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

From: Jeff L Jones (jeff@spoonless.net)
Date: Fri Mar 21 2008 - 11:32:10 MDT

On Fri, Mar 21, 2008 at 7:34 AM, Matt Mahoney <matmahoney@yahoo.com> wrote:
> 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.

Ok, I think I see why you got the right answer. It's because of two things.

First, you're using the formula for the entropy of a black hole. And
this is where the Bekenstein bound comes from. It's just the Area in
Planck units divided by 4. The reason they are the same is because a
black hole is the maximum amount of stuff you can store in a region of
space... it's the only thing that saturates the bound. So that part
makes sense.

Second, you're figuring out what the Scharwzchild radius is if you put
a whole bunch of matter in a volume that is the size of our universe.
It looks like you're using the Hubble radius to estimate that. I
don't think that's really the right radius to use, but it happens to
be within on order of magnitude away from the deSitter horizon... and
a related fact is that the amount of dark energy happens to be roughy
the same order of magnitude as the total amount of matter in the
universe... so that if you calculate the Schwarzchild radius for that
amount of matter, you get back roughly the size of the universe again.
Sometimes this is called the "coincidence problem". As far as I
know, the only known explanation for this is the anthropic
principle... the amount of dark energy and dark matter have to be
roughly equal in order for life to exist. And that only happens at a
special time during the universe's evolution. So we happen to be at
that special time.

> 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.

Same thing. You're dividing the volume of the universe by S to get
the volume of a baryon (or equivalently, dividing the volume of the
universe by the volume of a baryon to get S). Either way, this is not
meaningful since it's based on volume and not area. You can fit a
whole lot of bits inside the volume of a baryon (about 10^30 bits I
think). So that's a lot more than 1. It's just that you can't fit
that number of bits in *every* volume that size at once, because they
will have correlations between them due to their gravitational
interactions.

> 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.

As you say, the definition of "dark energy" is just "whatever is
making the universe accelerate". In the Standard Model of Cosmology
(aka Lambda-CDM), this is Einstein's cosmological constant (also known
as "vacuum energy"). Yes, it *is* a part of Einstein's field
equations. It's just that this constant is very nearly zero (in
Planck units, it's 10^-120) so for a long time people thought it was
zero and didn't bother to include it in some introductory GR courses.
But it is a part of GR... this has been discussed on this list before.

Is it possible that the dark energy is really something more than just
Einstein's equations? Perhaps, but I doubt it. The simplest
explanation is that it's just Einstein's cosmological constant, which
so far fits all observations and perfectly explains the acceleration.

> 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.

Inflation certainly explains the uniformity of the CMB... but if you
want to criticize it, you could argue that it also introduces a number
of problems of its own. Nevertheless, it's by far the best theory we
have and has made some impressive predictions which have come true and
show that it's on the right track. As far as I know, nobody thinks
that the big bang started at a single point. That's not a part of the
big bang theory.

Jeff

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