## Is an infinite series of random numbers possible?

 Quote by 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.
Thankyou Loren Booda. I'm actually learning a lot myself and you've motivated me to look at a few things as a result of this discussion.

I'll attack this question in a few parts.

I wanted to comment specifically on something first before I attack the rest of your post.

If this result is true, then the fact that the entropy is bounded for any finite sub-region tells us that there is indeed a mechanism used to make sure things don't get too disorderly and as a conjecture I imagine that the nature of gravity with regards to the black hole phenomena helps create this form of stabilization.

In terms of what we have talked about before regarding the idea of bounding entropy for a finite-subregion so that at the minimum you don't get a level of unmanageable chaos, this idea of a mechanism to make sure that this doesn't happen makes sense at least from this perspective. I'm not saying that it's necessarily the only reason for the result of these interactions, but I do think it is one plausible thing that could be used to analyze exactly why this is happening.

I have a few books on this kind of thing tucked away somewhere but I haven't really had the motivation to go into it in detail, so I'm considered looking at these results at a later date.

The interesting thing to take note of is how the radiation (from the black hole) varies not only with the area of the event horizon, but also with the temperature.

The reason for the above inquiry is that if you know roughly how the entropy of the information exchange is happening (not necessarily in an atomic way but in a macroscopic way), then what you can do is you can look at that exchange and understand what happens in the most chaotic circumstances.

Also with regards to the idea of it being based on area and not to do with volume, I am going to go make a wild speculation and say that because the black hole represents the situation with the most entropy for that particular region, then what you are looking at is a situation where for a given volume, the entropy has reached a maximum and therefore if a black hole maintains this entropy characteristic, the entropy itself will not change despite what is going on inside the black-hole, if indeed the black-hole scenario represents the situation of a maximum-entropy.

With regard to the area problem, what I would say for this is that if a black-hole has to have a spherical volume, then if the projection of the black-hole volume onto the surface where the horizon is measured is strictly proportional to the area, then it is no surprise that the entropy is in fact proportional to the area. If the region enclosed by the event horizon is circular, and the volume itself relates to a sphere, then you can see that the area is indeed proportional to the volume of the region that the black hole is enclosed in.

In fact, intuitively we would expect something that had the characteristics of a black-hole (i.e. if it was under a gravitational force so big where every part of the object would be accelerated towards the very centre of the body), that the black hole itself would be a spherical object. The only thing that remains is to see whether the event horizon is itself a circular object and if this is the case, then it is not surprising that the entropy is proportional to the area.

Also if the black-hole represents the state of maximum entropy for that particular configuration within that given space, then the entropy would be the maximum allowed.

Now the really interesting thing to take note of is how the entropy changes over time. The thing that I would pay attention to, is exactly how the area of the event horizon changes, how the radiation emission from the black hole changes, and also how the temperature changes under certain configurations.

The reason I say the above is that if the black-hole really is the state of maximum entropy, then understanding what happens in this case will tell you essentially how things become 're-ordered' again.

Again the motivation for this line of reasoning is the example of a system that is allowed to become too disordered and as a result so chaotic that it ends up destabilizing the whole system if it is allowed to propagate willy nilly. If the black-hole at least in part helps stop this situation from occuring, then what this phenomena will tell you is how for lack of a better word, God deals with this situation: in other words, how stability is maintained of the entire system.

In fact, the energy conservation rules for black-holes will tell an awful lot about how orders of all kinds are actually maintained.

 [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.
The only comment I have on this is that if the radiation (or any other information exchange between the black hole and other regions regardless of how it happens) gives us information about the entropy, temperature or other characteristics then I imagine this would give a lot of information about the black hole.

In terms of mass, if the radiation corresponds to containing temperature information, then mass information would be communicated. In terms of spin and charge, I don't know enough about these characteristics to answer this currently.

 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.
I'm going to take a look at this later.

 [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: (Mblack hole·Mquantum)1/2=MPlanck, where M is mass.
Before I comment on this can you point me to either a paper or an article (or something along those lines) that gives me a bit more background for what you are saying?

Doesn't have to be absolutely formal: I just need a bit of context and background.

 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.
One comment I have with this is the idea of 'random speeds'. Again this is from a computational perspective.

I would go further than what you have done and consider every entity relative to everything else and then draw conclusions from that rather than from just a measure of what you are saying.

The thing is, if there is some kind of interaction going on, then there will exist a transformation of your raw data with minimal entropy and if that entropy under a particular transformation is 0 then this describes your interaction in a completely deterministic manner. Even if it is not exactly zero, it's still good enough for most practical purposes to be useful.

Again I don't see things in a physical context: I don't see things in terms of particles, forces, electrons, space-time surfaces and so on: to me it's just information with various orders and also I am not completely aquainted with all of the definitions used in physics (I know some though). I would look at a system in a general way, try and extract various orders and then interpret what those mean in the context of the interpretation of the information presented. Personally, in my view, trying to understand something in a fixed constraint whether that's in terms of human sensory perception to me is not how I look at things: All I see is information.

If this conversation goes deeper that it is now (which is fine by me), then I will have to get acquainted with these definitions and constraints that are being talked about and I hope you bear with me if this is required.

Also finally with regards to 'random speeds'. I will mention this later in this post but the idea of things being 'completely random' doesn't make sense in terms of stability and enforced variability: I will talk about these soon.

 http://en.wikipedia.org/wiki/Cosmological_constant -- The cosmological constant Λ appears in Einstein's modified field equation in the form of Rμν -(1/2)Rgμν + Λgμν = 8∏G/c4Tμν 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, ρvac (and an associated pressure). In this context it is commonly defined with a proportionality factor of 8∏ Λ = 8∏ρvac, 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.
The thing about this analysis is that if you look at this in isolated macroscopic context, then you will probably miss it's role in the context of looking at it with respect to the other mechanisms that exist (in other words, the other forces besides gravity).

With regard to the first issue, I speculated that the behaviour of a black-hole might be the mechanism to not only control entropy for a given finite sub-region for a particular space-time configuration, but also to 'deal' with this situation by creating a situation that effectively either re-directs information in a manner that it can be become more stable or perhaps even to isolate this from other systems as to stop things from going haywire.

Now the thing is that in an accelerating universe where space for a lack of a better word becomes 'stretched', again this could well be designed relative to all other forces to make sure things either 'don't get haywire' or 'don't come to a halt'.

When I say 'don't get haywire' I am talking about a situation where things become too chaotic causes an overall system-wide chaos that is irreversible. The halting problem is the exact opposite: by this I mean that you want to have some kind of minimal gauranteed variation or evolution constraints that allow things to remain dynamic.

Both of the above issues need to be addressed and I also want to say that it is a better way to analyze not only systems in general, but also scientific systems like physics and so on.

The thing is, by comparing the complement of certain models and systems relative to the data and actual models that have been formulated, you can actually give a reason why a particular model is either better or even exists at all with respect to another model simply on the basis of whether one model is sustainable over its evolution vs one that is not.

To do this you need to consider the system as a whole and not as a separate isolated system of its parts and by considering say an accelerating phenomenon without the other things that give a reason why this occurs, then the speculation will be ill-founded with regards to the primary motivation of such a phenomena.

You might see me saying this again and again, but the reason I say this is because without looking at things in terms of both stability and gauranteed evolution constraints (i.e. the same doesn't become static enough to prevent proper evolution), then many things will either be missed or understood. Any system that has the properties of one that evolves properly must have these attributes.

For this reason, I think scientific investigation needs to undertake a major shift from how it currently is going to something more aligned to the above way of thinking. A system that is prone to any kind of unfair arbitrage in any way is something that is not well designed in my opinion and it wouldn't make sense to analyze a system in the context of where you have a situation of unfair arbitrage.

What this non-arbitrage situation corresponds to is something that will need to discussed, debated, clarified and tested against experimentally and otherwise, but in terms of system design I see it as critical that this be used in the most basic of analyses.

Ironically however science is formulated to try and create some kind of certainty out of the uncertainty that we face in which we investigate things more or less to control our surroundings. If a system has been specifically designed to enable a system wide level of non-arbitrage, then it means that this has already been considered in the system design. This philosophically raises an important question and if it holds any water, might make quite a few people depressed.

In fact you could use the above form of analyses to make an inference on whether a particular system has been 'purposely designed' for lack of a better word in comparison to something that has been 'randomly chosen'. This kind of thing would support evidence of whether we really are 'just an accident' or whether this whole thing has been engineered on purpose.

This is speculation, but I don't think this whole thing was random, but rather engineered on purpose due to the amount of evidence for stability in all forms and how everything just naturally 'works together' in many kinds of orders. If I wanted to provide evidence mathematically I would do it from a stability analysis using some of the concepts above, but for now I'm going to base it on observation and anecdotal inference.

 [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.
Could you elaborate on this please? What do you mean by propagation?

 [Question]: A Schwarzchild black hole of radius RB has entropy proportional to its surface area. Consider it within a closed ("Schwarzchild") universe of radius RH>RB. What is their relative entropy? Remember the universe as having radiating curvature relatively negative to that of the inner black hole.
For this particular question, what I would like to know is can you have a black-hole inside a black-hole where there is any freedom for the configuration of the inner-most black-hole inside the outer-most blackhole.

If the entropy conditions are fixed for 'any' black-hole with regards to characteristics like temperature and subsequently entropy, then the thing that I would ask is 'does every sub-region of any black-hole contain entropy corresponding to the volume of that sub-region?'

If the answer is yes, then the answer would be simply be equal to the ratio of the inner object with respect to the volume of the entire object.

The reason why I would say the above is that if a black hole is the realization of something with maximum entropy, then in terms of the conditional distributions, all of these would also have to yield maximum entropy (remember our conversation before on this).

As a result of this, if it is true, it means that every sub-region of the black-hole also has maximal entropy for that region: This means that we should get the proportional characteristic I have mentioned.

Now if for some reason there was a deviation of the maximum entropy principle for any sub-region of the space, this would mean that a black-hole with this particular configuration would not have maximal entropy which to me is a contradiction.

If for some reason this could happen, what this means is that in a black-hole you could pretty much create order in any way you saw fit if you understood the mechanism because the fact that there is a decrease in entropy for some sub-region in a maximal outer black-hole means that you can engineer directly everything if you understand how to lower entropy in various ways. This would correspond to the ability to create any kind of order that you wanted to if you knew how to do it.

 Quote by Loren Booda Do you feel that our exchange is productive? I appreciate your contributions.
I do feel that our exchange is productive, but I would be interested in more feedback from you if you could please. I don't want to end up doing all the talking: I appreciate any kind of feedback whether you think I'm out of my mind or making sense.

 Quote by micromass That is NOT true. Only rational numbers repeat eventually.
Just think about it in more abstract terms, this is only textbook knowledge. If a number is random, that means there is an infinite amount of possibilities, and if there is an infinite amount of possibilities, you have the chance of it repeating at one point, be it after 10 digits or after a googolplex of digits. Sometimes you are only limited by which what shows up on a calculator display before cutting off.

 Quote by chiro What about a number like the decimal expansion of pi?
Pi is a great example of my hypothesis. From what we have studied, pi is 3.1415926535. Or it could be 3.14159265358979323. Or even 1 million digits. In those 1 million digits, it doesn't repeat. Pi is a non terminating decimal, so who's to say that it will never reapeat? It has the "opportunity" to, so to speak. If someone could live forever, but in just one time period, let's say just today, they would eventually do everything possible and go everywhere possible in the world.
 Recognitions: Gold Member Science Advisor Staff Emeritus Yes, you have that possibility- you said earlier it must happen. But I think the reason chiro responded as he did is the interpretation of "repeat". If you mean some "pattern" of digits will repeat at least once further down the list of digits, that is pretty obvious- there are only 10 digits so obviously digits must repeat a lot! There are only $10^n$ possible n digit patterns so obviously some such must eventually repeat. But Chiro was interpreting "repeat" as meaning that at some point the digits become "AAAAAA..." where A is a specific finite sequence of digits that keeps repeating and there are no other digits. That is true only for rational numbers- and "almost all" real numbers are NOT rational.

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 Quote by AntiPhysics Just think about it in more abstract terms, this is only textbook knowledge. If a number is random, that means there is an infinite amount of possibilities, and if there is an infinite amount of possibilities, you have the chance of it repeating at one point, be it after 10 digits or after a googolplex of digits. Sometimes you are only limited by which what shows up on a calculator display before cutting off.
 Quote by AntiPhysics Pi is a great example of my hypothesis. From what we have studied, pi is 3.1415926535. Or it could be 3.14159265358979323. Or even 1 million digits. In those 1 million digits, it doesn't repeat. Pi is a non terminating decimal, so who's to say that it will never reapeat? It has the "opportunity" to, so to speak. If someone could live forever, but in just one time period, let's say just today, they would eventually do everything possible and go everywhere possible in the world.
Pi has been PROVEN not to repeat ever. The only repeating numbers are rational numbers.

 Mentor Blog Entries: 8 Try to read this: http://en.wikipedia.org/wiki/Repeating_decimal

 Quote by micromass Pi has been PROVEN not to repeat ever. The only repeating numbers are rational numbers. Please don't talk about something you know nothing about.
Give me some scientific evidence proving pi doesn't repeat. And no, not a Wikipedia page simply stating it doesn't. And sometimes, as I said before, you are only limited by the technology you are using. If someone calculates pi to the quintillionth digit, and it doesn't reapeat in that string of numbers, how do you know it doesn't start repeating later on, where you don't even know anything about the rest?

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 Quote by AntiPhysics Give me some scientific evidence proving pi doesn't repeat. And no, not a Wikipedia page simply stating it doesn't. And sometimes, as I said before, you are only limited by the technology you are using. If someone calculates pi to the quintillionth digit, and it doesn't reapeat in that string of numbers, how do you know it doesn't start repeating later on, where you don't even know anything about the rest?
Here are several proofs that pi is irrational: http://en.wikipedia.org/wiki/Proof_t..._is_irrational
 Mentor Blog Entries: 8 And calculators have nothing to do with the problem. You are right that by checking a calculator, that it could still repeat after a while. But the proofs do NOT use calculators, but rather they use mathematical reasoning.
 Hey Loren Booda. I have read your post and I will give my answers shortly as I am occupied with other things, but I will try and get back to you very soon.

I just want to make a few comments based on your latest post.

You implied a connection between black holes and black-bodies in that the black hole acts like a black body in how it emits radiation.

I don't know if I interpreted this right, but if a black-hole actually emits like a black-body then what I am interpreting to happen is that the black-hole is taking a chaotic situation and turning into something that is more ordered in an analogy of 'energy recycling'.

In terms of your speculation with regards to entropy, the big thing you would need to answer your question is to identify the information exchange between the region bound by the black-hole (whatever kind of geometry that may be) and anything else.

If you assume that the black-hole has its majority of information exchange with its surroundings (i.e. from its event horizon onward), then if you could measure the properties of the radiation and also the properties of the event horizon itself, then I would agree with your speculation.

Again this is from an information theoretic context but it still applies to physical systems.

One thing that needs to be asked is if the black-hole itself can only have one entropy measure in which this measure is always the maximum for the black hole with respect to its a) mass b) charge and c) spin.

If the entropy is always unique as a function of the above in the way that it is always maximal, then in terms of the information, you will have to first construct a distribution that given an internal state-space inside the whole (if things are quantized then you will have to take in the quantization mechanism into account which will probably initially come from various quantum gravity/unified theories that also quantizy gravitational effects and the space-time associated with the region) and from this you will be able to get the realizations of this distribution.

Remember that because the entropy is maximized, this corresponds to a kind of uncertainty or unpredictability of the system as a whole inside the black hole.

But the important thing will have to do with what is emitted: if the black-body emits radiation in some kind of 'white-noise' signal, then this would make a lot more sense if you had maximal entropy inside the region bound by the actual event horizon.

By seeing how the information changes over time (the radiation and the area of the hole), then it seems very probable that if you measured the signal in a kind of markovian manner over the duration of radiation emittance, then if the radiation was done in a way that it radiated information corresponding to the actual information inside the whole on a kind of 'rate of change' basis, then yes I would think that the information could be obtained if this was the case.

To know this though, you would have to actually measure the radiation and use statistical techniques to check if this was the case.

Also the other thing that is required before you answer this is to actually be able to describe the information in terms of your quantized structures for your different forces or other interactions.

My guess is that you could perhaps infer this structure based on a variety of techniques that are based on statistical theory, but if there are already developed quantization techniques that have some kind of inuitive argument for them (like for example the gauge invariance situation in string theory to yield the SO(32) representation) then it would preferential to use these.

This way you can actually say what the structure of the information 'is' and based on the radiation (if it exists) then you can measure this (over the frequency spectrum) and with the structure and quantization scheme in mind, actually see if your speculation holds water.

 http://en.wikipedia.org/wiki/Black_h...mation_paradox -- In 1975, Stephen Hawking and Jacob Bekenstein showed that black holes should slowly radiate away energy, which poses a problem. From the no hair theorem, one would expect the Hawking radiation to be completely independent of the material entering the black hole. Nevertheless, if the material entering the black hole were a pure quantum state, the transformation of that state into the mixed state of Hawking radiation would destroy information about the original quantum state. This violates Liouville's theorem and presents a physical paradox. More precisely, if there is an entangled pure state, and one part of the entangled system is thrown into the black hole while keeping the other part outside, the result is a mixed state after the partial trace is taken over the interior of the black hole. But since everything within the interior of the black hole will hit the singularity within a finite time, the part which is traced over partially might disappear completely from the physical system. __________ [Speculation]The "No-Hair" Theorem states that we may obtain only that information belonging to mass, charge and angular momentum of a black hole. However, a black hole is a maximally entropic entity. To correlate these theories, the black hole must have a physical and continuous (as opposed to an nonphysical and discontinuous) boundary which allows one dimensional isotropic propagators (electromagnetic and gravitational quanta) to escape. A black hole may be a unique entropic entity -- bounded by only one surface.
This is a very interesting question.

In the above I assumed that all information in the black-hole was not entangled, (or at most only entangled with other information in the region of the hole).

With regard also to the above question (which I want to say while it's on my mind) I want to say that you should consider the situation where at one point you have entangled states for say two particles in the black hole and then later radiated information where the information radiated is 'entangled' with information inside the event horizon boundary.

In the context of the above, if this is the case, then it needs to taken into account when analyzing not only the mechanism for radiation, but the information encoded in the radiation. Also one should also identify how information could get entangled in a black hole to identify how to also identify entangled states in terms of their detection (and also to identify the information inside the black-hole region itself).

Now with regard to the first of this question, understanding the above would also help you understand the situation you posted for the first part.

As for the second part, again assuming a continuous boundary needs to be referenced to the quantization procedure if you assume one must exist.

A quantization scheme for quantizing space-time can allow continuous surfaces, but the nature of the quantization itself says that you can only have a finite number of different realizations with respect to some subset of the configuration space.

If we assume that a finite-sub region (in terms of a volume measure) must have a bounded entropy, then this specifically implies a bounded configuration space which implies a requirement for quantization.

The nature of the actual quantization can be many things (i.e. space doesn't have to be 'jagged' like we would imagine it), but again the requirement doesn't change.

In terms of your entangled states in the context of entropy, then I don't see why this is really an issue.

The region bounded by the event horizon can be bounded even in the case of an entangled state and what I imagine would happen is something similar to a kind of 'action at a distance', which although Einstein called 'spooky' is something that you would expect to happen if entanglement still held for two information elements regardless of whether they were separated by space-time boundaries like the one you would have in the situation for a black-hole.

Physically I can understand that this might be hard to accept, even as something to be initially considered as opposed to accepted, but we know that this phenomenon exists in normal situations and I imagine that you could test it given the right conditions to see if it holds in the above kind of conditions.

Again I see things as information: the physical interpretation of the processes is not something I worry about. If you can show a mathematical argument (statistical or otherwise) to show that this kind of entanglement can happen even between situations like the inside of a black-hole and something beyond the event horizon, then it is what it is.

I understand that because light can't escape a black-hole that since this is EM information and it can't escape then it wouldn't make sense to consider that a situation can be possible and violates 'physical intuition'. But again, I don't care about trying to appeal always to physical intuition if there is a mathematical argument for 'spooky action at a distance' or some other kind of similar phenomenon.

Also one must wonder whether anything that travels at c can be at 'all points' at once instead of as opposed to something which 'needs to propagate through space' like you would expect with your intuition like a 'cricket ball being hit into the air' or something else.

 (Mblack hole·Mquantum)1/2=MPlanck, where M is mass. MPlanck=(hc/G)1/2 Mblack hole=c2R/G Mquantum=h/cr __________ [Speculation]Celestial objects tend toward sphericity (with the exception of angular momentum and charge) due to gravity primarily acting upon them as they freeze, coalesce or collapse. The entropy density of conventional objects lies between that of a black hole and that of the universe limit. Black holes are so symmetric that their horizon area is minimal, their interior entropy maximal and that their quantum fluctuations approach both.
I kind of assumed that this was the case where the object itself would be symmetric in the form of a compressed sphere and this was based on an intuitive understanding of how gravity works if the actual force mechanism is largely that everything gets drawn to the centre of the mass.

I can't comment on quantum fluctuations though.

 http://en.wikipedia.org/wiki/Penrose...arity_theorems are a set of results in general relativity which attempt to answer the question of when gravitation produces singularities. A singularity in solutions of the Einstein field equations is one of two things: 1. a situation where matter is forced to be compressed to a point (a space-like singularity) 2. a situation where certain light rays come from a region with infinite curvature (time-like singularity). __________ [Speculation]I believe that Hawking said the equivalent to a black hole is a white hole, i.e. the one which Hawking-radiates more than its infall of matter. If that is the equivalent, what is the inverse? Might it be that the concave universe is the relative inversion of each convex black hole?
Here are my thoughts on this:

If black-holes are a way to deal with situations of very high energy density for a finite region, then this should be seen to be a situation that is basically a 'stabilizer mechanism'.

Again I refer you to the previous conversation. For a system to be stable and also encourage variation, you want to take care of not only the 'spread of chaos' but also of the issue of staticity: in other words you don't want the system to have situations where they converge to some particular state and stay-there.

If you had no stabilization, then chaos would breed more chaos and the system would become so chaotic that nothing useful could be accomplished. But if the system converged in a way to promote staticity, then you would lose the dynamic behaviour intended for such a system.

So with regards to black-holes and 'white-holes' I would see a black-hole as a stabilizer. The white-hole for 'spitting stuff out' would at least to me be more of a process that takes a chaotic state and reorders the energy so that it can 'start again' so to speak. I know this is a very vague description, but the interpretation is that the energy is re-ordered so that it can be used in a context that is stable and not chaotic.

Now in terms of entropy if this is the case, the whole process if the black-hole scenario is a stabilizer. The reverse of a stabilizer would be a 'de-stabilizer'. But this doesn't make sense at least in the context of the argument of having a system that needs stability.

Being able to effectively 'manage' this situation would be able to control energy. If it ends up being the case that we get in this situation I really hope that we aren't stupid enough to realize the consequences of having this responsibility in terms of what it will actually mean.

 http://en.wikipedia.org/wiki/Black_hole_thermodynamics The Zeroth Law The zeroth law is analogous to the zeroth law of thermodynamics which states that the temperature is constant throughout a body in thermal equilibrium. It suggests that the surface gravity is analogous to temperature. T constant for thermal equilibrium for a normal system is analogous to κ constant over the horizon of a stationary black hole. The First Law The left hand side, dM, is the change in mass/energy. Although the first term does not have an immediately obvious physical interpretation, the second and third terms on the right hand side represent changes in energy due to rotation and electromagnetism. Analogously, the first law of thermodynamics is a statement of energy conservation, which contains on its right hand side the term T dS. The Second Law The second law is the statement of Hawking's area theorem. Analogously, the second law of thermodynamics states that the change in entropy an isolated system will be greater than or equal to 0 for a spontaneous process, suggesting a link between entropy and the area of a black hole horizon. However, this version violates the second law of thermodynamics by matter losing (its) entropy as it falls in, giving a decrease in entropy. Generalized second law introduced as total entropy = black hole entropy + outside entropy. The Third Law Extremal black holes have vanishing surface gravity. Stating that κ cannot go to zero is analogous to the third law of thermodynamics which states, the entropy of a system at absolute zero is a well-defined constant. This is because a system at zero temperature exists in its ground state. Furthermore, ΔS will reach zero at 0 kelvins, but S itself will also reach zero, at least for perfect crystalline substances. No experimentally verified violations of the laws of thermodynamics are known. Interpretation of the laws The four laws of black hole mechanics suggest that one should identify the surface gravity of a black hole with temperature and the area of the event horizon with entropy, at least up to some multiplicative constants. If one only considers black holes classically, then they have zero temperature and, by the no hair theorem zero entropy, and the laws of black hole mechanics remain an analogy. However, when quantum mechanical effects are taken into account, one finds that black holes emit thermal radiation (Hawking radiation).
I know this flies in the face of conventional wisdom, but again I am adament that there is a reason why you would not want a system to literally 'blow-up'.

Also for your violations, I point you to this:

http://rsc.anu.edu.au/~evans/papers/exptFT.pdf

Remember that if entropy always increased, things get more unpredictable, more disordered and for many things like life, become too unstable.

The fact that we are here today and that we have many harmonius processes working together with one another shows that there are orders that exist and we can see them if we open our eyes and collect data.

The fact physics can be described by equations shows we have order. Again it does not make sense to have things be allowed to get more chaotic as times passes, it's just crazy to have that if you want things like living creatures to exist.

Even if you didn't want living creatures to exist, why then do so many of the physics we know involved some kind of optimization condition? This tells me that things aren't just created willy nilly: there are constraints and more importantly there is a reason for them.

Here is how I see it: you have two extremes.

The first extreme is staticity: You don't want things to converge to static points under given constraints. To do this, one tool you can use are the things we see in thermodynamics. You can also uses minimal energy requirements as well to promote dynamic behaviour and of course you would use all kinds of mechanisms to promote this for a variety of situations.

But then you have the flip-side: you don't want things getting out of control. You want to have dynamic behaviour but not so dynamic that it is unstable.

To me these things help do the above and the argument can be made mathematical but the idea need not be.

The real understanding comes from seeing where these boundaries are in terms of the lower and upper bounds of dynamic behaviour and also with respect to what they mean in various situations.

The fact that we have everything working the way it is and for example black-holes not swallowing up the entire universe or trying to decide whether gravity is going to be +9.8m/s or -1000m/s or even trying to predict if we will get stuck in walls is a great testament to the design we know and experience as reality.

 You implied a connection between black holes and black-bodies in that the black hole acts like a black body in how it emits radiation. I don't know if I interpreted this right, but if a black-hole actually emits like a black-body then what I am interpreting to happen is that the black-hole is taking a chaotic situation and turning into something that is more ordered in an analogy of 'energy recycling'. In terms of your speculation with regards to entropy, the big thing you would need to answer your question is to identify the information exchange between the region bound by the black-hole (whatever kind of geometry that may be) and anything else. If you assume that the black-hole has its majority of information exchange with its surroundings (i.e. from its event horizon onward), then if you could measure the properties of the radiation and also the properties of the event horizon itself, then I would agree with your speculation. Again this is from an information theoretic context but it still applies to physical systems.

[Speculation]A black hole emits thermally, but has lost all of its infalling information except for "No-Hair" quantities. Thus Hawking radiation is a function of mass, angular momentum and charge (i.e., temperatures of a black body). All quantum numbers have been reprocessed into those three. It would be difficult to differentiate between any order from the black hole horizon itself and anomalies near it.

http://en.wikipedia.org/wiki/Cosmic_...hip_hypothesis "The weak cosmic censorship hypothesis asserts there can be no singularity visible from future null infinity. In other words, singularities need to be hidden from an observer at infinity by the event horizon of a black hole." If there were naked singularities, perhaps they would interact and share relative entropy between themselves, point-to-point. The calculated value of entropy for the black hole is actually the relative entropy bounded by the hole's event horizon and its singularity.

[Speculation] At http://dieumsnh.qfb.umich.mx/archivo...st/Hawking.pdf, Stephen Hawking includes the Hawking radiation spectrum (Equations 1-6) which is a "completely thermal" function of the "No-Hair" quantities, rather than a black body spectrum, which is just a function of temperature. I recall that random distributions (such as those of mass, angular momentum and charge) add to be random. Since angular momentum and charge distort the sphericity of the black hole, I believe that the event horizon area affects them when calculating entropy.

Quantum gravity does calculate the value of black hole entropy (i.e., on a 2-sided area). This area attempts to conserve symmetry in time. The Planck area composes the black hole area (entropy) but the Planck black hole is unique among quanta in not representing quantum numbers other than those "No Hair." Perhaps the Planck areas filter or re-radiate the "No Hair" quantities due to their Planck geometry.

[Speculation]Statistics of quanta in black holes relies on a supersymmetry there between fermions and bosons:

Conventional black hole physics has sole extensive measurable quantities charge, mass, and angular momentum (the "No Hair" theorem). From these, the Hawking temperature, T, can be found. The statistical distribution n[B. H.] is a function of T, and predicts the occupation of the hole's internal quantum states with unobservable quanta:

n[B. H.]=n[F. D.]+n[B. E.]=csch(ε/κT)

where it is assumed that T is much greater than the T<sub>F</sub> for this black hole.

The quantum within that normally designates Fermi-Dirac or Bose-Einstein statistics by its half- or whole-integer spin values has "lost its hair."

Note: Black hole equilibrium above requires the constraints put forth by Stephen Hawking in his seminal paper, Black Holes and Thermodynamics (Phys Rev D, 15 Jan 1976, p. 191-197).

http://en.wikipedia.org/wiki/Hidden_variable_theory -- (regarding encoded information), Bell's theorem would suggest (in the opinion of most physicists and contrary to Einstein's assertion) that local hidden variables are impossible. Some have tried to apply this to entangled states straddling the black hole horizon.

Pair production at the black hole horizon entangles an infalling virtual particle with its infalling (or escaping) antiparticle. Is this the only instance of either two entangled virtual particles annihilating each other or a escaped particle divulging the quantum numbers (other than "No Hair" quantities) of the fallen partner? The pair production creates opposite spins which do not measurably correlate for either the infalling-infalling particles or the infalling-escaping particles. Spin is macroscopically conserved by the hole in either case.

 Here are my thoughts on this: If black-holes are a way to deal with situations of very high energy density for a finite region, then this should be seen to be a situation that is basically a 'stabilizer mechanism'. Again I refer you to the previous conversation. For a system to be stable and also encourage variation, you want to take care of not only the 'spread of chaos' but also of the issue of staticity: in other words you don't want the system to have situations where they converge to some particular state and stay-there. If you had no stabilization, then chaos would breed more chaos and the system would become so chaotic that nothing useful could be accomplished. But if the system converged in a way to promote staticity, then you would lose the dynamic behaviour intended for such a system. So with regards to black-holes and 'white-holes' I would see a black-hole as a stabilizer. The white-hole for 'spitting stuff out' would at least to me be more of a process that takes a chaotic state and reorders the energy so that it can 'start again' so to speak. I know this is a very vague description, but the interpretation is that the energy is re-ordered so that it can be used in a context that is stable and not chaotic. Now in terms of entropy if this is the case, the whole process if the black-hole scenario is a stabilizer. The reverse of a stabilizer would be a 'de-stabilizer'. But this doesn't make sense at least in the context of the argument of having a system that needs stability. Being able to effectively 'manage' this situation would be able to control energy. If it ends up being the case that we get in this situation I really hope that we aren't stupid enough to realize the consequences of having this responsibility in terms of what it will actually mean.

[Speculation]The black hole acts as a stabilizer by virtue of its great symmetry. If you have a mass of "intermediate" symmetry (of "No Hair" variables) and collide it with a black hole, the symmetry of the black hole would at least temporarily decrease. If a "high" symmetry mass collides with another of "low" symmetry, their resultant symmetry would be "intermediate." Pure mass, angular momentum and charge are of "high" symmetry, whereas other quantum numbers would be of "intermediate" symmetry. So only the "No Hairs" impose their symmetry on the geometry of the hole, while others "can't get out of the hole." Thus a Schwarzchild black hole becomes more massive, rotating or charged.

 I know this flies in the face of conventional wisdom, but again I am adament that there is a reason why you would not want a system to literally 'blow-up'. Also for your violations, I point you to this: http://rsc.anu.edu.au/~evans/papers/exptFT.pdf Remember that if entropy always increased, things get more unpredictable, more disordered and for many things like life, become too unstable. The fact that we are here today and that we have many harmonius processes working together with one another shows that there are orders that exist and we can see them if we open our eyes and collect data. The fact physics can be described by equations shows we have order. Again it does not make sense to have things be allowed to get more chaotic as times passes, it's just crazy to have that if you want things like living creatures to exist. Even if you didn't want living creatures to exist, why then do so many of the physics we know involved some kind of optimization condition? This tells me that things aren't just created willy nilly: there are constraints and more importantly there is a reason for them. Here is how I see it: you have two extremes. The first extreme is staticity: You don't want things to converge to static points under given constraints. To do this, one tool you can use are the things we see in thermodynamics. You can also uses minimal energy requirements as well to promote dynamic behaviour and of course you would use all kinds of mechanisms to promote this for a variety of situations. But then you have the flip-side: you don't want things getting out of control. You want to have dynamic behaviour but not so dynamic that it is unstable. To me these things help do the above and the argument can be made mathematical but the idea need not be. The real understanding comes from seeing where these boundaries are in terms of the lower and upper bounds of dynamic behaviour and also with respect to what they mean in various situations. The fact that we have everything working the way it is and for example black-holes not swallowing up the entire universe or trying to decide whether gravity is going to be +9.8m/s or -1000m/s or even trying to predict if we will get stuck in walls is a great testament to the design we know and experience as reality.

[Speculation](Referring to the paper you cited)Loschmidt's Paradox would apply to Newtonian dynamics, statistical mechanics, quantum mechanics and general relativity, all being time reversible. Thus the paradox seems trivial, and as stated "one cannot prove" it.

The Fluctuation Theorem appears more plausible. In the manner of familiar statistical mechanics, two simple probabilities (of entropies representing antisymmetric processes) in the system limit yield the Second Law. It remains unvalidated.

Asymmetric time seems to be the sticking point with establishing violation of the second law. Simply put, we need a universal theory which incorporates time asymmetry to begin with. Building from limited theories, I believe, is putting the cart before the horse.

Staticity or chaos? First assume an Anthropic Principle. Next to the big bang, possibly the most powerful, turbulent entity of the universe is a supernova -- which leaves behind a black hole remnant! The black hole rebounds the might of the supernova. There is a point at which the supernova and black hole are sharing physics, Hawking radiation counteracting free quarks. Mass is fed into the nascent black hole, compressing even more the horizon, which most likely started as a plurality of such surfaces. As black holes merged from Planck to stellar, their entropy, and thus their temperature, accelerated as the sum of their radii squared. Where there once was a fluid of black holes and extreme turbulence is now a relatively cold gravitationally collapsed object within a ghostly nebula.
 I will answer your questions later on, I kinda want to chill out for a while since I had three exams in the past two days. I will also have to read the papers and get a bit of context for the things you are describing (although you have done a great job of putting in a conversational context which I really like). Very good conversations going on here: I really enjoy it.

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