = 0$ and $t_1 > 0$, i.e. $\phi(0, t_1): \{*\} \rightarrow M_1$. In what sense should I interpret a homotopy between initial state (big bang) and final state (the current universe)? Is it even true that the Big Bang started from a singularity?
Best Answer
The singularity at the start of the universe in the Big Bang model is not supposed to be understood as part of the smooth manifold of spacetime, precisely for this reason.
The time function on spacetime does not actually assign a "point" to $t = 0$. It's undefined (otherwise spacetime wouldn't be a Lorentzian manifold), and the same is true if you take an FLRW universe and try to keep the initial spatial slice 3d - since the scale factor goes to zero, the manifold is not Lorentzian there.
If you want to model the initial singularity of the Big Bang as part of spacetime, you need to consider more general models of spacetime than a Lorentzian manifold.