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Entropy, information and Omphalos cosmology

  1. Jul 15, 2006 #1
    Entropy, information and Omphalos cosmology

    Roger Penrose, in The Road to Reality, discusses the Second Law of Thermodynamics and the Universe’s entropy. He floats (section 27.13) an order-of-order-of magnitude argument that the entropy of our Universe has increased from mighty big (10^10^88) to humungous (10^10^101) from the time of decoupling to the present, as its available degrees of gravitational freedom (that were dormant in its initial state) are taken up. He seems to suggest that on a Universe-wide scale, the second law rules because of gravity. He also makes the point that there is plenty of scope for a further increase of entropy to monstrously vast (10^10^123) if degrees of gravitational freedom were to become fully taken up.

    Now there is a general inverse relationship between entropy and the rather difficult-to-specify concept of “information content”. The larger the entropy, the less the “information content”. A high-entropy jumbled string of 0's and 1's contains less information than the same-length sentence of ordered ASCII code. Folk who deal with this concept sometimes quantify information as the amount needed to specify the system, or to create it.

    The Second Law applied on the largest scale implies that the information content of the Universe as a whole (if such a “whole” exists) is decreasing as it evolves.

    Cosmologists may remember Philip Gosse, who in 1857 argued in Omphalos that the world (and the Universe?) was created only a few thousand years ago (say in 4004 B.C.); complete with all that was needed for it to function just as we find it does. According to Gosse the Universe was created with all the fossils needed to keep palaeontologists employed and alluvial gold deposits ready to be mined in California. Young-Earth theologists hold similar views to this day.

    It might be argued that the easiest Universe for a Creator to make is one that needs the least amount of information to specify it. Global operation of the second law would then ensure that this is a fully evolved Universe. The later you make it, the easier it gets! Perhaps Philip Gosse got the age of the Universe wrong, and it is only a nanosecond old.

    Or perhaps the Creator is Lazy and hasn’t made it yet.

    Is there evidence that the second law operates on a Universe-wide scale, or is it everywhere a local statistical phenomenon?
  2. jcsd
  3. Jul 15, 2006 #2
    I won't say why, because we're not supposed to get philosophical in cosmology, but you are so close to being exactly, perfectly, and/or concisely – right on the money.
  4. Jul 16, 2006 #3
    If the inflationary theory is correct then the primordial inflationary era following the Big Bang rapidly drove the universe to maximum entropy (in absence of gravity). If one wanted to “create the universe at the position of minimum order”, then the logical point would be the point towards the end of the period of primordial inflation, when the universe was almost completely uniform, and before gravity had “switched on”. This was the point of minimum order. It was only when gravity “switched on” after the initial period of inflation that the entropic arrow was effectively reset, and the universe has been trying to catch up ever since.

    Best Regards
  5. Jul 17, 2006 #4
    Thanks for this clear timeline of how entropy could have varied near "the beginning".

    The part that still puzzles me is exactly how the action of gravity, in curdling the Universe from its immediate post-inflation uniformity, increases "global" entropy, and if it is indeed gravity that now maintains the rule of the second law on a Universe-wide scale.

    Penrose invokes the Bekenstein-Hawking black-hole formula to estimate the potential that gravity has for generating entropy. But what about the myriad gravitational condensations that don't end up as black holes, but rather as ordinary stars and galaxies? If the second law does prevail globally (as I guess it must - or cosmology wouldn't be physics-made-large) one should be able to estimate the global entropy increase from a model condensation of this kind, and match it smoothly on to the entropy increase from a similar condensation that ends up as a black hole. Do you know if and where such estimates have been made?

    The problems I'm having here are related to the difficulties I expressed in another thread (The Virial Theorem and Cosmology) in which folk in this forum helped a great deal to clarify my difficulties with the energy dissipated from gravitational condensations. But it seems that I haven't yet got straightened out completely. Then it was energy. Now its entropy!
  6. Jul 17, 2006 #5
    Thanks for the compliment. But I'm more confused than right, hoping that others will clarify the difficult topic of entropy for me. I'm afraid that my remarks on Gosse's Omphalos cosmology were somewhat tongue-in-cheek, and not meant to be taken too seriously!
  7. Jul 17, 2006 #6


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    that seems like a nice way to think about it.
    when you say gravity "switched on" do you refer to the moment when the scalar field (presumed to be driving inflation) decayed?

    I don't know whether I agree or disagree with your scenario----no, clearly I NEITHER agree nor disagree, I don't understand early universe matters well enough to take a stand.

    But I can dimly fathom the idea that if you neglect gravity or consider it "turned off" then the U began with very HIGH entropy
    while if you include gravity in the picture then entropy began very LOW

    because using oldman's terminology, from the gravity point of view an "uncurdled" universe is extremly improbable. a uniform distribution of matter is astronomically unlikely.

    dark energy has a kind of Emperor's clothes feel to it. it is embarrassingly drafty. once one believes in a scalar field that doesnt clump and is constant and uniform throughout the universe then where does one stop----does one go on and believe the Arthurian Legend as well?
  8. Jul 19, 2006 #7
    The following is speculation.

    If there was initially some kind of repulsive “anti-gravity” force (during inflation) then this would not only explain the driving force behind inflation itself, but would also work to “smooth everything out” – anti-gravity would tend towards a universe with uniform distribution of mass-energy. In presence of anti-gravity, the maximum entropy state would be one of total mass-energy homogeneity/uniformity, with no clumping whatsoever.

    Assuming this was the case during inflation, and that anti-gravity switched to gravity towards the end of inflation, then the state of maximum entropy (mass-energy uniformity) would suddenly become a state of minimum entropy – because gravity would start the process of gravitational clumping (which then leads to formation of stars etc). In effect the “reversal of the gravitational arrow” at the end of inflation (turning anti-gravity into gravity) would suddenly switch the state of high entropy to a state of low entropy.

    If it were not for gravity, then the universe would have stayed pretty much as an ever-expanding but uniform and homogeneous mix of hydrogen, helium, photons, neutrinos and a few other particles, from the end of inflation to the present day. The force which powers the stars is gravity, the force which powers black holes is gravity, and the stars turn matter into energy (increasing entropy as they do so).

    They still increase entropy – by converting mass to energy and releasing it as photons. The reason Penrose fixated on black holes is I guess because the calculation is made a bit easier – the entropy of a black hole is simply proportional to its mass.

    A star like the sun will create, over its lifetime, about 10^6 photons for every baryon contained within it. The universe today contains about 10^10 photons per baryon (mostly remnants from the big bang). The entropy increase due to stellar processes, though these processes power all life on earth, is therefore extremely low.

    I’m not aware of detailed calculations, but I’m sure they exist.

    OK. I’m no expert on the Virial Theorem, but as a gravitationally bound system collapses (radius decreases) the total kinetic energy of the system increases, and it must therefore lose (dissipate, radiate) energy if it is to remain stable – this energy dissipation is effectively an increase in total entropy (ie gravitationally bound systems result in entropy increase as they collapse).

    I’m an ignoramus, sorry. I don’t know what a “scalar field” looks like – is it anything like a football field? My speculation is that there was an initial anti-gravity (repulsive) force (which drove inflation and eventually resulted in an homogeneous distribution of mass-energy), which then switched into a gravitational (attractive) force at the end of inflation.

    Exactly – and if the early universe started with an homogeneous distribution of mass-energy with no (or anti) gravity, and then gravity “switched on”, then you would suddenly flip from high to low entropy.

    In the presence of gravity, yes. As Penrose points out, the alleged “starting state” of the universe, just after inflation, points to a very low entropy, which EITHER implies an artificial “unreasonable aim of the creator” (to use Penrose’s words), OR it implies some mechanism which drove the early (pre-inflation) universe to that apparently low-entropy state. Penrose doesn’t like (= doesn’t believe in) inflation, so he’s stuck with the “unreasonable aim of the creator”.

    But switch gravity off, and that early universe is NOT in such a low entropy state after all - in the presence of anti-gravity then the uniform distribution of the post-inflation universe is in fact the most likely distribution – ie represents maximal entropy.

    In other words, whether any given state of the universe represents high or low entropy is a perspective based on the prevailing forces – it depends on what forces are available. If the nature of these forces changes over time, then a state of low entropy can suddenly become a state of high entropy, or vice versa.

    Dark energy, I believe, is supposed to be something like anti-gravity, isn’t it? It drives the universe to ever-faster expansion?

    Best Regards
  9. Jul 19, 2006 #8
    I'm a little confused, a force is something that causes the motion of particles through space. But cosmological expansion is simply more space. If particles expand during inflation, it is the space between them that is increasing and not particles moving through space away from each other. If gravity (positive or negative) is a force that causes particles to move through space, then what does gravity have anything to do with expansion or contraction of space itself?
  10. Jul 19, 2006 #9
    Your confusion is perhaps arising from the notion that there is some pre-existing thing called “space”, which exists independently of any mass/energy. This is the classical Newtonian idea.

    I suppose that depends on whether one believes the notion of space in absence of mass/energy has any meaning - I don't. The existence of mass/energy is what gives meaning to space - therefore a universal force of repulsion between all mass/energy would result in "expanding space".

    What's the difference? Mass/energy defines space - if mass/energy expands, then by definition so does the space defined by that mass/energy. What IS space but "the separation between instances of mass/energy"? (unless you believe, like Newton, that space exists "in and of itself", independently of mass/energy?)

    Gravity causes particles to move relative to each other. The separation of those particles we call space. It's not the case that space is "pre-existing" like some kind of cartesian graph-paper backdrop against which particles move.

    Best Regards
  11. Jul 20, 2006 #10
    Thank you for this and other comments, Moving Finger. Together with those of Marcus they have substantially clarified my understanding. For instance, I now see why Penrose labels the Big Bang “extraordinarily special”-- because it is an initial state with what at first glance seems to be an extraordinarily high entropy, whereas "initial" states usually have low entropy.

    I remark only that there are perhaps better ways of qualifying the word entropy than saying it is “high” or “low”. Indeed, as you point out, these adjectives depend on one’s perspective.

    I now appreciate that the evolution of the highly symmetric standard model universe is adiathermal (because of its symmetry) and, if reversible, adiabatic --- except for “microscopic” irreversible changes like galaxy and star formation. (Here I am paraphrasing Peacock’s Cosmological Physics, section 9.1). It is these microscopic processes, driven by gravity, which must lead to a monotonic increase in entropy from the initial homogeneous state. If one accepts the second law on a universe-wide “global” scale, then global entropy can never decrease. The end of inflation (if inflation happened) would mark a point of inflexion in the monotonic increase of entropy, rather than a sudden switch in perspective from "high" to "low" entropy.

    Two other comments:

    1 I find Penrose’s estimate of the maximum global entropy (10^10^123) as that of a black hole with mass equal to that of the observable universe still a bit puzzling. Is it not a strange fact that a black hole with the universe's mass has an event horizon with similar radius? It could then be argued that our flat universe is nothing but a black hole, and that it already has maximum global entropy. But, as Penrose points out, the entropy today is only 10^10^101. How can the amount of entropy estimated depend on whether one is outside or inside this black hole?

    2 About the device called inflation that, it is proposed, prepares the initial state of the standard model universe from some sort of primordial quantum chaos:

    Presumably entropy would also increase monotonically during this event, which is too brief for slow-acting gravity to operate "microscopically" as an irreversible entropy-producing agent.

    The net result of inflation seems to be to create an initial Universe, complete with all that is needed for it to evolve further, just as modern cosmology now describes it did. I comment that the Omphalos cosmology of Philip Gosse had a similar purpose, suited to the knowledge of the mid-1800's!
  12. Jul 20, 2006 #11
    Some people in the past have claimed that time would reverse when the Hubble constant becomes negative (i.e. Universe contracting towards the Big Crunch). This would basically be a reversal of the second law of thermodynamics. Thermodynamic entropy decreases as radiation is absorbed (which can only happen if energy comes from exterior sources), as angular momentum is being transferred from massless photons to massive particles so that work can be done among massive particles (such as the process of photosynthesis and the use of solar panels). In our region of the universe, evidently, the opposite is true. Residual mass is being converted to massless energy faster than the contrary, carrying away the angular momentum with it. Light cannot do "work" on light, only matter and light can do work on matter. No work can be done if there are no objects in the vicinity. Hence, gravity in the form of the big crunch would provides a period in which entropy may decrease. But gravity, as it appears in our universe is not strong enough to capture all the light (or even matter for that matter) so that it may be used to do work again. Only something with a very large surface area, and something that confines the entire universe, and something that can convert the the all the remaining massless radiation into matter, or allow it all to be absorbed by matter, would allow the universe to be conserved (and it cannot be a black hole). Obviously, this requires that we abandon the cosmological principle, since it predicts that there will not be any sufficiently large objects having a surface area of 10^18 + square light years that would feasibly return the energy (and a black hole of this size would have an inescapable singularity, meaning that it would have to be some other entity). If such an object were scaled down to the size of the earth, the sun would be a mere nanometer in length. The universe would have to be completely self contained, even at very high energies, as much as how quarks that make up the universe themselves are not subject to big crunch or heat death (although gluons may be emitted and absorbed between them). After 10^23 cycles, nuclei don't disappear nor diffuse, instead the simply exchange matter among baryons and mesons (albeit with a connection with an external environment). Why can't our universe do the same thing?
    Last edited: Jul 20, 2006
  13. Jul 20, 2006 #12
    Then how much mass equals how much space?
  14. Jul 21, 2006 #13

    It seems to me that Moving Finger and Mike2 are having trouble agreeing on what "space" is.

    Take heart. It's difficult to agree about space.

    The word "space' is so often used in cosmology, where it is often qualified as "expanding", "curved" or "distorted". Analogies are drawn between space and two-dimensional curved surfaces, balloons, elastic sheets and background fabrics; the notion that the Universe is filled with a curved, stretched and distorted medium called space must by now be firmly embedded in the minds of the public. See for instance the amazing drawings in the May 2005 issue of National Geographic.

    But "space" is quite a mysterious concept, not easily defined. Over the years many philosophers and other eminent folk have attempted without much success to describe space and to endow it with some physical reality, as it were. Here is an example of obfuscation by two eminent physicists , who wrote that space is:

    "..an abstract construct possessing those properties of rigid bodies that are independent of their material content".

    Newton used "absolute space" as a standard against which acceleration could be measured. His space was like an invisible backdrop hung from the reference frame of the "fixed" stars. Even in the late 19th century, after Maxwell had introduced abstraction as a fashionable modus operandi in physics, space was imagined to be filled with an all-pervasive but intangible fluid — the ether — whose main purpose was to sustain light waves. But these ideas have failed.

    Einstein himself was not entirely clear on how to define or describe empty space. In The Meaning of Relativity (1922), he wrote rather vaguely of Poincaré's views, theorems of congruence, and forming a space of reference by "continuing" one body until it met another — thus giving an impression of space as something that can be filled with contiguous bodies. He also warned of "the fatal error" of speaking of space in the abstract.

    In cosmology and relativity space has been described as an abstract "substratum" (Bondi) and reference is often made to "the fabric of space" (e.g. the popularisations of Brian Greene). Nowadays physicists call empty space "the Vacuum" and assign to it structure; "loops", "dark energy" and the ability to spontaneously erupt into virtual particles and even entire Universe(s).

    All this without defining "space " itself! Amazing.

    Here is an alternative way of defining things, which I like. It is simply to describe what can be done with them. Example: if you define a bicycle by listing the physical operations you can perform with it and on it — for example as something you can sit on, move by pedaling, and steer to where you want to go — you at least convey some idea of what a bicycle is.

    This is the method of operational definition, recommended in 1927 by Percy Bridgman in The Logic of Modern Physics. Along these lines, I maintain that for the purposes of physics and cosmology the simplest way to define "space" is to describe what you can do with, or in, "space".

    Like: Space is what you can swing a cat in.

    For the more serious purposes of cosmology and physics, one might describe the scheme you use to measure distance. An operational definition of space would then be to define space as that in which you implement this scheme. The scheme itself is embedded in the subject of relativity.

    As far as I am concerned there is no need to say more about the matter. But I'm sure others won't agree.
  15. Jul 21, 2006 #14
    The end of inflation point of the Big Bang corresponded (in the presence of gravity) to a very low entropy when compared with the present entropy of the universe. Penrose calculates the entropy of the universe today is about 100 orders of magnitude higher than it was in the very early universe (ie post-inflation), and it will be a further 23 orders of magnitude higher still when we approach “heat death”. The total difference of 123 orders of magnitude (entropy of post-inflation Big Bang compared to entropy of heat death) is what Penrose refers to as the “unreasonable aim of the creator” in creating the initial (post-inflation) conditions.


    “High” and “low” are relative terms. In a universe with universal repulsion, the homogeneous state at the end of inflation represents the “highest” entropy state attainable in such conditions (there is no higher entropy state available in the absence of gravity); switch to universal gravitational attraction, and the homogeneous state at the end of inflation represents one of the “lowest” entropy states attainable in such conditions. The fact that entropy tends to increase explains how inflation and universal repulsion combine to produce the low entropy starting-state (post inflation) of our universe. Hence no need for the “unreasonable aim of the creator”.

    Yes, I’ve wondered the same thing myself.

    The current estimate for average baryon density is about 10^-30 grams/cc. The mass-equivalent of photon density is about 4 orders of magnitude lower than this (therefore may be ignored). At this baryon density (assuming no dark matter), the universe would need (based on my calculations) to have a radius of approx 4 x 10^10 light years or more in order to be a black hole. The “observable” universe is I believe estimated at about 1.4 x 10^10 light years in radius? That does indeed seem very close to the size needed for being a black hole (especially when you factor in the estimate of dark matter density which may be many times more than the baryon density). But if the universe is a black hole, then how does this match up with the observation that the universe is expanding at an ever-increasing rate? Does this mean that at some time in the future the universe will stop being a black hole? Or does it just become an increasingly bigger and less dense black hole?

    The critical black hole density scales (I believe) as the inverse square of the schwarzchild radius, so a doubling of radius would result in one quarter of the critical density. But for a given volume containing a fixed mass, doubling the radius would result in a drop in density to one eighth (actual density for a given mass scales as the inverse cube of the radius). Thus if the universe is today a black hole, and it continues expanding, it must (if it is finite in size) reach a point at some time in the future when it stops being a black hole.

    It seems quite a coincidence that the present era corresponds to a density and size of universe which is just on the borderline of being a black hole…… go back to much earlier times (less than a billion years of age) and our universe was definitely a black hole, go forward to much later times (more than 100 billion years of age) and our universe is no longer a black hole. Is this right, or am I making some big mistakes somewhere?

    Interesting point. I guess the conventional estimate of black hole entropy assumes no structure inside the black hole – ie all the mass/energy is in the form of a singularity. If this is not the case, if some or all of the mass/energy is distributed as structure within the event horizon, then it’s possible the black hole would have a lower entropy than outside observers guess (but since the outside observers don’t know about the internal structure, all they can do is to guess). The Penrose calculated black hole entropy is therefore (I guess) a limiting maximum assuming all the mass is at the singularity (or at least structureless).

    Yes. Entropy increase during inflation (in absence of gravity) would drive the early universe to a homogeneous distribution of mass-energy.

    Best Regards
  16. Jul 21, 2006 #15
    I agree with all you say. I know very little about black holes, let alone the details and assumptions made when Bekenstein and Hawking arrived at their formula, which Penrose understands and uses. But your post is the first time I've heard anyone mention structure inside a black hole --- despite the naturalness of this if you imagine big enough (say universe-sized) holes. I see no reason why there should not be black holes within black holes within black holes... and so on ad infinitum, which is another interesting thought.

    Your suggestion that the universe could stop being a black hole (if it is one) as it expands is even more interesting. I fear that it may get buried in this rather odd-sounding post, and I've taken the liberty of extracting it into a new thread. I'll watch its progress/resolution with interest.
  17. Jul 21, 2006 #16
    Thank you, that is a good start. But if space is operationally defined as the background in which particles move, it should be keep in mind that we don't know that much about what particles are. Not only that, but some think that particles can not even be defined in space whose curvature is rapidly changing (Wald). While others think that particle are equated to space through differential structures (Torsten and Helge). So it seems some work still remains to be done in that area.
  18. Jul 22, 2006 #17
    What I wrote had nothing to do with particles, let alone defining space as the background against which particles move.

    And since you can't do anything to or with with a particle, it is a only a (useful but abstract) figment of the physics imagination.
  19. Jul 22, 2006 #18
    One problem with this is that for any agent subject to such time-reversal, their world would be indistinguishable from our world. If one reverses the arrow of time, then everything reverses, including our psychological arrow. So an agent experiencing this would notice nothing different to what we experience – to us (in our forwards timeframe) it would seem that the agent is living “backwards from the big crunch” - but the agent would think to itself that it was living forwards from the big bang.

    The REAL problem with this (as pointed out by Huw Price in his excellent book “Time’s Arrow and Archimedes’ Point”) is what happens in the middle, where the forward arrow half of the universe meets the reverse arrow half of the universe. If we are traveling forward in time from our big bang, what happens when we get to that transition point? Does the arrow of time suddenly flip into reverse at all points in the universe simultaneously? How does it know when to do this? And what happens then? Having flipped into reverse, it has to flip into forward again straight away (because time will start flowing back into the forward half of the universe), and then flip-flop “eternally”, being stuck at the transition point. The entire notion of having time flowing forwards from the big bang in the “front half” of the universe, and flowing backwards from the big crunch in the “back half” of the universe is naively appealing because of its symmetry – but it results in paradox or inconsistency at the juncture of these two halves – in simplistic terms its like having two rivers one flowing east and one flowing west which meet each other – with nowhere else for the water to go……

    Gravity is universally attractive, so if one has enough mass/energy within a certain volume of space it follows that nothing (not even light) can escape that volume. This is exactly what a black hole is. For a volume with radius equal to 15 billion light years (roughly the age of the universe), the average density needs to be roughly 10^-29 grams/cc in order for that volume to be a black hole. The present density of baryons is estimated at 10^-30 grams/cc, and if baryons make up 10% or less of the entire mass of the universe then our entire universe could be a black hole.

    Space is defined as the distance we measure between instances of mass/energy. Take away all mass/energy and what is left? I (and Ernst Mach) would say “nothing”, presumably you would say “space”?

    Agreed. “No you and no cat” = “no space”. (here we assume that there is no other mass/energy apart from you and cat)

    Agreed. How does one measure distance in total absence of mass/energy?

    Oldman – you omitted any reference to Mach’s principle, which to me is the most mysterious aspect of space that nobody (imho) has ever really got to grips with.

    Mach’s principle suggests that inertial frames of reference should be coupled to the distribution of mass/energy in the universe at large. Therefore that frame of reference in which the universe as a whole is at rest might be considered to be a preferred ’frame’, in which total energy is conserved, in contradiction to the spirit of the equivalence principle. Indeed such a preferred frame of reference does appear to exist; it is that in which the Cosmic Background Radiation (CBR) is globally isotropic.

    Best Regards
  20. Jul 24, 2006 #19
    One doen't have to! Operational definitions should be formulated in terms of what we can do, and humanity has lots of tools, for instance radar ranging equipment. In the same vein one could perhaps define mass/energy in terms of the gravitational accelerations it produces, and so on, until one has exhausted all the primary entities that need defining and come full circle, having laid the foundations for physics.

    I agree strongly, and find it most peculiar that, as you say:

    " a preferred frame of reference does appear to exist; it is that in which the Cosmic Background Radiation (CBR) is globally isotropic".

    I have not yet come across a full discussion of this peculiarity. It smacks strongly of an ether, and seems to be avoided like the plague by modern cosmologists, maybe because of this taint. But folk do seem to be puzzling about why the l = 2 part of the observed CMB spectrum does not meet theoretical expectations, and I wonder if this is somehow a related question. And especially the dipole!

    I have always believed Mach's principle, but have never understood how and if it fits into physics.
  21. Jul 24, 2006 #20
    None of those tools work in absence of mass/energy.

    Total absence of mass/energy means ……. No mass, no energy. No tools, no observers. Thus it is meaningless to talk of measuring space.

    Best Regards
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