Cycles of Time: An Extraordinary New View of the Universe. By Roger Penrose, 2012
I: The Law of Entropy
Entropy is the inequality of disorder or randomness for an isolated system. In other words, the randomness of a system in its second time frame would be greater than in time frame 1. The implications of such a statement suggest two things: 1) the entropy of a system increases as time moves further from its initial position, and 2) there is a degree of ambiguity or vagueness when measuring entropy over time. This means that we need to define what we mean by randomness and what it means when we say it is increasing. Point two is of particular interest because, in a general system like the universe, the forward passage of entropy is not homogeneous. There can be temporary moments where entropy can reverse, meaning that it can move from more random to less random.
Regardless of its vagueness, the progression of entropy is that it grants a notion of time to the mathematical models of physics that correspond to our perception of it. For example, in Newtonian mechanics, the models work just as well, explaining the trajectory of an object as it moves backward in time as it does moving forward in time. Penrose cites an egg falling off a table, which is then in freefall before smashing into the ground. In our perception of time, the reverse of the situation, the broken egg pulling itself together, zooming upwards in a parabolic arch, and coming to a stop atop the table fully formed, would be preposterous. Yet, in Newtonian mechanics, the model that explains the object's trajectory can work forward and backward in time and still make mathematical sense. To solve this contradiction between our perception of time and the a-temporal nature of our models, it becomes pertinent to add entropy to Newtonian mechanics. Returning to the egg, we can say that its entropy is a function of the model N, with its initial state being NI and its finished state being Nf, which follows the inequality Nf(e) > NI(e). Therefore, we have introduced temporal direction into our model, and changing its direction requires a means to reverse the entropy inequality by applying some force greater than Nf(e). Thanks to entropy, the temporality that is presented to us by our perceptions can now be preserved in our models.
Suppose we have a pot of paint made up of discrete units of blue and red. We say discrete because we imagine the paint molecules as either blue or red balls, which take up space in an N^3 cube, creating a subdivision of the cube. Each compartment of the cube holds a paintball of either colour. In gauging what colour ball occupies an individual compartment, we take a snapshot of the density of red balls to blue ones in a localised area of the cube. A subset of the total cubical space centred around the compartment where we wish to know the ball's colour. This subset n^3 of N^3 is a cube in miniature, where we will say that its fundamental unit is the compartment of 1, which holds either a blue or red ball: N>n>1. The reason we are doing the sub-division of cube N^3 is that we are assuming that each ball of paint in its corresponding compartment is too small to be seen or too small to observe its colour, hence we will apply a hue to the n^3 cube made up of blue and red balls, b+r = n^3. The hue is to be decided by the ratio of b to r. If r/b > 1, there is to be a redder hue and if r/b <1, there is to be a bluer hue. Now, we are to suppose that the N^3 cube is developing a purple hue or that each n^3 is developing a ratio of equal red and blue.
Entropy enters the picture as the measurement of the probabilities on how r and b are to arrange themselves or the configuration space of a system. Expanding on this, the configuration of space—the phase space P—has 6-dimensions, where each ball has 3-coordinate positions and 3-momentum positions—which gives us the initial position of every ball and its angular velocity—we encode all of this about individual balls as a singular point p in P. The state of the system is determined by where p is within P, or the evolution of the phase space is determined by the path of p as it curves in that space. This curve determines the dynamic passage of particles in the phase space through time.
Having defined phase space and how the balls move in it, it is time to relate it to entropy. We are assuming that the movement of any ball in n^3 is indistinguishable from any other ball in n^3, which is a property of macroscopic space. So that every p in P is modelled by the Boltzmann formula: S = KlogV, where v is the volume of P or n^3, and K is the Boltzmann constant.
As the particles move from their defined localised phase space to neighbouring phase spaces, the system as a whole moves into thermal equilibrium. Particles that exist in their own n^3 space at an initial time move outward and increase their volume. As this or that particle moves from n0^3—its initial point—into n1^3, and n2^3, the particle continues to move into neighbouring spaces till it reaches nmax^3. The maximum volume the particle can occupy in the system. It is at this point that we say that the system has reached its thermal equilibrium.
II: Gravity and Space-Time
Applying the big bang to Einstein's cosmological models allows us to map the history or evolution of space-time. The first to do so was the Russian mathematician Alexander Friedman, whose work, which was later to become part of the Friedmann-Lemaitre-Robertson-Walker (FLRW) model, showed the evolution of the universe under three conditions:
Space-time curvature is negative, and the geometry of the universe is hyperbolic.
The curvature is 0 the geometry is Euclidean.
The curvature is positive, and the universe's geometry is elliptic.
Each curvature presents its cosmological evolution, for if the curvature of space is elliptic, the result would ultimately be a big crunch. Which implies that the temporal and spatial dimensions of the universe are finite. While the other two cases imply that the universe will continue to expand outward forever. Observational data provided by Saul Perlmutter and Brian Schmidt on the behaviour of distant supernovas do not match up to the theoretical understanding of the FLRW model but showed that the universe is accelerating in its expansion.
By applying entropy to an expanding universe, where equilibrium is impossible, we have an adiabatic expansion, where the entropy remains consistent. The phase space, where the particles move into, is growing. So, while the particles are becoming evenly distributed in the phase space, the phase space itself is expanding. This is compounded further by exporting our understanding of entropy to the universe as a whole, which introduces the issue of gravity. If phase space is the universe, and if our particles of paint are, in fact, stars and other large bodies of matter, a classical model of entropy should see these star "particles" become evenly distributed in space. Because of gravity, these stars attract each other, forming clusters. Through the force of gravity, the universe introduces a force that acts as a self-regulating system that creates a level of cohesion into what should be an increasing level of randomness. However, such a mechanism, the reversal of entropy by the forces of gravity, is rendered mute with the introduction of black holes.
Black holes, as understood by Stephen Hawking's work from the 1970s, is when one applies quantum field theory to curved space-time, black holes have a tiny temperature T, which is inversely proportional to the black hole's mass. A black hole whose mass is 10 M☉ would emit a temperature of 6x10^-9k, and because it is an inverse proportion, the larger the black hole, the smaller the temperature. So, for the black hole at the centre of the galaxy, which has a mass of 4,000,000 M☉, its temperature would be only 1.5x10^-14k or -273.15°C. In contrast, the average temperature of the Cosmic Background Radiation (CMB) is -270..45°C, noticeably warmer than black holes! Now, if the universe is to expand forever, the temperature of the CMB will decrease and reach equilibrium with black holes. Upon reaching this equilibrium, black holes will start to radiate out their energy in the form of temperature and massless particles, and because of e=mc^2, black holes will lose mass. After an incredibly long time, Penrose estimates google years (10^100) for the largest black holes to have radiated away their entire mass and hence, cease to be.
III: The Big Bang
The Weyl conformal tensor C is the measure of curvature in space-time and introduces ellipticity of light rays—gravitational lensing—and because of this, allows for the measurement of null cone structures of Makowski space. At the conditions of the big bang, the Weyl curvature vanishes, C=0, which represents an initial-type singularity. Black holes, on the other hand, represent a final stage singularity, where the curvature is C=∞. This becomes the Weyl Curvature hypothesis or WCH. The initial conditions of the universe right after the big bang were incredibly hot. To the point that all particles would have enormous kinetic energy, and any rest mass would be irrelevant. Because particles lack any rest mass, the Higgs field would be incapable of producing particles with mass. Therefore, leading to particles becoming mass-less, and in short, behavies like photons.
As the early universe would be massless, the laws of physics would become dominated by conformally invariant laws, and the metric g, or the geometry of space-time would become irreverent, and space becomes a smooth 3-surface without a time dimension. Because of this, it is theorised, as a mathematical model provided by the Cambridge mathematician and student of Roger Penrose Paul Tod, that this surface extends beyond the event that was the big bang itself.
The cosmic evolution of the universe begins when black holes start to evaporate, with the largest disappearing ~10^100 years from now. After which, the universe would be home to only low-energy photons. As they are massless, they cannot make an impact or dent in space-time and deny the passage of time entirely. It would be a universe without a clock, as it is empty of all matter. Is it to remain so for eternity? With low-energy photons following the second law of thermodynamics and becoming more dispersed. Therefore, the conditions of the late universe, which very well could be called the eternal universe and the early days of the big bang have a similar smooth 3-surface space-time, as they contain no mass. It is here that Penrose openly speculates that the conformal manifolds of these two separate conditions of the universe are a loop, where each loop or aeon begins as it ends, as a smooth conformal space-time surface. What Penrose calls the Conformal Cycle Cosmology or CCC.
IV: Experimental Notes
As black holes in previous aeons evaporate, however slowly, expelling low-energy photons, could they survive into the present aeon? Or, at the very least, be detectable? The implication is that there are fingerprints from previous aeons that exist in our current universe. Further, gravitational radiation from black holes interacting with one another would result in a burst of large amounts of energy and reduce the motion of both black holes. Resulting in gravitational waves emitting from these black hole orbits and moving outwards forever at the speed of light. In essence, gravity waves become light cones, extending from a point e and extending to f^ or the following aeon. As these gravity waves move from one aeon to another, they can interact with matter in a new universe. It would look like gravity is radiating from some unseen point in the present universe. This is Penrose's explanation for dark matter, gravity that is seeping outward from a universe of a previous aeon and is now interacting in the present.
V: Concluding Notes
What can be said about Roger Penrose’s work on Cycles of Time? The book itself is confusing and full of technical jargon that makes it almost impossible to fully comprehend his arguments. I have tried my best to distil the essence of his work into a jargon-light review, and even still, I’m disappointed in the end result. That is, I’ve felt that I’ve missed critical points and junctions that are buried under his dense prose. For example, while the forces responsible for developing the cosmological evolution of the universe, entropy and gravity, are in contradiction with one another, I fail to see how they manage to develop the curvature of the universe that works with Penrose’s notion of the cycle. For example, using the FLRW model, the universe has three possible conclusions, one that is closing in on itself and two that are open. For Penrose’s model to work, that requires the universe to remain open and I feel that he does not go far enough in justifying why it should remain open and not close in on itself. Other issues include exactly how matter is to “spontaneously combust” as a big bang on a 3-smooth manifold, and what exactly causes that? As well as the dicey problem of if protons, which have mass, decay or not. The final criticism is that while intellectually engaging and thoroughly imaginative, the experimental notes leave something to be desired, and ultimately leaves the work feeling like the equivalent of medieval scholars discussing how many angels can dance on the head of a pin. Or how God abhors a vacuum, even in spite of experimental observation. That is, it generates fascinating and imaginative discussion, but has a certain futility to it.