From: Jeff L Jones (jeff@spoonless.net)
Date: Thu Mar 20 2008 - 20:54:30 MDT
On Thu, Mar 20, 2008 at 6:16 PM, Matt Mahoney <matmahoney@yahoo.com> wrote:
> I had independently derived an order of magnitude estimate of the information
> content of the universe a few years ago using a different approach. I
> estimated the number of possible standing wave patterns in a volume of size R
> as a function of the mass-energy E = mc^2 in that space, which depends on
> Planck's constant h.
I don't know how you estimated it, but if you really got the right
answer it seems like it must have been pure coincidence. The number
of standing wave patterns scales with volume, whereas the
Bekenstein-Bousso bound scales with area. The holographic principle
implies that there are a lot of hidden correlations between different
possible standing wave patterns, because of their effects on spacetime
itself... making any estimate based on standing waves meaningless.
> It is rather interesting that the volume of a bit is about that of a proton or
> neutron, but S does not depend on the properties of any particles.
The area of a bit is 10^-69 m^2, which is roughly 1 Planck length
squared. There is no fixed "volume of a bit", as the number of bits
is not proportional to volume. If you're just taking the volume of
the universe and dividing it by the volume of a baryon, it's not very
meaningful, because you can definitely have a lot more than 1 bit
inside the volume of a baryon. It's just that once you take into
account all of the weird gravitational correlations throughout
spacetime, you find that there are less bits total than you'd expect
from a volume-based count.
> It has always puzzled me why the universe didn't collapse into a black hole
> shortly after the big bang if it had the same mass as today.
A black hole is defined as a region of space from which light cannot
escape. Comparing the Schwarzchild radius to the radius taken up by
some amount of matter/energy is only a rule of thumb, not a hard "law
of physics". In particular, it only applies to static spacetimes,
where space is not expanding. When space is expanding or contracting,
it provides other ways for light to escape. In a universe with no
dark energy, where the density of matter is less than the critical
density, space will expand forever, and there will be no event
horizons (black holes) and light will not get trapped in any
particular region. If the density of matter is greater than the
critical density, then the universe will eventually collapse back onto
itself and "big crunch" which I suppose you could regard as similar to
a black hole but as far as I know it's not usually called that.
In our universe however, which contains dark energy, the situation is
different. There *is* an event horizon about 16-billion light years
away. (Note that it's a bit further away than 13-billion lightyears,
the Hubble horizon, but not as far as the current "radius of the
visible universe" which is 45-billion lightyears... the distance to
the furthest objects we can currently see). This event horizon is
created by the fact that the expansion is accelerating, which traps
light in a region of size 16-billion lightyears (and this region is
currently shrinking). So in a sense, we *are* in a black hole.
Although it's again usually not said that way.
> It also puzzles
> me that a black hole is not symmetric with respect to time (stuff goes in but
> not out) even though Einstein's general field equations are. Shouldn't there
> be another solution with the opposite sign, one where stuff goes out but not
> in, like an expanding universe? In that case, dark energy is just gravity?
Yup... good intuition. There is such a solution, and the technical
term for it is a "white hole". If you interpolate between a black
hole and a white hole, you get a wormhole that appears to dump matter
from one universe into another. It goes in the black hole on one
side, and out the white hole on the other. Unlike black holes, they
are purely theoretical... nobody has ever seen evidence for them in
the sky. But as you say, they do exist mathematically in GR.
However, I would urge caution to anyone getting excited about white
holes or wormholes because they are solutions to classical GR and are
not necessarily consistent with quantum mechanics (ie, quantum
gravity). Most theoretical physicists are very skeptical that they
could really exist.
Jeff
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