If I apply the relations given in http://xaonon.dyndns.org/hawking/hrcalc.js and http://www0.arch.cuhk.edu.hk/~hall/ag/sw/SpinCalc/SpinCalc.htm to the values for the mass of the observable universe (obtained from Wolfram alpha) as M=3.4 10^54 kg, then I get a diameter of 1.1 10^28 m, about 10 times the estimated diameter of the obs. universe at 8.8 10^26 m. Clearly in the same ballpark.
Tinkering with the mass, and excluding dark energy (for lack of 'rest mass'?) reduces the input to 28% of M, and that produces a black hole diameter only 3 times too big.
Now, is this coincidence?
Reading If the observable universe were compressed into a super massive black hole, how big would it be?, I find no other remark addressing this 'coincidence' than Johannes's (quote) The radius of the observable universe equals the Schwarzschild radius of the total mass of the observable universe (unquote).
Is Johannes' statement correct and generally accepted? If so, what is the logic here?

from"How can the Schwarzschild radius of the universe be 13.7 billion light years?":
"... we'd have shown that the Schwarzschild radius of the mass of the observable universe is equal to it's radius. Sadly it doesn't quite work. The dimension R is the current size of the observable universe, which is around 46.6 billion light years, while the size used in Hubble's law, re, is the current apparent size 13.7 billion light years."
So we are to conclude that the 'close fit' (apart from a factor 3 or so) between the Schwarzschild radius and the estimated mass (apart from dark energy) is indeed a numerical coincidence? Please note that John Rennie's argument introduces the 'critical density' and Hubble's Law, while the relation between mass and radius of a black hole allow us to input a mass and obtain a radius without that.

  • 1
    $\begingroup$ No idea what that java code is doing (not bothering to read ~500 lines of code to figure it out), but the Schwarzschild radius of a $10^{22}\,M_\odot$ black hole (converting 1e54 to solar masses) gives me 1.5e26 m, not 1e28. $\endgroup$ – Kyle Kanos Apr 27 '15 at 14:35
  • $\begingroup$ @KyleKanos: using 2e30 kg for solar mass? $\endgroup$ – Wouter M. Apr 27 '15 at 14:52
  • $\begingroup$ Google knows how to convert. $\endgroup$ – Kyle Kanos Apr 27 '15 at 15:13
  • $\begingroup$ Regardless of whether such a coincidence occurs, or not (it doesn't seem to fit well), it would be no different from the angular size of the moon and sun being so close that total solar eclipses are possible. Go a couple hundred million years forth and back and that wouldn't be happening, either. Did somebody "do it"? No. It just happens that we are lucky to live at the "right" time for it to happen. Sometimes you get lucky and when you don't (like most of the times), there is nothing to see to begin with. $\endgroup$ – CuriousOne Apr 28 '15 at 2:44