From: Matt Mahoney (firstname.lastname@example.org)
Date: Sun Aug 26 2007 - 21:15:38 MDT
--- Jeff L Jones <email@example.com> wrote:
> I get:
> mass of sun = 2*10^30 kg
> total energy in observable universe ~ 10^55 kg ~ 10^25 suns
> radius of observable universe = 78 billion lightyears = 7*10^23 km
> Schwarzschild radius of observable universe = 3*10^25 km
> So it looks like, just based on the usual linear scaling of
> Schwarzshild radius with mass, one would think our universe would be
> in a black hole with a radius about 42 times the current observable
> horizon. Interesting that they are so close (less than 2 orders of
> magnitudes away)... I'm not sure why.
It is actually much closer than that. According to
http://en.wikipedia.org/wiki/Universe#Composition the critical density of the
universe is 9.9E-27 kg/m^3 (consisting of 4% luminous matter, 23% dark matter,
and 73% dark energy aka cosmological constant).
According to http://en.wikipedia.org/wiki/Observable_universe the observable
radius of the universe is 4.647E10 lightyears. (The observable universe is
the size now of that part of the universe from which light has had time to
reach us since the big bang 13.7 billion years ago, accounting for spacetime
curvature). The volume is thus 3.56E80 m^3. (Note: I corrected a math error
on this page). Therefore the mass of the observable universe is m = 3.52E54
The Schwartzchild radius ( http://en.wikipedia.org/wiki/Schwarzschild_radius )
is 2Gm/c^2 = 551 billion lightyears, or 11.3 times the radius. But if you
take the *apparent* size of the universe (radius 13.7 billion lightyears), a
similar calculation yields a Schwartzchild radius of 13.5 billion lightyears.
I guess that this is not a coincidence, but rather falls out of the
calculation of critical density (although I haven't done the math). If you
neglect dark energy (allowing the universe to expand forever), the
Schwartzchild radius would only be about 1/4 of this value.
Nevertheless it is interesting that an expanding universe appears "dual" to a
black hole. The edge is the event horizon. Matter accelerates toward it.
Objects and information can only cross it one way. From our view, these
objects cannot be seen to cross, but rather appear to slow down and are red
shifted until they fade from view. When a black hole absorbs matter, entropy
decreases. Yet in our universe, entropy can only increase. Entropy breaks
time symmetry. It defines the direction of time. General relativity and the
Schwartzchild equation derivation assume time symmetry: you can replace t with
-t and get the same solution.
-- Matt Mahoney, firstname.lastname@example.org
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