Is an infinite series of random numbers possible?

chiro

Just as a general observations, I really couldn't imagine a situation where you would be able to completely nullify energy of a system, no matter what the magnitude or what the region of space.

Just as a thought experiment imagine if you could in some region, nullify the energy for that region. This would mean that for this region if this was the case everything would be completely static and there would be no possibility for any kind of dynamic behaviour.

With regard to virtual particles being used to 'borrow' energy, again in terms of state-space I would consider this as part of the system and not something that is isolated from it. The fact that it exists or at least the mechanism in some form exists means that it should be included in whatever way that is appropriate.

The big thing at least in my mind is this: how does the quantization of energy (as given by E = hf) relate to the quantization of the medium used to represent it? Moreover, how does all of this affect the supremum of the entropy measures in some finite space that I was talking about earlier?

Personally I think the nature and quantities related with the lowest states tell us a lot about the nature of the system. I'm going to have to at some point take a closer look at these kinds of things: you've got me interested now damnit!

chiro

The staircase algorithm was just an example to show how you could use entropy measures to deduce an order or pattern of some sort and this was always the intention.

I've heard about the nature of cyclic structures in physics like cyclic time and so on, but I can't really comment on the specifics.

Loren Booda

Cosmological entropy -- The blackbody spectrum is accurate to the finite number of radiating bodies which compose it. Heat exchange toward equilibrium moves the measured cosmic background radiation emissions to the perfect thermal curve, driving an increase in surrounding order. This balance avoids the "heat death" of the universe by limiting the blackbody radiation to countable radiators -- i.e. spacetime never realizes a maximally symmetric, boundless and randomized state approaching infinite entropy, but one which exhibits gains of statistical anentropy.

Microscopic entropy -- Vacuum mass-energy is paradoxically empowered by the action of observations - from the Copenhagen interpretation, I believe. (Without observers, would Maxwell's Demon work?) The likelihood of population by virtual quanta increases with more constant entropy density, assured by a random thermal distribution. Entropy density bounds are determined by their divergence there from the blackbody spectrum as ω/2∏ approaches 0, or ∞. You brought up that quantum energy being zero in space-time does not intuit comfortably, as I put it:

ΔE: E≠0.

Is this reminiscent of quantum number rules?

chiro

One thing I want to comment on in general so it may help you understand why I am even spending many posts talking about this topic.

Scientists study nature in the hope that they understand something at whatever level which at a minimum usually relates to figuring out how 'something works'.

If a scientist figures out some particular thing, they have found an order in that context. The larger the pattern applies to, the larger the order. It might be a small order like figuring out a particular cell or virus always acts the same way or it might be a large order like describing the general conditions or approximate conditions that gravity or electromagnetism follows. Both examples are types of orders but the more general one applies to a state-space much more broadly than the prior one.

When you keep this in mind, it becomes a lot more obvious that statistical methods are necessary because they can see things that any kind of local deterministic analysis would not and in fact unsurprisingly in many contexts, they do just this when you look at how these things are applied in data mining applications. Also you'll find that these kinds of statistical techniques and analysis are found when you have to analyze data from say the Large Hadron Collider or some other really highly data-intense scenario like you would find in astrophysics or even in miltary applications (think of all the people who use the really powerful supercomputers and then find out why they use them).

If you lose sight of this above aspect, then you have only constrained yourself to representations that give a very narrow local viewpoint, although albeit still a very important one but if you can not free your mind from this mental prison then you will be missing out on all the other things out there and not connecting all of the other isolated orders that have been discovered (like all the the other physical formulas and so on) and treat them largely as separate instead of as connected.

chiro

Taking a look at this:

http://chemed.chem.purdue.edu/genche...m.html#quantum

It seems to indicate that the energy of the electron requires x amount of energy in accordance with some mathematical constraints to go between orbitals and that it can never go below the orbital corresponding to n = 1. I have to point out that I am someone with mathematical training who has not studied physics specifically enough to put a lot of this into context in a specific way.

It seems that from this fact there is indeed a minimum energy level that is non-zero and in the context of what you are saying I am inclined to agree if this quantum number accurately reflects the attribute of the magnitude of the energy present.

For the cosmological part, again I am going to base my agreement on the line that if there were infinite entropy then there would be absolute chaos. Chaos in this kind of context is not good for anything especially life because for many things to function, orders of different kinds must be present. Imagine for example if gravity was just an ad hoc thing that decided when to be +9.8m/s^2 or when it decided to be -1000m/s^2. Think about what this would do to life: IMO it wouldn't exist.

There is also an argument by physicists that say that if the constant G were outside of a very narrow range then life including us would cease to exist. I don't know if its true or not but the kind of argument has a lot of important implications for science because what it does is brings up the issue of how order is so important for possibly not only us humans to exist or even all plant and animal life to exist, but possibly even for the universe as we see it to even exist.

It's not to say however that there do not exist finite subsystems with maximal or close to maximal energy for that subsystem. High levels of entropy in given situations are important IMO because a high level of entropy induces disorder which in a statistical sense equates to non-determinism or randomness. That element of randomness allows us to have the antithesis of what we could consider a 'Newtonian Universe' where a universal clock and absolute rules dictated the complete evolution of the system. If this were the case then we would be able to exhaust every possibility down to some conditional order and we would get a minimal entropy characteristic for the system just like the stair-case example I posted earlier but maybe in a more complicated manner.

So again the reason why I agree with you about having bounded entropy as a general property for all possible conditional distributions but for still having appropriate situations where entropy is maximal with respect to some sub-space is that it allows for things to still work (like life) but it also allows the case where there is 'real' evolution and for lack of a better word 'choice' at any kind of level for any scale given appropriate constraints (which we are in the process as human beings, trying to figure out).

If the above doesn't make sense to you, imagine the broken plate scenario happening with gravity, electromagnetism, the strong force, and even something more macroscopic like biological interactions. Imagine for an instant that people were splitting in half randomly and people's heads were dissappearing into outer space and then back again like a game of russian roulette. Imagine you picking up a gun unloading the chamber and then you fire the pistol and a bullet comes out.

To me, this is the above reason why there are constraints and understanding what these constraints are will probably give us humongous hints about why we even exist.

Loren Booda

Parallel Universes, Max Tegmark -- http://space.mit.edu/home/tegmark/PD...erse_sciam.pdf. What is not physically possible in an infinite universe? Can a finite universe have infinite possibilities? Do universal event horizons repeat without bound?

Are observers physically immortal?

A truly unified theory might transform the existing order in maximal ways, including entropy/anentropy reversal.

Thermal disequilibrium moves toward equilibrium by absorbed or emitted correspondent photons, with a decrease in entropy.

What is the most ordered universal structure possible? Is an empty universe interpretable as having both maximum and minimum entropy density? Can a maximally entropic universe have the same "complexity" as one of minimum entropy? Does an observer always impose order upon a more random universe? Can two or more disordered universes interfering together (e.g. through branes) reduce entropy overall?

Entropy, being scale dependent, sees an object like the Moon as being more ordered on many levels relative to the Earth.

Probability zero regions, found near atomic orbitals, are located in singular spacetime structures but quantum mechanically can be considered P>1, as they can not accommodate finite particles.

The cosmic background radiation -- containing the microwave background radiation -- includes photons, gravitons, WIMPS (like neutrinos) and perhaps Higgs particles which impinge anentropically (focused) from the event horizon upon an observer. The accelerating cosmos, with possible inflation, linear expansion, and dark energy provide an outward entropic divergence of energy.

chiro

This is an interesting question.

Frank Tipler has written a book trying to flesh out ideas about the physics of immortality. Just in case you are wondering, he has written pretty extensively about topics involved in General Relativity and even to some extent Time Travel with respect to space-times that allow theoretical paths to time travel.

But if I wanted to give a specific question for this, I would be asking this important question: what energy is involved for consciousness, what kind is it, where is it stored (in some kind of field for example) and how can it be transformed?

In my view, answering those questions will give a specific way to start thinking about this question in depth from a viewpoint that I think both scientific communities and religious communities can both appreciate and agree on as a basis for exploring this topic further.

Personally (IMO disclaimer), I think that there is some kind of other field that is not part of the known fields like EM, the nuclear forces and gravity that contains something that compromises of what we call 'consciousness'.

I am not saying that things like EM and the other forces don't play a role in how we behave, what we think, and so on, but I don't think that it is the whole story.

With the above aside in terms of immortality, if the energy that makes up consciousness can not be destroyed, and also can not be transformed away to something that loses or wipes information about conscious awareness then I would say that yes physical observers are indeed immortal on that argument.

But in order to argue the above you have to first define what consciousness actually is in terms of energy and also what kinds of energy forms they actually are and unfortunately I have a feeling it's going to take a while to even get close to even defining the specifics of this, let alone doing an experiment or having discussions about the veracity of whether the claim is wrong, right, or somewhere in between.

In terms of the infinite possibilities question, again this comes down to the discussion we had before about whether you can always construct a joint distribution that has random entropy for all conditional distributions for 'prior' events. In other words the entropy of each possible conditional distribution has maximal entropy. If this is always the case, then you should have infinite possibilities.

Also remember that the above is framed in terms of a finite state space. Think about it like constructing a process where no matter how you construct any conditional distribution for the next roll given every permutation of the previous rolls, all distributions will have maximal entropy. This means that you can construct a completely random system. If you can't do that but can do something in between minimal and maximal entropy then it is semi-random. If you can only construct a zero entropy distribution, then it means your system has become deterministic.

For the infinite universe question (what is not possible in an infinite universe), this will have to do with not only physical arguments but with philosophical arguments as well.

You see the reason that plates just don't assemble themselves from broken states and that gravity acts in a uniform way and even that quantum behaviour and all other physical interaction mechanisms work the way they work says to me at least that there is a reason why you can't just do 'anything you want', at least not currently.

Again my thought experiment would be to consider if people just randomly dematerialized and gravity just decided when it wanted to 'work' and 'not work' and the kind of chaos that would create for life in general. This tells me that there is a reason for the constraints at least in the context that you want an environment that supports and promotes the situation for living organisms in any form.

In terms of possibilities, this can be formed if you have a clearer idea of the nature of the different joint distributions. The big caveat though is that we don't have these yet. Science is very young for earthlings in the current state it is in and the amount of data we have and also the tools to effectively analyze it are not mature enough to really make all of these connections.

It's not just actually having the data: it's also having the computational hardware and technology, the algorithms, the mathematical techniques, and all of this to actually do all of this. These areas are evolving quite rapidly, but it's going to be a little while at least before it gets to a stage where we can give a more specific quantifiable answer using the above to answer 'what's really possible'.

For now we have to rely on experimental results, theoretical ideas and discussions, and the inquisition of scientists to help push this boundary and thankfully this is happening on a scale that probably never would have been imagined even a hundred years ago.

The ironic thing about humans is that we crave certainty.

While I don't think this is necessarily a bad thing, the effect that it can have is that in a scientific perspective, we want as much certainty as possible both in its predictive power and subsequently in the mathematical representations that are used to both describe and predict things.

Quantum mechanics has come along and destroyed this notion and I think it's a thing that we should embrace at least in the idea that at some level, things will not be able to be predicted.

Here is one idea I have about why this kind of thing is good.

Consider that you have the complete set of laws that allow you to take the state of the complete system and engineer it in such a way that you can create whatever state you want at a future point of time.

Now consider what the above would do to the stability of the system. This situation creates situations where the stability of the system itself can be for lack of a better word, destroyed.

If situations exist like this, then what this would mean is that you would get all these possibilities where you would get these situations where things just literally blow up and create a situation where the evolution of a system is essentially jeopardized.

In a situation where this doesn't happen, you would need some kind of non-zero minimal entropy for all conditional permutations to avoid this very scenario which means you need to resort to a statistical theory of reality and not a deterministic one.

A situation where levels of stability in different contexts are 'gauranteed' or at least probabilistically high enough to warrant enough confidence would result in a kind of collective design so that this kind of thing would either not happen, or at least happen with a tiny enough probability so that it can be managed.

In fact if things had some kind of entanglement, then this mechanism could be used to ensure some kind of stability of the entire system and localize instabilities of the system if they do occur as to ensure that the system as a whole doesn't for lack of a better word 'blow up'.

The real question then if the above has any merit, is to figure out how you balance this kind of stability with the system both locally and globally having the ability to evolve itself in a way that is fair?

I don't know the specifics, but in the context of what I've been saying in this thread it would not be good for system stability to move towards a state of either maximal entropy or complete minimal entropy for reasons discussed above.

To me, the situation where you have the most ordered universe is where all conscious forms work together in a way that doesn't create instability.

Some might see this as a religious theme or some kind of 'new age' comment, but an ordered system would look more like something that works in unison for each and every element rather than having elements working against one another.

If I had to characterize it, I would characterize it as every conscious form working with another to create the scenario where everything would be supplementing everything else in a way that creates a system where the energy ends up being directed in a way that everything works together as a whole which results in a kind of unification of all conscious beings which means that everything becomes a unified system which in terms of information means that it can be described as such which results in a decrease of entropy.

Remember entropy in this context is synonymous with not only order but also with the amount of information to describe something.

Remember that if you have a collective system that reaches some set of unified goals or constraints, then instead of having all these separate set of constraints to describe something, you end up having a situation where they end up merging which will result in requiring less information to describe the system. This lessening in the amount of information to describe the system translates in a reduction of entropy including the overall measures for all conditional entropies.

To me, the observer has the choice to either decrease or increase the entropies that end up contributing to the system as a whole but I would estimate that for a collective system to evolve in a positive manner, you would always want a system to at the very least decrease it's entropy over its evolution within any sub-region and collectively to find some kind of order for the system as a whole that reduces it's entropy from a previous state.

In terms of what that actual order is, I can't say but I imagine that there are many different kinds orders that could be formed just like there are many different functions that can be described once you have a dictionary and language structure that is minimal enough to describe a complicated system in a minimal form.

If this sounds like BS or foreign you should note that these ideas are a huge part of information theory including the area known as algorithmic information theory. If you want more information about this you should look up Kolmogorov complexity: it's not something that has been clarified in terms of algorithmic methods but the idea has been clarified to some respect.

A very good observation.

The thing is however, you need to define the order being used and this is really the heart of what makes language interesting.

The nature of the order could be to do with geometry and color variation. Describing a filled circle with a color spectrum that has little variation in one language is ordered.

But in another language it is not ordered. In another language something like the Mandelbrot set is highly ordered, but describing the moon in that language is highly disordered and requires a tonne of information.

This is why we have so many languages, jargon, structures, codings and so on. They all have a purpose in a given context. One language will represent something with minimal order but when you convert it to something else, it would take a ridiculuous amount of information to represent that same thing.

The question then becomes, how do we create languages in the best way possible? This is not an easy question and it is something that we are doing both consciously and unconsciously every single day.

The ultimate thing is that there are many different orders and not just one which makes it very interesting because we as scientists want to find 'the universal order' but my guess is that there are many orders that are just as valid as any other at the scope that they are presented at (i.e. the actual state space that these orders correspond to: think in terms of cardinality of the set).

I don't know what this means, can you give me a link to a page that describes this?

Can you point somewhere where this is described mathematically (and possibly in summary in english)? I'm for most purposes a mathematician/statistician and not a physicist.

chiro

By the way I haven't read the article for multiverses so I'll read that shortly.

Loren Booda

The (quantum) wavefunction condition ψ(x)=0 holds continuously only when it is everywhere continuous.

Hypothesis: at a given x, the probability P(x)=ψ*ψ (assumed continuous and smooth) of locating a singular particle is assumed zero at the singular point ψ(x)=0. So ψ_min(x₀)=0 implies (dψ/dx)_min(x₀)=0, unless ψ=0 for all x.

__________

If ψ_min(x)=A(exp(2∏i(xp/h)))_(x=0)=A(cos(2∏(xp/h))+isin(2∏(xp/h)))_(x=0)=0

Eigenvalues: x=(N+1/2)h/2p

and (dψ/dx)_min=-2∏(p/h) A(-sin(2∏i(xp/h)))+icosA(exp(2∏i(xp/h)))=0

Eigenvalues: x=N(N+1/2)(h/2p)²

__________

P=probability=ψ*ψ

x=spatial dimension

A=constant

N=integer

h=Planck's constant

p=momentum

Conclusion: if ψ_min(x₀)=0, its first derivative derives a singular, local maximum or minimum there, but its neighboring points do not, unless ψ(x)=0 for all x.

chiro

Geez Loren Booda, you'll really stretching me! I love it! :) I'll give an answer shortly.

chiro

The thing though is that with physics, the discussion is about what to do with regards to the issue of having one theory in continuous space (General Relativity) and another in discrete space (Quantum Field Theory).

Now I've been reading a little bit about this lately and one approach that is being used is to 'quantize' GR in which you basically get the field of Quantum Gravity.

This approach in my mind makes more sense than trying to make QFT continuous. My reasons for thinking this way is that we already know that all of the interactions and subsequently all the energy calculations work in a quantize way so at least to me it doesn't make sense to have an embedded set that describes the space to be continuous either.

For the above, it's like for example taking a Diophantine system and then describing the sets for describing the domain and codomain to be real numbers. This is completely un-necessary because you know that for this kind of thing you are only going to deal with finite numbers of states when you look at a finite subregion of the entire state-space for that particular process.

So based on this line of reasoning (which may be right or wrong and I'd love to hear your comments), then the next thing to do is to find a quantization scheme for space-time which is what many people are working on currently in many ways.

What this will do is essentially force the probability distribution to be non-continuous, but the real question lies in the way that it will be discontinuous.

See the thing is that you can't just quantize the space in the regular way that you would say quantize a 3D cartesian geometry by quantizing each axis individually. The problem with doing that is that not only are dealing with non-euclidean space-times, but we are also dealing with quite a number of interactions that ultimately will define the actual quantization procedure of space-time itself.

Personally one way I would approach this quantization is from a number-theoretic viewpoint because if a quantization scheme had to exist for a completely quantized system, then it means that for this quantization scheme the solutions to the Diophantine equations that specify that system would have to make sense in the way that all the solutions that are meant to exist corresponding to results in this physical reality actually do exist and also just as equally important, all the results that do not exist also don't exist in the Diophantine system.

So if you were to go this route, then the first thing would be to think about ways of expressing a Diophantine form of the system (it will have probabilistic properties of course) and then through the probabilistic description of the Diophantine system, then generate some useful probability definitions of a specific part of the system, like a particle like an electron.

One of the tricks to model the kind of behaviour you find in Diophantine systems that take place in continuous systems is to use the Dirac Delta function. This 'infinite-spike' allows you to model the behaviour of a finite field when you are dealing with a continuous state-space. When you have a natural space that is discrete, this isn't needed and you can get all the kinds of discrete behaviours when you consider something like a Diophantine system to model a process (and it's important to note that it can be made probabilistic).

So my question to you is, will you continue to work in a continuous framework meaning that you have to deal with all these issues related to Dirac-Delta spikes, discontinuities of every sort and the consequences of such, or are you willing to go the other way and assume a completely discrete framework and as a result use number theory (and it's probabilistic variant) to do physics instead?

Loren Booda

Special relativity imposes a relative speed limit of light speed c. General relativity, Georges Lemaître posited, has no relative speed limit for the universe. Particle horizons proceed toward us from a theoretical big bang in reverse order of their creation. The singular big bang, relative to us, may actually stretch across the celestial sphere. The distance of the singularity from us could well determine our physical universe. Whether the big bang is now out to infinity or at a finite horizon has affected particle creation, the evolution of forces, physical constants and the (local) geometry of our spacetime.

Think of cars accelerating from a stop. The cars behave much like galaxies moving according to the Hubble distance expansion, approximately r=c/H₀, where r is the relative distance a galaxy is from us, c the speed of light and H₀ the Hubble constant, about 70 (km/s)/Mpc. (That is, kilometers per second per megaparsec.) The farther one travels outward, the faster one expands relative to home base. If the law holds, eventually the traveler reaches the event horizon, where, like a black hole, Earth-light does not have the energy to continue (but there the traveler might find himself in a sea of Hawking radiation thanks to his investment).

Close to home we observe some rotational, then somewhat peculiar (random) expansion of the galaxies, farther on the moderate "Hubble law" escape, then the many named accelerative outward expansion, first found by supernova measurements. While our universe rushes away from us (and does so wherever we happen to be) the big bang remnant, singular as ever, has rained particles (albeit diminished) upon us. The microwave background is one remnant -- recombination of electrons and protons to create hydrogen. This happens in the lab at 3000K, which when divided by 2.7K, just happens to yield the redshift (Z≈1000) of the MBR.

The question remains, how does the ultimate outward cosmic background radiation (CBR, not just from microwave horns) correspond to the inner one of particle accelerators? When we look to the sky we see a rain of photons, when we look to the ground we feel the pull of gravitons. What might be interesting to measure is the entropy of the outer flow against that of the inner. Pointing our telescopes farther unravels earliest times; nearer do our microscopes enable uncertainty. We learn that out of high energy condense the quanta of fundamental forces.

chiro

Can you give me some specific equations to look at?

Again I am not a physicist, but I do know a little bit about mathematics.

One thing that is interesting is that there is an idea that the universe is actually holographic. Now if this is the case structurally (like the interference pattern you get when you look at a real holographic film itself), then this has huge consequences for entropy.

In order for a hologram to retain its structural integrity (in terms of the actual information it represents), what this means is that there is basically a form of global entanglement. The effects on entropy are very big since if we are able to reduce some or all of the information for some finite sub-region of our state-space, then it means that changes will propagate through the entire system in both microscopic and macroscopic manners.

Now again, I have to point out that I am not a physicist you will have to give me equations and if possible, also a bit of extra context behind your question to give me some physical intuition.

Also the holographic nature if it exists in a kind of 'space-time' manner also means that the entanglement is not prevalent for things at one 'slice' of time, but rather across space-time as a whole. The effects of this kind of entanglement, if it existed, would mean that not only would it be seen in entropy calculations, but also that if it had the properties of a hologram information packet, that you could experimentally check whether the entropy pattern matches that of a hologram as well. This would be a nice physics experiment ;)

With regard to the evolution of forces, to put this into context of entropy, again you have to see where conditional entropies are minimized not only under the raw data, but also under transformations as well.

The thing is that if there is an order that is being created (remember there can be many many different orders in a highly complex system with many interactions going on) then what you would do is to extract a significant order and make an inference about what is happening. You would want to extract orders that minimize entropy in a maximized state-space for the highest conditional order possible (when I say conditional order I mean with respect to a joint distributions that has a higher number of initial states with respect to the rest of the states.)

In terms of the evolution of not only the physical state itself in space-time but also the forces, again you have to see where the order is.

If you want to conjecture why a particular set of 'forces' have been chosen, then again relate these to state-space in terms of the best orders that can be obtained. If it turns out that the orders vanish, or if the system 'blows up' and becomes 'unstable' with respect to existing orders that are extrapolated from the current system, then you have a way of contextually describing when you interpret what the orders mean 'in english' from their mathematical counterparts why the forces 'are what they are' vs 'are what they could be'. This kind of thing would strengthen what you know as the 'Anthropic Principle' and other ideas similar to it.

For the Hubble stuff, it would be helpful to give some equations and if possible some extra context to what you are saying. Again I'm not a physicist.

Finally with respect to your last statement, again I don't see things in terms of gravitons, or other force communicators required to make physical intuition: I see things mathematically in the most general non-local manner possible. In terms of physical intuition, it is not preferrable to do it this way because physics is a very specific endeavor that is rich of complexity at even the smallest scales and for specificity and clarification, requires one usually to see things in a local context.

Now the above might sound arrogant, but the reason I say this is because with my background and experiences, do not for whatever reason see things this way. I see things from a different perspective which can be beneficial and not so beneficial, just as every perspective has its benefits and limitations.

It would be interested to also get your feedback as well on my responses if you don't mind just to get some relativity for my comments. :)

AntiPhysics

Technically, no. Eventually, if it is truly infinite, after all the googolplexes of combinations of numbers, it will repeat. Randomness is only based on the time that you study it for. If you have 0.1256627773728172818918268162, that obviously doesn't repeat. But if you let it continue, it will repeat eventually.

chiro

What about a number like the decimal expansion of pi?

micromass

That is NOT true. Only rational numbers repeat eventually.

Loren Booda

chiro,

I respect that this is a mathematical forum, so I will try to remain conscious about the topic of this thread. My apologies for the lack of hard equations. Such relations below will often be expressed in "English." I struggle to provide the best descriptions possible. Coding is an area which I am not familiar with. Do you feel that our exchange is productive? I appreciate your contributions.

__________

http://en.wikipedia.org/wiki/Holographic_principle -- Black hole entropy
The holographic principle was inspired by black hole thermodynamics, which implies that the maximal entropy in any region scales with the radius squared, and not cubed as might be expected. In the case of a black hole, the insight was that the informational content of all the objects which have fallen into the hole can be entirely contained in surface fluctuations of the event horizon. The holographic principle resolves the black hole information paradox within the framework of string theory.
. . .
An object with entropy is microscopically random, like a hot gas. A known configuration of classical fields has zero entropy: there is nothing random about electric and magnetic fields, or gravitational waves. Since black holes are exact solutions of Einstein's equations, they were thought not to have any entropy either.

But Jacob Bekenstein noted that this leads to a violation of the second law of thermodynamics. If one throws a hot gas with entropy into a black hole, once it crosses the horizon, the entropy would disappear. The random properties of the gas would no longer be seen once the black hole had absorbed the gas and settled down. The second law can only be salvaged if black holes are in fact random objects, with an enormous entropy whose increase is greater than the entropy carried by the gas.

Bekenstein argued that black holes are maximum entropy objects—that they have more entropy than anything else in the same volume. In a sphere of radius R, the entropy in a relativistic gas increases as the energy increases. The only limit is gravitational; when there is too much energy the gas collapses into a black hole. Bekenstein used this to put an upper bound on the entropy in a region of space, and the bound was proportional to the area of the region. He concluded that the black hole entropy is directly proportional to the area of the event horizon.

Stephen Hawking had shown earlier that the total horizon area of a collection of black holes always increases with time. The horizon is a boundary defined by lightlike geodesics; it is those light rays that are just barely unable to escape. If neighboring geodesics start moving toward each other they eventually collide, at which point their extension is inside the black hole. So the geodesics are always moving apart, and the number of geodesics which generate the boundary, the area of the horizon, always increases. Hawking's result was called the second law of black hole thermodynamics, by analogy with the law of entropy increase, but at first, he did not take the analogy too seriously.

Hawking knew that if the horizon area were an actual entropy, black holes would have to radiate. When heat is added to a thermal system, the change in entropy is the increase in mass-energy divided by temperature:

dS = dM/T

If black holes have a finite entropy, they should also have a finite temperature. In particular, they would come to equilibrium with a thermal gas of photons. This means that black holes would not only absorb photons, but they would also have to emit them in the right amount to maintain detailed balance.

Time independent solutions to field equations don't emit radiation, because a time independent background conserves energy. Based on this principle, Hawking set out to show that black holes do not radiate. But, to his surprise, a careful analysis convinced him that they do, and in just the right way to come to equilibrium with a gas at a finite temperature. Hawking's calculation fixed the constant of proportionality at 1/4; the entropy of a black hole is one quarter its horizon area in Planck units.

The entropy is proportional to the logarithm of the number of microstates, the ways a system can be configured microscopically while leaving the macroscopic description unchanged. Black hole entropy is deeply puzzling — it says that the logarithm of the number of states of a black hole is proportional to the area of the horizon, not the volume in the interior.

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[Speculation]: Regarding the "black hole information paradox," a black hole's "singularity" may be a composite of quantum black holes. Information about the "singularity" would manifest at the black hole horizon as the only variables we may know about a black hole: mass, spin, and charge (and derivations thereof). The extreme symmetry of the Schwarzschild black hole transfers coherently (much like an "isotropic laser" or "holograph") such information that is allowed about the singularity.

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Remember the Heisenberg uncertainty principle applies for all quanta: a very small mass complements a very large radius: ΔrΔp≥h, or ΔrΔcm≥h. In other words, small measurements relate to large ones through their action, or units of Planck's constant.

r=radius of action, p=momentum of action, c=speed of light in vacuo, m=mass of quantum, h=Planck's constant.

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[Speculation]: M* is the characteristic mass of quantum gravity. This Planck mass demarcates exclusively black hole masses above from those of quanta below. Symmetry between these regions implies a duality for the two classes of entities. The Planck (quantum) black hole, with its mass M*, itself shares and interrelates properties of black holes and quanta. Since inverting the mass scale around M* compares black holes and quanta one-to-one, a black hole could be a real quantum "inside-out" - in terms of that scale - and vice versa:

(M_{black hole}·M_quantum)^1/2=M_Planck, where M is mass.

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http://en.wikipedia.org/wiki/Peculiar_velocityIn physical cosmology, the term peculiar velocity (or peculiar motion) refers to the components of a receding galaxy's velocity that cannot be explained by Hubble's law.

According to Hubble, and as verified by many astronomers, a galaxy is receding from us at a speed proportional to its distance. The Hubble distance expansion, approximately r=c/H₀, where r is the relative distance a galaxy is from us, c the speed of light and H₀ the Hubble constant, about 70 (km/s)/Mpc. (That is, kilometers per second per megaparsec.)

Galaxies are not distributed evenly throughout observable space, but typically found in groups or clusters, ranging in size from fewer than a dozen to several thousands. All these nearby galaxies have a gravitational effect, to the extent that the original galaxy can have a velocity of over 1,000 km/s in an apparently random direction. This ["peculiar"] velocity will therefore add, or subtract, from the radial velocity that one would expect from Hubble's law.

The main consequence is that, in determining the distance of a single galaxy, a possible error must be assumed. This error becomes smaller, relative to the total speed, as the distance increases.

A more accurate estimate can be made by taking the average velocity of a group of galaxies: the peculiar velocities, assumed to be essentially random, will cancel each other, leaving a much more accurate measurement.

Models attempting to explain accelerating expansion include some form of dark energy. The simplest explanation for dark energy is that it is a cosmological constant or vacuum energy.

http://en.wikipedia.org/wiki/Cosmological_constant -- The cosmological constant Λ appears in Einstein's modified field equation in the form of

R_μν -(1/2)Rg_μν + Λg_μν = 8∏G/c⁴T_μν

where R and g pertain to the structure of spacetime, T pertains to matter and energy (thought of as affecting that structure), and G and c are conversion factors that arise from using traditional units of measurement. When Λ is zero, this reduces to the original field equation of general relativity. When T is zero, the field equation describes empty space (the vacuum).

The cosmological constant has the same effect as an intrinsic energy density of the vacuum, ρ_vac (and an associated pressure). In this context it is commonly defined with a proportionality factor of 8∏ Λ = 8∏ρ_vac, where unit conventions of general relativity are used (otherwise factors of G and c would also appear). It is common to quote values of energy density directly, though still using the name "cosmological constant".

A positive vacuum energy density resulting from a cosmological constant implies a negative pressure, and vice versa. If the energy density is positive, the associated negative pressure will drive an accelerated expansion of empty space.

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Thus the expansion "ladder" is largely determined by peculiar velocity, the Hubble expansion and a parameter like the cosmological constant.

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[Speculation]: Entropy of a black hole is proportional to its surface area. Entropy of conventional matter is proportional to its volume. I assume entropy of a concave spherical cosmological horizon, of reciprocal geometry, to be that of an inverted Schwarzchild black hole -- thus differing in their sign of curvature -- that is, with geodesics converging rather than diverging.

Aside: a simple dimensional argument considering conventional entropy (three dimensional) and black hole entropy (two dimensional) yields individual quanta having entropy proportional (one dimensional) to their propagation.

[Question]: A Schwarzchild black hole of radius R_B has entropy proportional to its surface area. Consider it within a closed ("Schwarzchild") universe of radius R_H>R_B. What is their relative entropy? Remember the universe as having radiating curvature relatively negative to that of the inner black hole.