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.



A/Prof Russell Standish                  Phone 0425 253119 (mobile)
UNSW SYDNEY 2052                         [EMAIL PROTECTED]
Australia                                http://www.hpcoders.com.au

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