general relativity - How does the friedmon solution to Einstein's equations resolve paradox of bounded infinities?

This article talks about a potential explanation of dark matter based on something called the "friedmon." I have no interest in the dark matter question, but the article has made me curious about this friedmon thing. Googling "friedmon" turns up very little.

The article says the friedmon is a solution to Einstein's equations of GR wherein an infinite expanse (field or space-time, I guess you would call it) sits as an object within another infinite expanse. Well, let me just quote the article:

"To an external observer, a friedmon looks like a micro blackhole with an electric charge the same as that of the electron. However, the friedmon's interior can be macroscopically large — up to the size of the known universe."

I believe the notion of an infinite universe sitting as a finite object in another infinite field is also central to the multiverse theory.

I'm just curious as to how you express this idea mathematically. At face value there appears to be a contradiction in the idea of an infinite universe which is also a finite object situated in some broader expanse. I would also appreciate if someone can point out a paper or something which shows the details of the friedmon solution to Einstein's equations of GR.

Answer

Some Googling later I have an answer for you. The relevant papers are:

• M. A.Markov, “Elementary particles of maximally large masses (Quarks and Maximons),” Soviet Physics (Journal of Experimental and Theoretical Physics), vol. 24, p. 584, 1967.


• V. I. Man’ko and M. A. Markov, “Properties of fridmons and the early stage of evolution of the universe,” Theoretical and Mathematical Physics, vol. 17, no. 2, pp. 1060–1063, 1973.



• M. A. Markov, “The maximon and minimon in light of a possible formulation of the concept of an ‘elementary particle’,” JETP Letters, vol. 45, pp. 141–144, 1987.

Start with the usual Schwarzschild metric describing a static black hole. As you fall in the curvature (Kretschmann scalar) increases steadily and goes to infinity at $r = 0$ i.e. at the singularity. Starting with (I think) Bardeen in 1968 there have been various proposals that quantum effects replace the singularity by a minimum radius around the Planck length, and that at this metric is smoothly joined to a de Sitter metric i.e. an expanding universe.

So if you fell into a black hole (and you were smaller than the Planck length) you'd pass through a point of maximum curvature in the black hole and emerge in a region of spacetime that looked like an expanding Friedmann universe - hence the epithet Friedmon. This is also the basic principle in Smolin's idea of black hole evolution as described in The Life of the Cosmos, and it's not dissimilar to Wheeler's Bag of Gold spacetimes (there's a description of Bag of Gold spacetimes in this PDF).