The Invisible Universe: Temporal Asymmetry, Entropy and Ergodic Dynamics (pt. 2)



I have never liked laziness. But I love the rainy, cold weather so much, with the wind whistling gently through the eucalyptus tree next to my bedroom window, sounds of heavy droplets pounding against the gutters above from the residue of rain peeling off the leaves as I curl myself up in foetal position in the middle of my bed, covered with multiple layers of soft blankets and my thick woollen doona, that in my state of blissful warmth and contentment it must be that I am the centre of the universe. It is a principle of mine to formulate a moral system encompassed by the homogeneity of all things and infinity of God, to see myself as a mere particle interacting with other particles not through kinetic energy but rather a gravitational one in a perfectly harmonious formulation, that is, to become aware of my capacity to consciously observe through freewill the pre-determined system and transcend toward oneness within this perfectly designed order of nature, by letting go of all that had been socially learned and to embrace the simplicity of the principles that preserve moral goodness and stand superior to anything I or others desire either emotionally or in the flesh. As they say, it is better to suffer alone than to suffer in the hands of others who are indifferent to you, just as much as it is better to suffer not being with someone you love because you know they are not good for you than to submit to one who has no admiration or respect for you, since in doing so your principles and honour remain intact and the balance within becomes impenetrable. Suffering is ignorance, dishonour a failure of adhering to your principles. This is the mathematics of morality. As with all things physical, certain principles are required to ensure that calculations are axiomatic leading to an impeccably smooth result applied informally or intuitively. Isotropy and homogeneity, for instance, principles that everything observable is the same in every direction leads to a sophisticated cosmological model where the primordial distribution of matter is impeccably smooth and by extension ensures coherent formulations of the universe, a universe  filled with 10^11 galaxies and stars all pulled together and stretched not by energy but by gravity.

Perhaps, whilst warmish under all these blankets, it must mean that I am the centre of the universe? That is to say that if I am lying in bed all mushy as I am now, or somewhere inside the Large Magellanic Cloud, or even further still in EGS-zs8-1, would calculating distance, velocity, or any coherent equation such as the Minkowski metric remain the same [trick question]? That is, would the universe still be isotropic and homogenous; would I be able to synchronise clocks at large distances wherever I was in the universe even over extended magnitudes [despite how cosmic time views distant galaxies since when the light from that galaxy reaches us, millions of light years has already passed and the galaxy itself would have changed]? There are several foreseeable problems when contemplating as such, namely that motion affects the arrow of time and that cosmic time itself is dependent on homogeneity, the latter contending space requires uniformity. Before I begin, let me reiterate the previous post regarding inflation, that the total energy at the beginning of the universe started at very close to zero, whereby the negative contribution to the energy of the gravitational field cancels the energy of matter and thus repulsive gravity drives inflation with the growth volume faster than the decay, allowing the physical universe to expand exponentially. We are able to confirm relative homogeneity and isotropy through the fluctuations imprinted in the anisotropy of the cosmic microwave background and gives light to the conditions of the early universe, which was once filled with plasma but where photons themselves – whilst moving at the speed of light – remained immobile in the density and so velocity stood at zero. As the universe expanded, the plasma cooled and became a gas and as such cosmologists began to question thermal equilibrium, the second law of thermodynamics and entropy, the latter allegedly being low during the early epoch of the universe. Thus in continuation, the problem we face here is that as the universe expands and progresses over this time, from an ordered state – namely that of low-entropy – toward a disordered high-entropy, the latter itself dependent on the arrow of time, how exactly can the early universe in the past, where it was hotter and denser and had a stronger gravitation pull, be perfectly smooth?

Hubble expansion, which is about 70km per megaparsec, is the expansion rate that we see at present with the inflationary epoch ending 10^-32 seconds after the big bang to expand at the rate of the Hubble constant.[1] If the universe was thus once condensed to a very small size until it expanded at a factor of 10^26 due to inflation and eventually ending that lead to a fixed or steady expansion as we know is now taking place, the process itself nevertheless preserves the subatomic smoothness that the initial conditions held. This is particularly coherent when assuming that we are a part of a multiverse. In Einstein’ GR field equations, he applied the cosmological constant Λ in an attempt to explain a static universe prior to Hubble’ expanding one and thus later rejected it, however for both inflation and dark energy, the ubiquitous Λ becomes a necessary algorithm that binds the theory together as the energy density of the latter in particular causally drives expansion and a flat universe that can expand infinitely. With Riemannian geometry, cosmological observations of the CMB radiation through the Wilkinson Microwave Anisotropy Probe (WMAP) have measured angles that add to exactly 180 degrees, which in a Euclidean space purports a universe that is k=0 or flat[2] and as its density remains constant as it expands, dark energy or the energy of empty space itself plays a vital role. The horizon problem also shows that the temperatures at different directions of the CMB radiation are uniform to almost 1 part in 10^4 [accounting a minor electric dipole] or 1 part in 10,000 and therefore almost the same – something that should not actually be possible – purporting that the only solution to this thermal equilibrium is inflation. That is, for example, regions billions of light years in opposite directions must communicate or interact in some manner to reach this symmetry and the explanation is that they – at one point in time – were interacting and the process of inflation has stretched them out into altered directions, thus favouring the model of an isotropic and homogenous universe.

As there is an arrow of time and as the universe is expanding, in the past the universe would have been infinitesimally smaller particularly as we reach the beginning of time. As such, the density and heat would have been higher – something clearly attributable to the CMB radiation – and the fact that perfection or a state of low-entropy is requisite should we adhere to thermodynamic laws and the direction of time, the conditions of the big bang becomes formidable. In addition, if the initial conditions were not perfectly ordered and smooth, it would have fizzled away. As mentioned, assuming the universe is geometrically flat because of the ratio between the mass density and the critical mass density being very close to Ω =1 and stabilised through the force of repulsive gravity as illustrated by the cosmological constant, is the fabric of the universe smoothing as it expands.

Ergodic theory itself relates to the properties of dynamical systems and thus is relevant concerning statistical mechanics and in all likelihood – whilst not used very often and perhaps even frowned upon – when analysing the early universe. The likelihood of this disregard is perhaps based on statistical mechanics as a study itself rather than erodicity as a theoretical formulation, which can be particularly useful when honing down on the problem of the arrow of time and entropy because of the fact that thermodynamics avoids examining the microscopic constituents of a system. I would also assume that it is perhaps because atoms are coupled throughout the universe through time-dependent electromagnetism and scalar fields, thus the implication undermining the required dynamical mode since a dynamical system is characterised by the Hamiltonian point in phase space illustrated by generalised coordinates (qk) and moment (Pk) with the path converging on a time-average value. That part is still a bit fuzzy for me, though. Boltzmann’s entropy formula S = k log W describes this statistical domain of thermodynamic systems [log itself aids with minimising the size of the universe with W being the number of microstates giving a probability at macro-level][3] and that entropy is conserved by the monotonic function of ordered sets as the microstates increase. Any corresponding change at macro-level is causally connected to a change with a microstate within the system, but this interconnected dependence between the two states naturally shows that the macro-state itself contains maximal entropy. Duh. That is, thermodynamic equilibrium of system is consistent with the constraints of the second law of thermodynamics and that the formula is the statistical evidence of this. In its simplest, ergodicity is merely analogous to ascertaining the averages of behaviour within a system, measuring transformations, recurrence, arbitrary convergence etc &c. or quite simply the dynamics and that over time the probability of visiting every required state occurs. A macro-state in equilibrium is largest in size, thus over ‘time’ [that is, time-average probability] the system spends visiting the phases as it reaches this equilibrium and thus maximum entropy is, well, ergodic. Even so, it is still incredibly difficult resolving arbitrary estimates coupled with the fact that cases involving the second law of thermodynamics do not necessarily require erodicity at all. But when considering the constituency of time in this framework, the idea that the direction or arrow of time will eventually lead a system toward maximum entropy and ergodicity may, in reverse, explain time.

Ergodicity itself is somewhat Epicurean, not to say that it has any connection with Epicurus’ Nature of the Cosmos, but rather the philosopher himself – more notably adhered by Lucretius – believed that it is a mark of an intelligent mind to think of multiple possibilities – from the absurd to the rational – so as to identify and explain a solution to a cosmological problem;[4] even occasionally, multiple descriptions can prove a theorem adequately at the same time. That is, it is a mark of an intelligent state of mind that it thinks of every possibility on a subject in question that will lead to the correct or highly probable answer; ignorance is to believe in assumptions. Lucretius’ cosmological phenomenology is based on his thought experiment regarding infinite space, whereby should one suppose to travel to the end of the universe and throw a spear through it, what would happen to the spear?[5] Either it will hit it and fall, or it will go through the boundary – that the boundary of a finite universe is ultimately illusory – and toward another space that we are not aware of; what this would mean is that all possibilities and possible worlds outside of the finite space that we understand is actually possible. Therefore the universe is infinite and according to Newton this must be true; his failure, however, was the proposition that the universe was static purported by his assumption of the stars being fixed relative the inertial frame, namely because the distribution of mass would be unstable. The problem of symmetry, however, regarding a state where the universe is accelerating, is how the direction for which this acceleration is determined. In order to substantiate the validity that there is actually a physical system, the universe requires isotropy and since we can acknowledge that when we look out to the universe that in every direction we can observe the CMB radiation, one can conclude that it may very well be a symmetric space. Nevertheless, for the sake of avoiding the likelihood of falling down the existential rabbit hole before becoming overwhelmed by the vanity of, well, everything, let us assume that the universe can be modelled as a dynamical system, contained in an isotropic, homogenous and maximally symmetric but statistically within a finite structure and governed by an arrow of time, and in doing so the analysis of erodicity and entropy within such a model seem almost possible. The problem is that, if the arrow of time purports that time itself is moving forward in one direction, that the universe is expanding alongside time as it reaches its maximum state of high entropy, it would mean that therefore the universe had a past and so to not defy the second law of thermodynamics, the early universe would have to be at a state of low entropy. In an environment where the observable universe is much denser or smaller in the past – since the universe is expanding – it would logically imply that it was hotter and the pull stronger. How is it that in that macroscopic parameter consisting of a hot and dense environment instead was smooth and cool? This does not make sense since the early universe was in a state of equilibrium which, given the calculations above, must uphold the thermal law of being in a state of high entropy. In addition, temporal asymmetry works in contradiction to the second law of thermodynamics; motion cannot function without time, it would be like matter frozen in a dimensionless space or swallowed in a blink beyond the event horizon. Time’ arrow works in a manner that directs motion forward, evolutionary of sorts and adapts to the processes within its environment in an attempt to find a state of equilibrium.

In classic thermodynamics, the joule [free] expansion – where within an adiabatic container enclosed with monatomic gas molecules and no energy or thermal properties – the gas densely kept to one side of the container with a closed partition between another empty container that has been vacuumed of any properties at all and therefore completely empty,[6] when the partition is opened and consequently the gas in one container increases in volume and expands into the other, the pressure of the gas that had been densely kept in the other compartment diminishes [like blowing up a balloon with helium gas and then letting it go; the gas is released from the balloon with the rubber shell flying about the place in an awkwardly loud and flatulent manner]. There is no pressure or work, ΔU = q + w = 0 but nevertheless there were changes [in consideration of ideal gas][7] in temperature and therefore PV=nRT whereby the pressure and volume equates to a constant of the gas and the temperature, so the first law regarding the conservation of energy in thermodynamics remains valid. The ergodic hypothesis by Boltzmann was formulated to prove in principle the determination of the distribution of gas molecules and their kinetic speeds in his equipartition theorem, which is mathematically ascertaining the energy of any given physical system through the distribution of generalised coordinates and momenta. The second law of thermodynamics contains the interesting problem vis-à-vis this very blog post, that the law governs the exchange of thermal contact and gradual arrangement toward a fixed equilibrium; that is, the natural evolution of any given system is determined to a state of equilibrium. Once the partition is open and the gases are dispersed, they spontaneously find a state of equilibrium and do not randomly paste themselves to the ceiling of the container etc &c. How can a hot cup of tea become lukewarm as it cools to room temperature and thus asymmetric as it reaches a state of equilibrium with its environment? Or is that a deductive fallacy? Could travel back in time? One of my favourite paradoxes from All You Zombies[8] is as follows:

“A baby girl is mysteriously dropped off at an orphanage in Cleveland in 1945. “Jane” grows up lonely and dejected, not knowing who her parents are, until one day in 1963 she is strangely attracted to a drifter. She falls in love with him. But just when things are finally looking up for Jane, a series of disasters strike. First, she becomes pregnant by the drifter, who then disappears. Second, during the complicated delivery, doctors find that Jane has both sets of sex organs, and to save her life, they are forced to surgically convert “her” to a “him.” Finally, a mysterious stranger kidnaps her baby from the delivery room.
Reeling from these disasters, rejected by society, scorned by fate, “he” becomes a drunkard and drifter. Not only has Jane lost her parents and her lover, but he has lost his only child as well. Years later, in 1970, he stumbles into a lonely bar, called Pop’s Place, and spills out his pathetic story to an elderly bartender. The sympathetic bartender offers the drifter the chance to avenge the stranger who left her pregnant and abandoned, on the condition that he join the “time travelers corps.” Both of them enter a time machine, and the bartender drops off the drifter in 1963. The drifter is strangely attracted to a young orphan woman, who subsequently becomes pregnant.
The bartender then goes forward 9 months, kidnaps the baby girl from the hospital, and drops off the baby in an orphanage back in 1945. Then the bartender drops off the thoroughly confused drifter in 1985, to enlist in the time travelers corps. The drifter eventually gets his life together, becomes a respected and elderly member of the time travelers corps, and then disguises himself as a bartender and has his most difficult mission: a date with destiny, meeting a certain drifter at Pop’s Place in 1970.
The question is: Who is Jane’s mother, father, grandfather, grand mother, son, daughter, granddaughter, and grandson? The girl, the drifter, and the bartender, of course, are all the same person. These paradoxes can made your head spin, especially if you try to untangle Jane’s twisted parentage. If we drawJane’s family tree, we find that all the branches are curled inward back on themselves, as in a circle. We come to the astonishing conclusion that she is her own mother and father! She is an entire family tree unto herself.”

Quantum entanglement is an interesting method of understanding the arrow of time in this context. The uncertainty principle in quantum mechanics[9] asserts that measuring the position of a particle and its momentum is never accurate, furthered in confusion with evidence that sometimes interaction between two particles merge or entangle to form a ‘oneness’ that dictates the momentum and position one to the other that they are no longer two separate particles – though physically it is so –nevertheless communicating invisibly one to the other as a combined force. The theory for me is extra cool when I think of telekinesis where I remember in high school was asked what superhero power I would want and would instantly be resolved on telekinesis, thinking that in order for me to move objects using my mind, all I would need to do is calculate the state of the object and its relationship with the wave of consciousness as a physical yet invisible system and once I was able to calculate the particles as it – from object to my mind – penetrates my brain, I become capable to control or move it because I understand the math behind it. Of course, I have also dreamt that I was a butterfly [awkwardly stares out with one eyebrow raised]. Nevertheless, it is of interest to me where the interaction prior to the amalgam between the particles peaked at a derivative equal to zero, namely the very point where particles enjoin to become a state where they can no longer be classified as autonomous. Reaching this balance of connectivity between particles from a pure particle state to a combined oneness in perfect equilibrium as it relaxes into its new and unchanging form is the real parameter that works comparatively to the notion of thermal equilibrium and thus the evaluation of thermodynamic properties.

It appears that no matter where I am in the universe, I will still get the same answers to the same equations and the physical world would appear to me, well, to be the same in every direction. That is, the symmetry of the expansion rate is homogenous confirmed to a degree through Hubble’s Law, which is the velocity between two galaxies being equal to the Hubble parameter times the distance V=Hod and verifies that objects would appear to be expanding outward relative to the observer; the measurement of the radial velocity determined by the redshift. Thus galaxies are moving away and galaxies even further still at a much faster rate explained by the fact that should both the source and the observer be stationary, there would be no time differentiation or delays viz., the time for the wavelength to reach the observer, hence the Doppler effect. When thinking about the cosmological redshift, whereby light that has been emitted from a distant galaxy reaches us on earth, calculations of the spectral features of photons namely λ =h/p requires attentiveness on how the light itself will shift from the frequency it had when emitted to the frequency we measure when receiving it, that is, the momentum and time it takes for the wavelength to reach the observer, the evolution of this process changes as the photons are stretched. Physicists have thus determined that the universe is not only expanding but seemingly also accelerating.

So, as a particle that I have subjectively simulated into human form lies comfortably in bed with the rain trickling down this peaceful Saturday evening knowing that the universe would not appear closer to, say, the cosmic microwave background radiation should I be viewing space from a telescope on a habitable planet in EGS-zs8-1, it would naturally follow that the universe is both isotropic and homogenous. If not, then a universe on a large-scale that is homogenous but not isotropic would be designed like a brick wall, or conversely, isotropic but not homogenous would find galaxies aligned. It would also mean that the universe could also be static yet, which we know is not true as galaxies appear to be moving away. It would seem that the universe is expanding whilst galaxies themselves remain static. When photons emitted from a distant galaxy reach us the observer, the distance and velocity of the wavelength with the time it takes to us from the source is quantified by the Hubble constant times the distance between galaxies. The asymptotic nature of ∆t would purport that the atomic properties in space and thermal energy interact with time in a manner that will continuously interfere in the process of reaching absolute zero, and as stated previously, even a vacuum state still contains energy even though extremely low. Measuring time during the inflationary epoch remains questionable, even with the capacity to measure the smallest possible unit of time through Planck [5.39 × 10^-44 s] whereby probabilities are the only reality attributed to the questionable state of time. Perhaps the total entropy in the universe is already infinite, in which case it was always infinite. It is possible that we observe as a physical system and experience as time are merely arrangement within this universe as part of a multiverse, a pocket universe and therefore if the macro-state as a whole – the singularity – has reached an eternal state of equilibrium, perhaps there is no ‘beginning’ or ‘end’ except the singularity itself. God who is ‘alpha and omega.’ If, in the end everything can be doubted, that while the universe exists it doesn’t exist, the choice is left as to how we can ascertain whether the singularity exists and what time is leaving one strange phenomenon; prophesy. I will get to that in a later post.

[1] Cesare Emiliani, Planet Earth: Cosmology, Geology, and the Evolution of Life and Environment, Cambridge University Press (1992) 68

[2] Carlos I. Calle, Einstein For Dummies, Wiley (2005) 309

[3] Don S. Lemons, A Student’s Guide to Entropy, Cambridge University Press (2013) 72

[4] See Lucretius’ cosmology and use of the Principle of Plentitude briefly explained in Michael J. White, Agency and Integrality: Philosophical Themes in the Ancient Discussions of Determinism and Responsibility, Springer Science & Business Media (2012) 4

[5] Philip de May, Lucretius: Poet and Epicurean, Cambridge University Press (2009) 27

[6] Clement John Adkins Equilibrium Thermodynamics, Cambridge University Press (1983) 162

[7] Peter Atkins, Julio de Paula, Ronald Friedman, Physical Chemistry: Quanta, Matter, and Change, OUP Oxford (2013) 576

[8] David Darling, The Universal Book of Mathematics: From Abracadabra to Zeno’s Paradoxes, John Wiley & Sons (2004) 139. See Robert Heinlein’ All You Zombies

[9] K.V.S.Gnaneswara Rao, Engineering Physics, S. Chand Publishing (2008) 38

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