'twas perfectly readable to me, since it was bog-standard LaTeX
notation which is a defacto standard for mathematical notation in
email.
Until someone figures out a way of getting all email clients to read
and write mathML (which will probably be never), this is as good as it
gets.
Cheers
On Mon, Nov 05, 2007 at 03:34:52PM -0800, George Levy wrote:
Sorry the nice equation formats did not make it past the server. Anyone
interested in the equations can find them at the associated wiki links.
George
Russell Standish wrote:
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 [EMAIL PROTECTED]
Australiahttp://www.hpcoders.com.au
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