'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] Australia http://www.hpcoders.com.au ---------------------------------------------------------------------------- --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
