On Fri, Nov 02, 2007 at 12:20:35PM -0700, George Levy wrote:
> Russel,
>
> We are trying to related the expansion of the universe to decreasing
> measure. You have presented the interesting equation:
>
> H = C + S
>
> Let's try to assign some numbers.
> 1) Recently an article
> <http://space.newscientist.com/article/dn12853-black-holes-may-harbour-their-own-universes.html>
> appeared in New Scientist stating that we may be living "inside" a black
> hole, with the event horizon being located at the limit of what we can
> observe ie the radius of the current observable universe.
> 2) Stephen Hawking
> <http://en.wikipedia.org/wiki/Black_hole_thermodynamics> showed that the
> entropy of a black hole is proportional to its surface area.
>
> S_{BH} = \frac{kA}{4l_{\mathrm{P}}^2}
>
> where where k is Boltzmann's constant
> <http://en.wikipedia.org/wiki/Boltzmann%27s_constant>, and
> l_{\mathrm{P}}=\sqrt{G\hbar / c^3} is the Planck length
> <http://en.wikipedia.org/wiki/Planck_length>.
>
> Thus we can say that a change in the Universe's radius corresponds to a
> change in entropy dS. Therefore, dS/dt is proportional to dA/dt and to
> 8PR(dR/dt) R being the radius of the Universe and P = Pi. Let's assume
> that dR/dt = c
> Therefore
>
> dS/dt = (k/4 L^2) 8PRc = 2kPRc/ L^2
>
> Since Hubble constant <http://en.wikipedia.org/wiki/Hubble%27s_law> is
> 71 ± 4 (km <http://en.wikipedia.org/wiki/Kilometer>/s
> <http://en.wikipedia.org/wiki/Second>)/Mpc
> <http://en.wikipedia.org/wiki/Megaparsec>
>
> which gives a size of the Universe
> <http://en.wikipedia.org/wiki/Observable_universe> from the Earth to the
> edge of the visible universe. Thus R = 46.5 billion light-years in any
> direction; this is the comoving radius
> <http://en.wikipedia.org/wiki/Radius> of the visible universe. (Not the
> same as the age of the Universe because of Relativity considerations)
>
> Now I have trouble relating these facts to your equation H = C + S or
> maybe to the differential version dH = dC + dS. What do you think? Can
> we push this further?
>
> George
>
I think that the formula you have above for S_{BH} is the value that
should be taken for the H above. It is the maximum value that entropy
can take for a volume the size of the universe.
The internal observed entropy S, will of course, be much lower. I
don't have a formula for it off-hand, but it probably involves the
microwave background temperature.
Cheers
--
----------------------------------------------------------------------------
A/Prof Russell Standish Phone 0425 253119 (mobile)
Mathematics
UNSW SYDNEY 2052 hpcoder.domain.name.hidden
Australia http://www.hpcoders.com.au
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Received on Sat Nov 03 2007 - 02:56:00 PDT