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So, I was thinking about white holes and one theory in particular about white holes, and that is that white holes, the theoretical opposite of black holes, are each "Big Bangs". That is to say on the other side of each black hole there is created a new Universe. So, in this particular flavor of white holes theory, the Big Bang is actually the opposite end of a super massive black hole with the mass of our entire Universe.

However, if this were the case, what would explain inflation. By inflation, I mean the period of time, in the early Universe when the Universe expanded exponentially. My question is, is there some phenomenon in black holes that would account for an inflation-style expansion of space? Is there any theories that have been proposed?

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The concept of the white hole normally comes from the maximally extended Schwarzschild solution in the Kruskal-Szekeres coordinates. In this solution, the white hole singularity is spacelike as a moment of time extended in space infinitely in one dimension (see: Is the Schwarzschild singularity stretched in space as a straight line?). From this moment, according to the Schwarzschild metric, space inside the white hole expands as a 3-cylinder (a 3D hypersurface of a 4D cylinder of the spherinder type) until its radius reaches the Schwarzschild radius.

This geometry is very different from the Friedmann geometry of the current $\Lambda\text{CDM}$ cosmological model. In this model, space either is flat (or negatively curved) and always infinite or else it is positively curved and has a zero extension in all three dimensions at the Big Bang.

The only black hole / white hole model potentially compatible with the Friedman geometry is the model in the Oppenheimer-Snyder collapse. In this model, the inner space of the black hole (or the white hole as a time reversal) is assumed to have the Friedmann geometry, the same as in the current cosmological model. In this case, the reasons for inflation would be the same as in the $\Lambda\text{CDM}$ model.

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