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
Russell Standish wrote:
>On Wed, Oct 31, 2007 at 05:11:01PM -0700, George Levy wrote:
>
>
>>Could we relate the expansion of the universe to the decrease in
>>measure of a given observer? High measure corresponds to a small
>>universe and conversely, low measure to a large one. For the observer
>>the decrease in his measure would be caused by all the possible mode of
>>decay of all the nuclear particles necessary for his consciousness.
>>Corresponding to this decrease, the radius of the observable universe
>>increases to make the universe less likely.
>>
>>This would provide an experimental way to measure absolute measure.
>>
>>I am not a proponent of ASSA, rather I believe in RSSA and in a
>>cosmological principle for measure: that measure is independent of when
>>or where the observer makes an observation. However, I thought that
>>tying cosmic expansion to measure may be an interesting avenue of inquiry.
>>
>>George Levy
>>
>>
>>
>
>There is a relationship, though perhaps not quite what you think. The
>measure of an OM will be 2^{-C_O}, where C_O is the amount of
>information about the universe you know at that point in time
>(measured in bits). The physical complexity C of the universe at a point
>in time is in some sense the limit of all that is possible to know
>about the universe, ie C_O <= C.
>
>C is related to the size of the universe by the equation H = C + S,
>where S is the entropy of the universe (measured in bits), and H is
>the maximum possible entropy that would pertain if the universe were
>in equilibrium. H is a monotonically increasing function of the size
>of the universe - something like propertional to the volume (or
>similar - I forget the details). S is also an increasing function (due
>to the second law), but doesn't increase as fast as H. Consequently C
>increases as a function of universe age, and so C_O can be larger now
>than earlier in the universe, implying smaller OM measures.
>
>However, it remains to be seen whether the anthropic reasons for
>experiencing a universe 10^9 years and of large complexity we
>currently see is necessary...
>
>
>
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Received on Fri Nov 02 2007 - 15:20:56 PDT