As discussed in this question, as far back as the 1960s it was suggested that the Schwarzschild metric can be smoothly joined to a de Sitter metric. For example the idea has been used by Lee Smolin in his ideas idea for black hole evolution.
If the suggestion is true it means that matter falling into a black hole would emerge into a de Sitter spacetime arguably comparable to our own.
So has the universe we live in started as a black hole of this type?
Is it possible to prove that the big bang was a black hole of this type?
Answer
Although there are suggestions that a black hole could lead into another expanding universe you should regard these as highly speculative. Although the metrics can be stitched together it requires a contribution from currently hypothetical quantum gravity effects. Until we have a working theory of quantum gravity there is no way to comment on how likely the scenario is.
To date the constructions link the Schwarzschild metric to a de Sitter universe, and our universe is not de Sitter (though in the future it will tend asymptotically towards de Sitter geometry). So on the face of it our universe does not look as though it is the far side of a black hole. Having said this, the effect of matter wasn't included in the original models as it makes the calculation too hard. It is conceivable that matter falling into the black hole would emerge to produce a universe that looks like ours. At the current state of the art it just isn't possible to comment definitively.
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