An infinite number of calculations is not possible. There are three
instances I can think of. Actually to be honest these are I think a part of
a single system. The universe I think obeys the Church-Turing thesis, which
means all that is dynamical or computable can be demonstrated on a Turing
machine. Symmetries that swap QCD color charge or the flavor charge, the
strong and weak nuclear forces respectively, are at least locally
algorithmic in nature. I will wrap this up at the end.
On the largest scale there is inflation, which stretched out space
enormously to 60-efolds or about 10^{26}, which means early data is
difficult to measure. A graviton in the extremely early universe, say
around 10^{-30} seconds to 10^{-35}seconds into the big bang has a
wavelength of around 10^{-30}cm. By expansion and inflation a coherent
state of such gravitons could be stretched into a classical scale
gravitational wave of millions of kilometers to billions of light years. An
eLISA type gravitational interferometer would imply a change in wavelength
by a z factor of z ≈ 10^{42}. For even longer say billions of light years
these could be detected as polarizations on the CMB and this is a z ≈
10^{55}. This z factor has an exponential dependence on the distance out,
and so this is around 50 times the CMB distance or 20 trillion light years
out. In other words, the sources of these observed gravitational waves are
now on the Hubble frame, a frame more or less simultaneous everywhere in
time, are now around 2 trillion light years out. This has a further
multiplier effect of around 1,250,000. So how about up to 7.5
billion-trillion galaxies. That would mean around a billion moles of stars,
if you remember Avagadro’s number of atoms in a gram molecular weight or
6.02×10^{23} atoms. If a water molecule represented a galaxy this would be
as much water as in a million tons of water --- about a lake’s worth of
water.
This is large, but it is the ultimate boundary. Anything beyond this is
lost. The e-LISA and increasingly it is thought fluctuations in pulsar
timing will detect early coherent gravitons as long wavelength
gravitational waves. These may have fingerprints on the CMB. Anything
further out than this is unobservable. Their fingerprints in the early
universe are longer than the cosmological horizon scale. Inflation enforces
a rule that the observer cannot witness an infinite universe --- even if it
is infinite.
Quantum mechanics enforces a form of this. Local hidden variables would
indicate that as the action S → 0 there is a UV divergence of degree of
freedom for hidden variables. In fact it would be infinite. Quantum
mechanics further eliminates infinite observable content.
Then there are black holes. The event horizon prevents observers from
witnessing a divergence. With the Kerr black holes and that the inner
horizon is Cauchy, which has been suggested as a way hypercomputation can
be accomplished. This would be a work around the Church-Turing thesis.
However, black holes are quantum mechanical, and the decay of a black hole
prevents the infinite condition necessary for an observer to perform a
hypercomputation from data piling up on the inner horizon. This actually
has the effect of enforcing a quantum form of the Bekenstein bound.
In effect the theorems of Turing and Gödel raise their heads and prevent
any observer from witnessing or performing an infinite computation. Any
attempt to perform hypercomputation, an infinite computation without
problems with Gödel, is prevented by what I call a general horizon
condition. This means it is not physically possible to acquire data about
observables in such as way as to loophole around axiomatic incompleteness.
This applies to any physical system, that by virtue of its interacting is a
sort of “observer.”
This means the universe is a fundamentally open system. This limitation
means there is not possible way to account for all quantum information in
the universe. The conservation of qubits may hold for type D, II, III, and
N Petrov solution types, here black holes, Robinson-Trautman solution and
finally gravitational waves, because they have asymptotic conditions that
allow for localization of mass-energy, momentum and angular momentum. These
solutions have Killing vectors that as isometries establish Noetherian
conservation rules. However, this does not apply for cosmologies.
LC
On Thursday, July 14, 2022 at 10:25:14 AM UTC-5 [email protected] wrote:
> The Bekenstein bound says if a volume of space has access to a finite
> amount of energy then the amount of information necessary to describe it is
> also finite, and that implies Bremermann's limit which says there is a
> maximum rate of information that can be processed in that volume, and it
> works out to be c^2/h= 1.4*10^50 bits per second per kilogram of
> mass/energy. However I think it should be possible, at least in theory, to
> extract work out of the expanding universe (see next paragraph), and if
> the expansion of the universe is accelerating then it seems to me the
> amount of energy you could have access to in that volume of space could
> potentially be infinite, not finite.
>
> Suppose you had 2 spools of string coiled in opposite directions connected
> by an axle and you extended the 2 strings to cosmological distances 180
> degrees apart from each other. As long as the Dark Energy force between the
> atoms in the string that were trying to force them apart was not stronger
> than the attractive electromagnetic force holding the atoms of the string
> together the string would not expand as the universe expanded, so there
> would be a tension on the strings, so there would be torque on the spool,
> so the axle would rotate. The axle could be connected to an electric
> generator and you'd get useful work out of it. Of course you'd have to
> constantly add more mass-energy in the form of more string to keep it
> operating, but the amount of mass per unit length of string would remain
> constant, however because the universe is accelerating the amount of energy
> per unit length of string you'd get out of it would not remain constant but
> would increase asymptotically to infinity. If the theories about the Big
> Rip turn out to be true and the acceleration of the universe is itself
> accelerating then it should be even easier to extract infinite energy out
> of the universe, provided we take care to continually shorten the string to
> keep it from breaking. So it would all just be a simple case of
> cosmological engineering. What could go wrong?
>
> And If you have infinite energy then you can perform an infinite number
> of calculations, so you could have an infinite number of thoughts, so you
> would have no last thought (the definition of death), so subjectively you
> would live forever. Of course the objective universe might have a different
> opinion on the matter and insist that everything including you had come to
> an end, but that hardly matters because subjectivity is far more important
> than objectivity; or at least it is in my opinion.
> John K Clark See what's on my new list at Extropolis
> <https://groups.google.com/g/extropolis>
> tif
>
>
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