# Re: Bekenstein bound (Re: A model of consciousness)

From: Matt Mahoney (matmahoney@yahoo.com)
Date: Fri Mar 21 2008 - 08:34:41 MDT

--- Jeff L Jones <jeff@spoonless.net> wrote:

> 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.

Suppose you have some photons trapped in a torus of circumference R or a tube
of length R/2 with reflecting ends. Each photon must have a frequency which
is an integral multiple of c/R, or an energy which is a multiple of hc/R or a
mass which is an integral multiple of h/cR. If the total mass is bounded by
m and you count the number of possible states and take its log, you get a
messy calculation whose result is about S = mcR/h (ignoring small constants).
If you increase this to 3 dimensions (as a mirrored cube or hypertorus) the
result is multiplied by 3, which I'll ignore because it is unimportant.
Adding particles with mass also does not substantially change the result. A
massive particle can have more states (because it moves slower) but you can
have fewer of them. Of course the universe is not a cube or hypertorus, but
this is only an order of magnitude estimate. Also, the universe is expanding
so it doesn't have standing waves. If you prefer, the number of states is
bounded by Heisenberg's uncertainty principle over time period T, and the
result is the same.

Now if you let R = cT be the "size" of the universe of age T, and set R =
2Gm/c^2 to the Schwarzchild radius and solve for the mass of the universe m,
you get S ~ T^2 c^5/hG ~ 10^122, a unitless number.

> > 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.

No, I divide the volume of the universe by S and get the volume of a baryon.
S depends only on T, c, h, and G, not on the properties of baryons or any
other particles.

>
> > 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.

Dark energy doesn't explain why the expansion of the universe is accelerating.
It is just an observation. It seems to me this behavior should be explainable
in the framework of Einstein's field equations.

Likewise, inflation does not explain the uniformity of the cosmic microwave
background. To me, it is a kludge. Maybe the big bang didn't start with a
single point. I mean, how do we know it did.

-- Matt Mahoney, matmahoney@yahoo.com

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