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How fast does Entropy travel?

  1. Apr 21, 2005 #1
    How fast does entropy travel? When something falls into a black hole, and the surface area increases proportional to the entropy of the object that falls into it, how fast does the surface area of the black hole change? Are there ripples of entropy change that travel across the surface? Or does the size of a black hole change instantly when things fall into it?
     
  2. jcsd
  3. Apr 21, 2005 #2
    Hi Mike2

    I don't know. But here is what I would guess: change in entropy has the same spacetime limit (c) as any other process. This has nothing to do with the rest of your question.

    Black holes are regions in which spacetime approaches singularity, which is another way of saying that a black hole is a single quantum state. I should think that due to this fact any process which affects any part of a black hole must affect the entire thing. But what is the size of a black hole? I mean, of course, the naked hole itself, and not the Schwartschilde horizon.

    The surface area of a black hole depends on its radius, and because of spacetime compression in the region of the singularity, distances, like the radius, become irrational. You could for example devise a giant set of calipers and measure the black hole from outside edge to outside edge, and get a measurement, but it would not necessarily be the same measurement you would get by sinking down a measured line into the black hole until it reaches the singularity region.

    Imagine if you like a whirlpool, that is the black hole, and you want to measure the diameter of the swirl. You decide that some angle of deflection of the surface of the water is the edge of the whirlpool, and you use your giant calipers to measure how far it is from one side of the whirlpool to the other. Because you are using calipers, it is as if you measured from outside edge to outside edge, which would be straight across the middle. But it is in the nature of whirlpools that the center is depressed lower that the average surface of the body of water. If you want to know the radius to calculate the surface area of the whirlpool you have to stay on the surface as you measure it. I am sure you can easily satisfy yourself that the distance straight across edge to edge is not the same as the distance along the surface through the deep center region of the whirlpool.

    A black hole is like a whirlpool in three dimensions of space instead of two dimensions of the surface of water. And of course the funnel of a whirlpool has a bottom, while there is no bottom to the depth of the funnel of a black hole. If you try to draw a line across the surface you will never reach the other side. It goes in forever. Well. At least forever as far as you and I are concerned. And you must double that to come back out again on the other side.

    I suspect there are no ripples on a black hole surface. I remember reading a paper about what would happen to a mountainous bump on a black hole, and the conclusion was that it would not persist, but would quickly be reduced to the unblemished sphere. I'll try to run that down if it is not too difficult. IIRC it was by some Russians.

    The gravitational collapse at the surface of a black hole would not support ripples, would it? I think your second answer is the better one. Since it is difficult to imagine the radius of a black hole I should think it would be just as difficult to imagine the surface area. The size of a BH would have to change instantly if at all, due to it's condition as a single quantum entity.

    I hope someone with the formulas and numbers will come by and give us both a better analysis, and I am certainly open to correction if my answer is antiquated or just plain not even.

    Be well,

    nc
     
  4. Apr 21, 2005 #3
    Entropy is the number of ways the system could arrange itself, given that is has a certain definite energy, volume, pressure and quantities for other observable characteristics.

    Basically, your question does not make sense because entropy does not travel. The entropy depends on the state of the system, so you are asking "How fast can the state of a sytem change?". Except that you dont measure this rate in the units of speed, distance/time, which is what you are asking.
     
  5. Apr 21, 2005 #4

    marcus

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    I too hope someone with formulas will appear (perhaps a gnumber-gnome) and I too am open to correction about this. I had the notion that entropy was only defined for equilibrium states. Like in thermodynamics temperature is only defined when a system has reached a steady state. So you have to wait until the gross overall see-sawing and surging back and forth stops and things settle down---and only then is something like temperature (or, I think also, entropy) rigorously defined and physically meaningful. Though I am not sure this is right, Mike's idea of entropy traveling around seems somehow paradoxical to me.
     
  6. Apr 21, 2005 #5

    marcus

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    Is it not the logarithm of the number of ways? or is that being unnecessarily pedantic? (the log helps make entropy additive)

    I think I agree with you that we are asking "How fast can the state of a system change?" and for the question would be how fast can it settle down into a stable condition----how fast can it transition from one equilibrium state to another, after being given a kick. because the entropy wouldnt be measurable, i fancy, in between the equilibria
     
  7. Apr 21, 2005 #6
    Entropy is the measure of disorder in a system. We can't easily solve three-body and higher wave equation systems so we use a measure of their entropy to describe them as a whole system. A whole system may take on or lose entropy over time locally, but when averaged over larger scales entropy always tends to increase. This means we can build orderly systems of materials, but over time they always fall apart. Order requires maintenance, and maintenance is work and attention.

    Left alone, over a long enough period of time, every object we know tends to work its way toward one of two fates: cold dust, or gravitational collapse. The universe, in so far as it can be thought of as a single object, must do one or the other eventually, but we cannot know which it will do. The outcome could depend on the whisper of a butterfly wing.

    Or, as the multiverse, each region must deserve some fate as utterly chaotic or utterly orderly, but we cannot predict which spacetime will tend at last to one or the other. I suppose there must be a fractal rule for the division of the multiverses into radient event universes. I suspect it may look very much like the Julia set, or perhaps fractal dragons.

    But the op question is toward the change in entropy that may occur in a wavelike pattern, or ripple, in the surface of some region. We add entropy to a region, say we store up a silo of corn, you can see right away that the region defined by the silo has much more order in it when full of corn than it does when it is full of dust. There has been a change in order for all our work of growing and storing corn. You could measure and map the entropy stored up in the corn, and thus in the silo, couldn't you? You could map the changes as some kind of wave, a ripple perhaps. The farmer comes every day all Winter and shovels ninety scoops of grain down for the sheep and the cattle. You could map that over time, make a wave of the rhythm of the scoop, the slow sweet munching of the grazers.

    It seems, whatever size the black hole is, it is going to be very nearly smooth and spherical, probably hyper-spherical, and so is the general form of the Schwartzchilde horizon that surrounds it. Do objects occupying space near the horizon get crushed by the jam up of infalling debris? Do tortuous tides and hideous screaming radiation fields ripped out of the torn fabric of hyperspace shriek the quintessense out of any doomed form that happens to fall into them? Probably nothing so dramatic, IMHO, but it was fun to imagine it.

    Now Marcus you have been showing us that the singularity is removed, so that the black hole way is more like, maybe, a tunnel to another universe, maybe one very similar to our own. If it is a tunnel it doesn't really have a spherical surface area, but a more cylindrical one. Then the surface area of the tunnel depends partly on the diameter of the tunnel, but also partly on the length of the tunnel. For black holes, the length of the tunnel is, well, a pretty long time. And the universe on the far side? Is it growing and expanding and does it have an age we can speak of or even a duration?

    Here is a ringer for you. What if each black hole branches out not to just one new baby universe, but to a whole tribal family of full grown Manly and Womanly Universes?

    Lets not go there. I feel a sense of karma sutra coming on.

    Be well,

    nc
     
  8. Apr 21, 2005 #7
    CHOP index:

    "entropy
    .Boltzmann formula
    .change in Joule expansion
    .experimental
    .fluctuations
    .from partition function
    .Gibbs formula
    .of a monatomic gas"

    also,

    "entropy (dimensions)"

    I have omitted the page numbers but can give them if needed. But just at a glance I see that there is something about entropy fluctuations.

    nc
     
  9. Apr 22, 2005 #8
    CHOP
    Thermodynamic fluctuations:
    Entropy fluctuations

    [tex]var=kT\frac{\partial S}{\partial T}|\sub p=kC\sub p[/tex]

    S entropy
    k Boltzman constant
    T temperature
    p pressure
    [tex]C\sub p[/tex] heat capacity, p constant
     
    Last edited by a moderator: Apr 22, 2005
  10. Apr 22, 2005 #9

    Chronos

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    Crossons answer is appealing. Entropy is a change of state, no boson exchanges required. Observing the consequences would, I think, be time constrained. The universe could, in principle, undergo an instantaneous change of state, but communicating that information to distant observers would travel at the same old snail speed of light. So basically, all observers in the universe would be the first to know it had changed states - and feel priveleged.
     
  11. Apr 22, 2005 #10

    marcus

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    is karma sutra even better than kama sutra?

    A professional Philosophy of Science guy, who also moonlights as a science journalist, has done an expert job of criticising Smolin's CNS as of the state of knowledge in 2002

    http://arxiv.org/gr-qc/0205119
    Is there a Darwinian Evolution of the Cosmos? - Some Comments on Lee Smolin's Theory of the Origin of Universes by Means of Natural Selection

    he raises some of the issues you raise and finds some weaknesses and gives a kind of balanced assessment, as is appropriate for a tentative proposal like this

    and (rather surprisingly I thought) he raises this point about multiple litters and he even cites a paper by two guys who studied it. it is his reference [2] by Barrabes and Frolov.

    anybody who wants to critique CNS, this paper by Vaas would make the job easy, but altho it raises many questions it does not totally reject
     
  12. Apr 22, 2005 #11

    marcus

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    tinkering with the LaTex, for purely notational (cosmetic) reasons, saying "quote" on somebody elses text will get you the source

    LaTex does subscripts with a dash _
    as in C_p for "C sub p"


    sample notation adapted from your post

    [tex]var=kT\frac{\partial S}{\partial T}|_p=kC_p[/tex]

    S entropy
    k Boltzman constant
    T temperature
    p pressure
    [tex]C_p[/tex] heat capacity, p constant
     
    Last edited: Apr 22, 2005
  13. Apr 22, 2005 #12
    I like Crosson answer. Entropy is not something physical or an observable like position momentum or energy.

    The surface area(if we can call it a surface) increases with the amount of mass absorbed by the black hole.

    The radius can be found by setting the classical escape velocity to c.

    [tex]
    \frac{1}{2}mv^2 = \int_{R}^\infty \frac{GMm}{r^2}dr
    [/tex]
    [tex]
    R = \sqrt{\frac{2GM}{c^2}}
    [/tex]

    Well the surface area just changes with the radius. As far as weather or not the surface ripples when mass enters at a particular point, I do not know. Get some one who know GR and Black holes and ask them. Ripples make sense though, since if you allowed an instantaneous transition it would open the door for information to travel faster then the speed of light.
     
  14. Apr 22, 2005 #13

    marcus

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    the surface is believed to ripple and to have resonant frequencies ("quasinormal modes") which depend on the mass of the hole

    there have been several dozen papers about the BH vibrations frequencies
    here is a 2004 review
    http://arxiv.org/abs/gr-qc/0411025
    but there have been more posted since then
    BH vibrations have attracted quite a lot of research interest

    the more massive, and bigger the hole is, the lower the resonant frequencies as you would expect. they have formulas to calculate
    (but nobody has ever seen a black hole jiggle or heard one ring)
     
    Last edited: Apr 22, 2005
  15. Apr 22, 2005 #14
    I ment a ripple of the event horizon from the mass passing through the horizon. This is what Mike2 seemed to be refering to, though the question was ill formed to start out with.
     
  16. Apr 22, 2005 #15

    marcus

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    exactly
    dropping some mass into a black hole is considered to be a possible way to excite these vibrations

    I think we are talking about the same thing
    for more info you might want to look at Berti's review article, it is recent (november 2004) and concise and sort of covers the subject
     
  17. Apr 22, 2005 #16
    That's easy then. Nothing passes through the horizon so it does not ripple.
     
  18. Apr 22, 2005 #17

    marcus

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    touché NC, I dont know right off how to resolve that apparent contradictions.

    I believe that we can observe a BH gaining mass (indeed the process of crud spiralling down into the hole drives quasars and makes Xrays and we observe that stuff, that is astronomers do, and tell us about it)

    and yet we are also told that if we watch Captain Kirk of the Enterprise being dropped into a black hole it would be like an endless slowmo re-run of Star Trek where his image fades out as it moves slower and slower but never actually passes through

    or better, Donald Rumsfeld.

    so when do the ripples start?

    also this Quasi-Normal Mode "ringing" of black holes that they talk about and do research about is not actually very vibrant. Someone I read said it was more like the ringing of a marshmellow than a bell. Somehow the ripples get damped out very very quickly, a quick "dup" instead of a long "doioioinnngggg"

    but what can i say? that review article by Berti has several dozen research articles about the ripple frequencies of black holes, including computer simulations and all kinds of respectable expertise and people getting into arguments and so on. I have to take it seriously.
     
    Last edited: Apr 22, 2005
  19. Apr 22, 2005 #18
    OK, if you must know, I was thinking more in the lines of changes in the cosmological event horizon, where at a certain distance from us, space is receding faster than the speed of light so that we cannot know what is happening there. This cosmological event horizon shares a lot of characteristics with a black hole event horizon. As object approach the cosmological event horizon, they will appear to redshift, slow down, and freeze.

    So just as with a black hole event horizon, if objects should cross it, then the entropy contained in objects crossing it would disappear. And so that entropy does not decrease, there must be an entropy associated with the cosmological event horizon just as there is with a BH event horizon. And just as with a BH horizon, the entropy on the surface of the cosmological event horizon is a constraint on the entropy inside the cosmological event horizon. Some have calculated that the entropy of the cosmological event horizon is quite large and there is no danger of events inside exceeding it. But then again others think there is some entropy associated with space itself. They consider the entropy of space might be calculated as the Weyl curvature divided by the Riemann curvature, or something like that.

    That being the case, I consider what happens with an accelerating universe where the cosmological event horizon is shrinking; its surface area from our perspective is getting smaller. And the constraint of the entropy inside is being reduced. Suppose that the entropy inside the cosmological event horizon were equal to the entropy calculated for the surface of the cosmological event horizon. Would the decrease in the entropy of the cosmological event horizon have immediate implications for the observable universe? Or would the change on the surface of the cosmological event horizon take time to permeate throughout the observable universe?

    I ask this because it seem curious that the universe started accelerating about the same time that life arose on earth. Is this just a coincidence? Or is there some power or principle beyond the observable universe imposing its will of creation in the universe. :wink:
     
  20. Apr 22, 2005 #19
    Mike,
    I am not happy with the intelligent designer any more than with the happy anthropos. Both are dead end philosophies. Why are they dead end? Because they dismiss the question. Imagine the two of them together! G-d did it that way so that anthropos would have a nice place to live. That answers everything and nothing, so it is useless to us as scientists trying to grow in understanding. Worse for the fate of humans, it gives us an excuse to quit trying to answer the question. G-d must know what HE is doing, and look what a great job HE has done already. Trust in G-d. Don't think anymore, just do what the Morlocks tell you and everything will be just fine. Well. For the Morlocks, anyway.

    However your question does have merit. I only want you to keep asking it and not get hoodwinked by opportunists who want you to rely on their opinion of what it says in some book.

    And, as rephrased, you question is much better than before. If, as now seems possible, the passage through the time warp of a black hole leads to regions of spacetime which may be similar to but disjoint from our own "universe", then it follows that our own universe is arrived at from somebody else's black hole. In short, when looking out at our universe, we are looking out from the inside of a black hole. This is a startling revelation, and if true, answers many questions about what goes on inside black holes horizons. Plus, it is a beautiful inversion of symmetry, worthy of Klein or even M.C. Esher.

    Back to Entropy for a moment. Lets say that we can define a scalar field that represents the condition of entropy in some region. Some locations in the region may have a higher entropy, some may have a lower entropy. Over time, the entropy in any location in the region may change. In fact, if we examine the changes, we may see that a drop in entropy in one location is always accompanied by a rise in entropy in immediately adjacent locations. Entropy, like temperature, obeys dispersion rules, and always tends to expand to fill any space available to it. We could measure the rate of this expansion and talk about the velocity of entropy change, rather like we talk about the progression in meteorology of a cold or warm front. It isn't precisely a velocity, and lots of assumptions have to be made, but generally it is useful enough to predict the weather. It is not very precise, but still meaningful, to talk of a storm front as moving in a certain direction at a certain velocity. Entropy should obey the same rules. As such, there seems to be no reason to believe that entropy is free of the usual rules, nor that it has more rigid restrictions.

    In a case in point, we humans value things that have a low rate of change of entropy. We like automobiles that last for a few years at least, and gold is gold because it doesn't readily fall down the entropy slope to become gold oxide. So we manipulate conditions to protect our possessions from the downward slide. We build houses, and close the drapes on the windows to protect the carpeting from the effects of bright sun. We make all kinds of efforts to keep entropy out of our lives, but it is only a delay tactic. Eventually we and everything we know around us will go either to dust or the cosmic fires of the black hole. In the between times, we get to play.
     
  21. Apr 22, 2005 #20
    I have to add that this idea of a surface with an area is extremely mushy. Ask yourself what the radius of a black hole is. Time and space are meaningless at the singularity so how can we speak meaningfully of a radius and a surface area?

    The problem seems to me to arise from Einstein, who gave us the two dimensional "rubber sheet" model of gravity. It is ok as a transitional model, and makes the necessary point, but it isn't the whole picture. There is not a two dimensional timespace surface, that is only part of the model, and must be discarded to think of timespace as it 'really' is, in more than two dimensions.

    Look at the rubber sheet model. Every textbook on relitivity has a sample. You start with a flat elastic sheet. You make lines on it , an x,y grid. You fix the edges in a horizon frame, normal to an accelerative force, and add one or more spherical objects the sheet and observe their behavior. They seem to attract each other. I am sure you know the drill.

    But now look at some features of the model. Look at the area of an x,y square near the edges, out where the sheet is fixed to the frame. You could cut out a slip of paper that would just fit into an xy square in that region. Then you could slide your sheet of paper over to the region where the central weight is located. You find that the square of your sheet of paper taken from the edge of the model is not the same shape as the distorted squares near the weight. This has to do with the topic called gauge invarience.

    The surface area of the square near the edge is not the same as the surface area of the square near the weight. Which gauge will you use to calculate surface area?

    Now we have to abandon the elastic sheet metaphore and go on into higher dimensionalities. We have three dimensions of space, not just two. Think now of the sun instead of a bowling ball, in space instead of on a sheet. You see immediately that the sun is not surrounded by elastic sheets. But the stretching effect, which is the main point of the model, is still there. Space, good old three dimensional space, is not the same near the sun as it is far away from the sun. And neither is time. Space and time are compressed by the gravitational field.

    With the sheet we are interested in what a small ball does when placed near the large ball. It falls inward, on the sheet, toward the large ball. The same effect is what we are interested in in three dimensions. Objects in space tend to fall inward toward each other, as if there were a depression in space "like" the depression in the elastic sheet. But there is no elastic sheet. Instead, spacetime itself is distorted. And, anything embedded in spacetime will be distorted also. That would include light, and you and me in our rocket ship.

    The effect of gravity in three dimensions might better be thought of as a kind of density rather that a kind of rubber sheet. Imagine a dust cloud in free space, made entirely of tiny particles, all the same size, all the same distance apart from each other. Now add a gravitational field in the center of the cloud. The effect of the gravitational field is to make the timespace distances between the particals smaller at the center, and further apart out toward the edges of the cloud. This is analogous to the depression of the elastic sheet at the center, only now we are in three dimensions. Instead of stretching and compressing the sheet, we now see a change in density of dust.

    nc
     
    Last edited by a moderator: Apr 22, 2005
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