From the best-selling author of The Emperor’s New Mind and The Road to Reality, a groundbreaking book that provides new views on three of cosmology’s most profound questions: What, if anything, came before the Big Bang? What is the source of order in our universe? What is its ultimate future?
Current understanding of our universe dictates that all matter will eventually thin out to zero density, with huge black holes finally evaporating away into massless energy. Roger Penrose - one of the most innovative mathematicians of our time - turns around this predominant picture of the universe’s “heat death,” arguing how the expected ultimate fate of our accelerating, expanding universe can actually be reinterpreted as the “Big Bang” of a new one.
Along the way to this remarkable cosmological picture, Penrose sheds new light on basic principles that underlie the behavior of our universe, describing various standard and nonstandard cosmological models, the fundamental role of the cosmic microwave background, and the key status of black holes. Ideal for both the amateur astronomer and the advanced physicist - with plenty of exciting insights for each - Cycles of Time is certain to provoke and challenge.
Intellectually thrilling and accessible, this is another essential guide to the universe from one of our preeminent thinkers. - Cycles of Time (Amazon)
- Part 1. The Second Law and Its Underlying Mystery
- Part 2. The Oddly Special Nature of the Big Bang
- Part 3. Conformal Cyclic Cosmology
- Epilogue - Appendices
Prologue: Rain and Mill
- Tom admires the power of the wildly tumbling water at an old mill
- Discusses with Aunt Priscilla how the energy from the Sun is harnessed to run the mill
- Feels puzzled by the concept of heat causing water to evaporate and lift up into the air
- Priscilla explains that the Sun's heat causes water molecules to gain more energy, leading them to rise and form clouds
- Water falls as rain, which can be harnessed for energy in a mill
The Energy of Random Motion
- When water molecules reach the top of the mountain, their energy gets converted into gravitational potential energy
- This energy is then used to run the mill when the water flows downhill
- Even in cold water, there is more heat energy in the motion of individual molecules than in the currents of water
Idea for a New Mill
- Tom proposes building a mill that directly uses the energy of water molecule motion in an ordinary lake
- This would involve using tiny windmill-like structures to harness the energy and drive machinery
- Priscilla explains this idea is not feasible due to the Second Law of Thermodynamics, which states that energy cannot be retrieved from random motions without becoming more disorganized
- She advises against pursuing such an idea, as it would be unlikely to work
Second Law of Thermodynamics Discussion between Tom and Aunt Priscilla
Tom's Skepticism:
- Not convinced by Second Law due to its "unpleasant" nature
- Believes laws can be circumvented with clever ideas
- Argues Sun's energy alone cannot heat Earth effectively
Explanation of Second Law:
- Requires colder upper atmosphere for water vapor condensation above mountain
- Earth doesn't gain overall energy from the Sun
- Sun's energy must go back into space eventually, only a small part is retained by Earth (global warming)
- Manifest organization in Sun's energy allows keeping Second Law at bay
Tom's Confusion:
- Struggles to understand concept of organization in Sun's energy and its origin
- Wonders where the original "organization" comes from
- Skeptical about theories explaining organization, such as Big Bang or previous collapsing phases
Cosmological Theories on Organization:
- Attempts to explain origin of organization in universe:
- Previous collapsing phase bounced to become Big Bang
- Black holes' bits collapsed into new expanding universes
- Universes sprung out of false vacuums
Tom's Reaction:
- Finds these theories "crazy"
Key Takeaways:
- Second Law of Thermodynamics explains the relationship between energy and entropy in a system, requiring colder environments for heat to be absorbed effectively.
- The Earth doesn't gain overall energy from the Sun as all gained energy eventually goes back into space.
- Organization in the universe remains a mystery despite various cosmological theories attempting to explain it.
The Second Law and its underlying mystery
1.1 The relentless march of randomness
- The Second Law of thermodynamics is a fundamental concept in physics, asserting that the entropy of an isolated system increases over time.
- This law challenges our intuition because it contradicts the idea that systems can return to their initial state after some disturbance, as observed in Newtonian mechanics.
- Understanding this mystery may lead us to new insights about the universe and its history.
1.2 Entropy, as state counting
- Entropy is a measure of disorder or randomness in a system.
- It can be calculated by counting the number of microstates (possible configurations) that correspond to a given macrostate (observable property).
- For example, when an ice cube melts, its entropy increases due to the increase in possible configurations for the water molecules.
1.3 Phase space, and Boltzmann's definition of entropy
- The phase space is a mathematical construct that describes all possible configurations of a physical system.
- Boltzmann defined entropy as the natural logarithm (log base e) of the number of microstates associated with a given macrostate, divided by the product of the Planck constant h and the temperature T: S = k ln W/hT.
- This definition relates entropy to information content and the number of available configurations in phase space.
1.4 The robustness of the entropy concept
- Despite its vagueness, the entropy concept has proven to be a useful tool for understanding physical systems and their behavior.
- It applies equally well to classical as well as quantum mechanical systems, and provides insights into thermodynamics, statistical mechanics, information theory, and cosmology.
1.5 The inexorable increase of entropy into the future
- The Second Law asserts that the entropy of an isolated system will always increase over time.
- This means that all natural processes tend toward increasing disorder or randomness, which is irreversible in the sense that it cannot be reversed by any physical process without violating the laws of thermodynamics.
- However, there are exceptions to this rule in the form of rare fluctuations where entropy temporarily decreases.
1.6 Why is the past different?
- The Second Law only applies to isolated systems; it does not apply when energy or matter is added or removed from a system.
- This raises an important question: why is the entropy of a system lower in the past than it is in the future for open (non-isolated) systems?
- Some theories, such as the Big Bang singularity or quantum cosmology, suggest that entropy might have been infinitely low at some point in time and has been increasing ever since.
Second Law of Thermodynamics
Description:
- Consistent with Newtonian dynamics, but depictions of self-assembling objects contradict this law
- Things are getting more random all the time
- Entropy increase: an inevitable feature of everyday existence (not a mystery)
- Second Law expresses commonplace experience
Implications:
- The Second Law is a separate principle added to dynamical laws, not derived from them
- Entropy definition symmetrical regarding time direction
- Newtonian dynamics are symmetrical in time, leading to non-constant entropy
- Reversing situations doesn't violate Newtonian dynamics but contradicts the Second Law.
Understanding:
- The Second Law states that entropy, or disorder, increases over time (approximately).
- This increase in randomness is observed as things evolving into more complex systems and eventually breaking down into simpler ones. It's a natural feature of the world around us.
- While the second law doesn't seem like a mystery at first glance, it does present some challenges when trying to understand how life emerges from seemingly chaotic processes, which will be explored in later sections.
- The Second Law is separate from dynamical laws (Newton's laws) and cannot be derived from them alone because they are symmetrical regarding time direction. Instead, the Second Law must be added as an additional principle to explain why entropy always increases under certain conditions.
Entropy and Randomness: Understanding Entropy in Physics through an Example
The Second Law of Thermodynamics:
- Asserts that certain processes, like self-assembling eggs or mixing paint, are overwhelmingly improbable
- Explained through the concept of entropy, which quantifies randomness
Entropy: Counting Possibilities
- Idealizing a pot of paint example with distinct regions of red and blue
- Entropy is a concept related to counting possibilities
- Boltzmann's insights on entropy involve subtle complexities
Measuring Entropy in a Paint Pot
- Divide the pot into cubical compartments, each occupied by one ball (red or blue)
- Define hue: average of red and blue balls' locations within a box
- Consider the number of arrangements giving a uniform purple hue vs. original configuration
- With large numbers of balls, probability of uniform purple is high, while original configuration is extremely low
Implications:
- Randomness increases over time, as entropy tends to increase in closed systems
- The Second Law guarantees that certain processes are improbable and tend towards less order (more entropy)
Entropy as a Measure of Probabilities
- Entropy is a measure of probabilities or different arrangements with the same overall appearance
- The natural logarithm of these numbers is used instead of the numbers directly, as it provides a more appropriate measure
Logarithms and Additivity Property
- Logarithms convert multiplication to addition
- For independent systems, entropy is additive if defined proportional to logarithm of number of ways state can come about
Configuration Space
- Configuration space: d-dimensional space representing all possible locations and internal degrees of freedom for particles in a system
- Point particles: 3q dimensions (3 coordinates per particle, q total)
- Visualization limited to lower dimensions (2 or 3), abstract mathematical concept not physical space or time
Entropy Definition Clarification
- Infinite number of arrangements in high-dimensional configuration space
- Volume measurement used instead of counting discrete things
- Region in configuration space representing indistinguishable states by macroscopic measurements (coarse graining)
- Robustness of entropy due to large volume ratios between coarse-graining regions.
Phase Space and Boltzmann's Definition of Entropy
Inadequacy in Previous Definition:
- Previous definition only addressed half of the issue
- Example: A bottle containing water and olive oil that separates over time
- Olive oil molecules strongly attract each other, leading to increased motion/velocities
Phase Space Concept:
- Need to account for both position and motion of particles/molecules
- Configuration space not adequate
- Phase space: A space with twice as many dimensions as configuration space
- Each position coordinate has a corresponding momentum (or angular momentum) coordinate
- Encodes the instantaneous positions and motions of all particles in the system
Dynamical Laws:
- Governed by Newton's laws of motion
- Deterministic, unique evolution according to dynamical laws
- Evolution curve in phase space represents the entire system's state
- Traveling along the evolution curve describes future evolution
- Reversing direction describes past evolution
Important Features:
- Phase space has a natural measure (dimensionless numbers)
- Important for Boltzmann's entropy definition
- In ordinary terms, higher dimensions have smaller measures than lower dimensions
- Quantum mechanics provides measures of phase-space volumes as just numbers
- Extremely small values in standard units
- Granularity and discreteness of quanta in quantum mechanics
Black Body Radiation and Quantum Theory
- Max Planck's analysis of black body radiation in 1900 explained the observed phenomenon
- This launched the quantum revolution by providing a theoretical explanation
- Discussion of equilibrium situations with different numbers of photons requires considering phase spaces of different dimensions
- Proper discussion is outside the scope of this book, but will be revisited in 3.4
Coarse Graining and Entropy
- Phase space: region in which a system can exist
- Two points belonging to the same coarse-graining region are indistinguishable with regard to macroscopic parameters (temperature, pressure, etc.)
- Entropy S of a state is defined by Boltzmann formula:
- Where V is volume of coarse-graining region containing p
- k is a small constant (3.1791023 JK-1), not to be confused with Boltzmann's constant k=1.380649x10^23 JK-1
- To be consistent, author reverts to natural logarithms and writes formula as klogV
Entropy as a Measure of Specialness
- Lowness of entropy is not a good measure of a state's specialness
- For example, the relatively high-entropy state of an egg after it has become a mess on the floor still contains very particular correlations between particle motions
- If reversed, this state could resolve itself into a perfectly completed egg projecting itself upwards
- This subtle correlation is not captured by low entropy, which only considers macroscopic parameters
- The Second Law demands that some states of high entropy can evolve to lower entropy, though this is a tiny minority of possibilities
Liouvilles Theorem and Entropy Conservation
- Liouvilles theorem: time evolution preserves volumes in phase space for standard classical dynamical systems
- This does not contradict the Second Law because coarse-graining regions are not preserved by evolution
- If the initial region V is a coarse-graining region, it may spread out to larger regions as time goes on
Use of Logarithm in Boltzmann's Formula
- Entropy value depends on the choice of system boundaries:
- For a small laboratory experiment, the relevant phase space and coarse-graining region are limited to the system of interest
- Including more degrees of freedom (e.g., entire Milky Way galaxy) expands the phase space and coarse-graining region
- Entropy now applies to the larger system as a whole, not just the local experiment
Phase Space in Physics:
- Experimenter considers a small fraction of total external degrees of freedom (galaxy) for analysis
- External phase space: huge, defined by complete set of parameters (space + internal)
- Coarse-graining region: characterizes state external to laboratory
- Product space concept: combination of two or more spaces with coordinates from each space
- Fig. 1.9 illustrates product space as a plane and line
- If external degrees are independent, coarse-graining regions in product space are products of constituent regions (Fig. 1.11)
- Volume element in a product space is the product of volume elements in each constituent space
- Boltzmann entropy: sum of entropies within and external to the laboratory
- Entropies of independent systems add together (product-to-sum property)
- Ignoring external degrees for experiment analysis, but crucial for universe entropy balance.
The Robustness of the Entropy Concept
Boltzmann's Formula:
- Provides an excellent notion of what the entropy of a physical system should be defined as
- Boltzmann put forward this definition in 1875, representing an enormous advance over earlier ideas
Aspects of Vagueness:
- Concerns the notion of a "macroscopic parameter"
- Example: Fluid system with unmeasurable details vs. future, more detailed measurements
- Detailed measurement might result in entropy reduction, but overall increase due to measuring apparatus
- Issue of subjectivity in defining macroscopic parameters remains enigmatic
Maxwell's Demon and Entropy:
- Maxwell imagined a "demon" violating the Second Law on a microscopic level
- However, when considering the entire system, including the demon, the Second Law is restored
- Issue of subjectivity in defining macroscopic parameters remains unresolved
Entropy Robustness:
- Entropy values of a system are relatively unaffected by technological advancements and precision
- Coarse-graining regions have vastly different volumes, so detailed changes make little difference to entropy
- Example: Model of red/blue paint mixture shows that increased precision has negligible effect on assigned entropy
The Entropy Increase in a Bath
- Considering entropy increase in mixing water for a bath
- Hot water: around 50C, volume: 75 liters
- Cold water: around 10C, volume: 75 liters
- Entropy increase: about 21407 J/K ( 1027 times larger coarse-graining region)
- Boundaries of coarse-graining regions are not well defined
- Fuzziness at boundaries separating regions
- No significant difference in assigning entropy to a state close to the boundary
Limitations and Subtle Situations
- Inadequate for situations like spin echo phenomenon (NMR)
- Nuclear spins lose order but then regain original state
- Seems to violate Second Law, but hidden order revealed with sophisticated measurements
- Similar issue occurs with stored information on CD/DVD or a viscous fluid experiment.
Entropy and Second Law
Figures:
- Fig. 1.13: Two snug-fitting glass tubes with viscous fluid between, line of red dye
- Fig. 1.14: Handle turned to spread out line of dye, then back, line reappears
Discussion on Entropy Definition and Second Law:
- Common viewpoint: no violation of the Second Law, just an issue with entropy definition refinement
- Debate over demanding a precise objective definition of physical entropy in all circumstances for universal applicability
- Questioning the need for a well-defined, physically precise notion of entropy that never decreases as time progresses
- Entropy as a useful concept but not necessarily fundamental or objective
- Uncertainty regarding why macroscopic quantities differ by stupendously large factors in our universe
Issues with Entropy Concept:
- Subjectivity involved in the concept of entropy, clouding central mystery
- Profound issue: enormous differences between coarse-graining volumes in actual universe reveal an objective fact about it.
My Opinion:
- No fundamental need for a well-defined and objective physical notion of entropy that never decreases as time progresses.
- Entropy is useful due to large differences in macroscopic quantities, revealing a remarkable fact about our universe.
Inevitable Increase of Entropy into the Future
Understanding Entropy's Increase:
- System starts in a state of reasonably low entropy
- As system evolves, it enters larger coarse-graining regions
- Each new region is more likely to have a greater volume than the previous
- Entropy value increases as point moves through phase space
- Eventually, point will reach the largest coarse-graining region (thermal equilibrium) and remain there with occasional fluctuations
Random Evolution vs. Deterministic Mechanics:
- The evolution curve describes a continuous evolution
- Coarse-graining volumes are unlikely to differ by such an enormous amount as a direct leap to the largest region
- Entropy increases gradually, not discontinuously
Questioning the Egg's Past Evolution:
- We can also consider the likely past behavior of the egg
- The Newtonian laws work equally well in the past time direction, giving deterministic past evolution
- Finding the most probable past history requires examining coarse-graining regions adjoining the starting point
- Most probable path involves entering larger and larger regions as time goes back
Second Law of Thermodynamics: Evolution Curves and Reasoning
Evolution Curves:
- Numerous curves leading up to p0 from smaller volumes, like -3, -2, -1, 0
- Volumes would be greatly increasing from smaller ones in the direction of time
- Consistent with Second Law
Reasoning Conclusion:
- Evolution curves seem to lead to continual gross violations of Second Law
- Egg perched on edge of table could have started as mess on floor, self-assembled into egg
- In conflict with what presumably actually happened: careless person placed egg on table
Problem with Retroactive Application:
- When applied in past time-direction, argument gives completely wrong answer
Why is the Past Different from the Future?
- Reasoning for Second Law in future evolution seems convincing, but issues with assumption of randomness towards coarse-graining regions
- Dynamical laws not random, and bias evident in past behavior (eg. egg balancing on table)
- Acceptance of past teleology vs. rejection of future teleology: familiarity and experience play a role
- Origins of universe with low entropy can explain Second Law's validity
- Key issue: extraordinary specialness of Big Bang state
- Points of clarification regarding arguments for Second Law based on time perception
- Argument from experience of time progression irrelevant to explaining the need for tiny coarse-graining region in our universe.
Critique of the Argument for a Second Law as Necessary for Life
Anthropic Reasoning:
- Argument that the presence of a Second Law is essential for life to exist
- This reasoning is anthropic and will be discussed further in 3.2 and 3.3
Limitations of this Argument:
- Physical requirements for life are not well understood compared to consciousness
- Even if natural selection requires the Second Law, it does not explain why the same law holds everywhere in the observable universe
Improbability of Life's Origin Without a Prior Assumption of the Second Law:
- The production of life from random collisions would be less probable than a miraculous creation
- Examining the evolution curve in phase space, the most probable way to reach the Earth's state (with life) would have been through a sequence of coarse-graining regions, including some that violated the Second Law
Potential Future Reversal of the Second Law:
- Observationally, the Second Law holds true in our universe and no reversal effects are seen
- However, we cannot rule out the possibility of a reverse Second Law eventually holding in the very remote future
- Such an eventuality is not intrinsically absurd but not plausible based on current knowledge.
Part 2: The Oddly Special Nature of the Big Bang
Our Expanding Universe (2.1)
- Observational evidence for an explosive origin of the universe comes from Edwin Hubble's observations in 1929 that distant galaxies are moving away from us with speeds proportional to their distances, implying everything came together at a single point - the Big Bang
- Subsequent observations and experiments have confirmed Hubbles conclusions
- The redshift of spectral lines emitted by atoms in distant galaxies is consistent with a Doppler shift, indicating recession from Earth
- Expansion rate of the universe has been detailed through time, providing a widely accepted picture
The Ubiquitous Microwave Background (2.2) [This section is missing]
Space-time, Null Cones, Metrics, and Conformal Geometry (2.3) [This section is missing]
Black Holes and Space-Time Singularities (2.4) [This section is missing]
Conformal Diagrams and Conformal Boundaries (2.5) [This section is missing]
Understanding the Way the Big Bang was Special (2.6)
- The Big Bang represents a state of extremely low entropy, which is difficult to explain in terms of thermodynamics
- Einstein's general theory of relativity provides a framework for understanding the expansion of the universe
- Analogy: Expansion of the universe is like blowing up a balloon; there is no central point from which everything expands
- Dark matter and dark energy are unexpected ingredients needed to explain the observed time dependence of the universe's expansion
Our Expanding Universe (2.1, continued)
- The Big Bang encompassed the entire spatial spread of the universe at the time it occurred
- Space itself was very tiny at that point, including all of physical space
- General relativity is a well-tested theory, especially in modelling binary pulsar systems with high precision.
The Original Cosmological Models
- Russian mathematician Alexander Friedmann proposed cosmological models in 1922 and 1924 based on Einstein's theory
- These models are called Friedmann-Lematre-Robertson-Walker (FLRW) models
- Assumes the spatial part of the geometry is completely uniform and homogeneous
Spatial Geometry in Cosmology
- Three main cases to consider for the spatial geometry: positive, zero, and negative curvature (K>0, K=0, K<0)
- Examples of these geometries depicted in Figure 2.3, based on Maurits C. Escher's artwork
- All three models originate with a Big-Bang singular state where the density and curvature become infinite
- Behavior of these models mirrors their spatial behavior:
- Spatially finite case (K>0) is also temporally finite, with both an initial Big Bang and a final "Big Crunch" singularity
- Spatially infinite cases (K=0, K<0) are not only spatially but also temporally infinite
Observations of the Universe's Expansion
- Observations since around 1998 suggest the universe's expansion does not match standard Friedman cosmologies
- Evidence indicates an exponential expansion characteristic of a Friedmann model with positive spatial curvature
- This exponential expansion occurs not only in the spatially infinite cases (K=0, K<0) but also in the spatially closed case (K>0), if the cosmological constant is large enough to overcome recollapse tendency
- The presence of an early cosmic inflation stage would not affect the appearance of Figs. 2.2 and 2.5
The Conformal Representation of Hyperbolic Plane
- Figure 2.3(c) illustrates a point that will be significant later: the conformal representation of hyperbolic plane, discovered by Eugenio Beltrami in 1868 and Henri Poincar around 1882
- This representation is based on the smooth finite boundary representing "infinity" in this geometry
- The ideas from conformal geometry will be addressed in more detail later, particularly in Sections 2.3, 2.5, and 3.2
The Ubiquitous Microwave Background
The Steady State Model:
- Proposed by Thomas Gold, Hermann Bondi, and Fred Hoyle in 1948
- Required continual creation of material throughout space at a low rate
- Material to be in the form of hydrogen molecules (proton-electron pairs) created out of vacuum
- Rate of creation to replenish density reduction due to universe's expansion
Observational Evidence Against the Steady State Model:
- Detailed counts of distant galaxies by Martin Ryle at Mullard Radio Observatory
- Accidental observation of microwave electromagnetic radiation (Cosmic Microwave Background, CMB) by Arno Penzias and Robert W. Wilson in 1964
The Cosmic Microwave Background:
- Predicted by George Gamow and Robert Dicke based on Big Bang theory
- Originally observed as a "flash of the Big Bang" with a red-shift effect due to universe's expansion
- Radiation from surface of last scattering, 379000 years after Big Bang
- Initially opaque universe due to plasma (charged particles), became transparent at decoupling
Significance of CMB Observations:
- Matches the frequency spectrum explained by Max Planck's black-body radiation in 1900
- Extremely uniform nature over the whole sky, with only slight deviations from uniformity
- Signals something fundamental about the nature and origin of the Big Bang
Modern Cosmology:
- Focuses on subtle deviations from uniformity in CMB, rather than its uniformity
Black-Body Curve and Thermal Equilibrium
The Black-Body Curve:
- Represents the radiation spectrum of thermal equilibrium for a particular temperature T
- Given by a specific formula [2.18]
- Quantum mechanics tells us this is the radiation spectrum in thermal equilibrium
Observed Spectrum vs. Planck Curve:
- Observed intensities lie within error bars (exaggerated by a factor of 500)
- Even the observations with greatest error concur with the Planck curve to within the thickness of the ink line
- The CMB provides the most precise agreement between an observed intensity spectrum and the calculated Planck black-body curve in observational science
Implications of Thermal Equilibrium:
- Suggests what we are looking at comes from a state that must effectively be thermal equilibrium
- But what does "thermal equilibrium" actually mean?
Thermal Equilibrium and Entropy:
- Recall the argument that the initial state of the universe (Big Bang) must have extraordinarily tiny entropy to explain the Second Law
- The observations appear to show the opposite: a state of maximum macroscopic entropy
- However, this may not be an equilibrium in the traditional sense, as the universe is expanding
- Tolman pointed out that such an adiabatic expansion could preserve the thermal state of the early universe's expansion
Resolving the Conundrum:
- The resolution lies in questioning the assumption that the universe conforms to relativistic cosmology
- Einstein's general theory of relativity accurately describes gravity, but it may not fully explain thermodynamics
- A difference arises when comparing a gas confined in a box vs. stars moving in a "galactic-sized" box under gravitational attraction
Newtonian vs Einstein's Theory:
- Newtonian theory cannot properly address clumpiness and rapidity of motion in a system with massive point particles
- Satisfactory solution lies in Einstein's theory with black holes
- Entropy increases through gravitational condensation
- CMB temperature is extraordinarily uniform over the sky
- Deviations are only a few parts in 10^(-5) due to Earth's motion
Cosmological Uniformity:
- Universe's early spatial uniformity, or cosmological principle
- Implies huge suppression of gravitational degrees of freedom
- Low initial entropy of the universe
Second Law and Egg Example:
- Commonplace instances of Second Law unrelated to early universe's uniformity
- Puzzle lies in how the egg ended up in a low-entropy state (perched on table)
- Explained by human intervention or other highly organized systems
- Highly organized structure of the egg is part of life's grand scheme, which keeps entropy low.
The Fabric of Life on Earth
- Requires maintenance of profound and subtle organization to keep entropy low
- Intricate, interconnected structure has evolved through natural selection and chemistry
- Biological complexity prevents system from violating fundamental physical laws like conservation of energy
- Sun is a powerful low-entropy source upon which almost all life on Earth depends
The Role of the Sun in Life on Earth
- Provides equal amounts of energy during the day and at night
- Energy received from the Sun has lower entropy than energy returned to space due to higher photon frequency
- Green plants convert high-frequency solar photons into lower-frequency ones through photosynthesis, providing low-entropy source for life
- Animals use this source of low entropy to keep their own entropy down
The Importance of Gravitational Clumping in the Formation of Stars and Planets
- The Sun's existence is due to gravitational clumping that produced it, allowing for thermonuclear reactions to occur
- These reactions are crucial for life on Earth as they provide a source of heat and water vapor
- Potential for stars to form comes from the Big Bang's initial low-entropy state, which allowed for gravitational degrees of freedom not to be activated
Space-time, Null Cones, Metrics, Conformal Geometry
Hermann Minkowski's Contribution:
- Demonstrated special relativity using an unusual type of 4-dimensional geometry in 1908
- Essential ingredient for Einstein's general theory of relativity
- Incorporated space and time into one indivisible whole, encoding special relativistic structure
Minkowski Space-Time:
- Points are referred to as events with temporal and spatial specifications
- Does not naturally separate into a time dimension and Euclidean 3-spaces
- Geometric structure provides an overall geometry to space-time
Geometry of Minkowskian 4-Space:
- Not built out of a succession of 3-surfaces, each representing space at different times
- Simultaneity depends on observer's velocity in special relativity
- No absolute notion of simultaneous for distant events
Minkowski's Revolutionary Idea:
- Space-time is one indivisible whole, making it an objective geometry independent of arbitrary viewpoints
- Provides a firm picture by introducing a structure to replace temporal succession of 3-spaces
Null Cones in Minkowskian 4-Space:
- Describes how light propagates at any event p
- Tells us the speed of light in any direction at p
- Particles' world lines are directed within null cones, with tangent vectors lying within them
- Massless particles (e.g., photons) must lie along the null cone at each event
- Null cones determine causality by indicating which events can influence others
Relativity Theory and Null Cones:
- Tenet: signals not allowed to propagate faster than light
- Event p can influence event q if there's a world-line connecting them within future null cones
- Arrow indicates past-to-future direction of causation (Fig. 2.12a, 2.13)
- Uniformity lost in general relativity but continuous assignment of time-oriented null cones remains
- Massive particle world-line lies within future null cones; massless particle (photon) in null cones (Fig. 2.14)
- Rubber sheet metaphor for non-uniform null cones, smooth deformations allow symmetries and diffeomorphisms
- Principle of general covariance: formulate physical laws so they remain unaltered by rubber-sheet deformations
- Manifold: smooth space without further assigned structure beyond topology (2D example: Fig. 2.3(c))
Geometry and Metrics in General Relativity:
- Manifolds: smooth spaces of definite dimensions, often referred to as n-manifolds
- Geometry may include metric assignments like g providing length notions and geodesics (Fig. 2.15)
- Hyperbolic geometry example: Escher's picture represents straight lines as circular arcs meeting the boundary at right angles (Fig. 2.16)
- Distance between points p and q in hyperbolic space calculated using g, pseudo-radius (2.30)
- Length units can differ from Euclidean geometry.
Hyperbolic Conformal Geometry
Straight Lines in Hyperbolic Plane:
- Straight lines (geodesics) are circular arcs meeting the boundary circle at right angles
- Different from Euclidean plane due to different conformal structure and metric
Conformal Structure:
- Provides measure of angle between smooth curves, but no fixed notion of distance or length
- Determines infinitesimal shapes through ratios of length measures in different directions at any point
- Can be rescaled without affecting the conformal structure (Fig. 2.17)
Escher's Fig. 2.3(c):
- Conformal structure of hyperbolic plane identical to Euclidean space interior, but different from entire Euclidean plane
Space-Time Geometry:
- Differences due to Minkowski's change of signature in metric
- Lorentzian space-time: 1 timelike direction and 3 spacelike directions (orthogonal)
- Orthogonality between spacelike and timelike directions symmetrically related to null direction between them
Measuring Spatial Separation in Space-Time:
- Ruler as a strip, not immediately obvious for measuring spatial separation (Fig. 2.19)
- Observer's rest frame required for distance measurement: using light signals and clocks
- Key fact about space-time metric: more directly related to time measurement than distance
Metric of Space-Time:
- Assigns time measure only to causal curves (timelike or null)
- Distinguished as chronometry by John L. Synge
Importance in Physics:
- Precise clocks exist at a fundamental level, central to General Relativity theory
- Individual massive particles play a role as virtually perfect clocks.
Matter and Energy: Mass, Rest Energy, and Quantum Theory
Rest Energy of Particles (Fundamental to Relativity Theory):
- Mass 'm' constant
- Einstein's famous formula: E = mc
Particle Behavior as a Quantum Clock:
- Stable massive particles behave like precise quantum clocks
- Each particle has specific frequency v of quantum oscillation (h is Planck's constant)
- High frequencies cannot be directly harnessed for practical use
- Need multiple particles combined in concert to build a clock
Massless Particles and Clocks:
- Massless particles (e.g., photons) have no rest energy or quantum oscillation frequency
- Cannot be used to make a clock as their frequencies would be zero
- This fact will be significant later
Bowl-shaped 3-Surfaces:
- Mark off the successive ticks of identical clocks
- Analogous to spheres in Minkowski's geometry
- Massless particles never reach the first bowl-shaped surface, agreeing with above statement
Geodesic Notion:
- In a space-time, geodesics are the world lines of massive particles in free motion under gravity
- Timelike geodesics: longest curve from one point to another on the same line
- For null geodesics (length is zero), only the null-cone structure of space-time determines them
Black Holes and Space-Time Singularities
Gravity's Effects:
- In most situations, the effects of gravity are small
- For a black hole, however, the "null cones" (future and past) deviate significantly from their Minkowski space positions
The Collapse of an Over-Massive Star:
- An over-massive star collapsing inwards
- Reaches a stage where the escape velocity becomes the speed of light
- Causes the "inward tilt" of the null cones to become extreme
- Locates the "event horizon" - the outer part of the future cone becomes vertical
Implications of the Event Horizon:
- Particles or light signals originating inside the event horizon cannot escape to the outside
- An external observer's light ray entering the eye cannot cross the event horizon into the interior of the black hole
The Fate of Material Inside a Black Hole:
- The original model suggested the material falls inwards and becomes infinite density at the center ("space-time singularity")
- Physicists view this as a fundamental conundrum, the "time-reverse" of the Big Bang origin
Trusting the Models:
- Questions about the trustworthiness of models like Oppenheimer-Snyder and Friedmann
- Concerns with assumptions like spherical symmetry and pressureless material
- The author notes these issues, as they stimulated their thinking on gravitational collapse in 1964.
Wheeler's Insights on Gravitational Collapse and Singularities
Background:
- Wheeler was inspired by Maarten Schmidt's discovery of a remarkable object with brightness and variability indicative of black hole presence.
- Common belief was that space-time singularities would not arise in general gravitational collapse due to theoretical work by Lifshitz and Khalatnikov.
- Wheeler had doubts about the mathematical analysis used and started thinking about problem geometrically.
Approach:
- Studied global aspects of light ray propagation, focusing, and singular surfaces.
- Investigated steady-state model's consistency with general relativity.
- Used conformal space-time geometry for understanding focusing properties in various situations.
Discoveries:
- Reasonable departures from symmetry don't help to avoid inconsistencies between steady-state model and general relativity without negative energies.
- Deviations from symmetry can't save the steady-state model unless negative energies are present.
- Trapped surface as a criterion for unstoppable gravitational collapse.
- Established theorem: When a trapped surface forms, singularities cannot be avoided provided certain conditions are met.
- No assumption of symmetry or simplifying conditions required; only the weak energy condition is assumed (energy flux across any light ray must never be negative).
Theoretical Results on Singularities and Black Holes
Strengths of Oppenheimer-Snyder Theorem:
- Applies to physically realistic classical materials considered by relativity theorists
- No information about the detailed nature of the singularity, including geometrical form and infinite density/curvature
- Does not specify where singular behavior will begin to show itself
Belinski-Khalatnikov-Lifshitz (BKL) Conjecture:
- Provides a plausible case for an extraordinarily complicated chaotic type of activity approaching a singularity
- Anticipates "mixmaster universe" behavior, first proposed by Charles W. Misner
- Likely the general case for singularities in relativity theory
Chandrasekhar Limits:
- Chandrasekhar (1931) showed limit on mass that can sustain itself against gravity
- Raises profound conundrum for larger, more massive stars
- Evolution of a star like the Sun leads to a white dwarf
- For larger stars, white-dwarf core could collapse into a neutron star or black hole
Trapped Surfaces and Black Holes:
- Trapped surfaces arise when sufficient mass is concentrated in a region smaller than the event horizon of a neutron star
- The existence of a trapped surface does not necessarily imply a black hole, depending on "cosmic censorship" conjecture
- Observational evidence favors the presence of black holes in certain binary star systems and at galactic centers
Conformal Diagrams and Conformal Boundaries
- Useful for representing space-time models with exact spherical symmetry (e.g., Oppenheimer-Snyder, Friedmann space-times)
- Two types: strict conformal diagrams and schematic conformal diagrams
Strict Conformal Diagrams:
- Represent a 2-dimensional subspace of the full 4-dimensional space-time (denoted by )
- Each interior point represents an entire sphere worth of points in the 4-dimensional space-time
- Can be imagined as a region rotating about an axis to visualize the 4-dimensional picture
- Axis of rotation is part of the boundary, with points representing single points rather than an S2 (single line in )
- Inherits conformal space-time structure from and has its own time-oriented null cones
Schematic Conformal Diagrams:
- Used to represent general space-times without exact spherical symmetry or when there is no exact symmetry axis.
Friedmann Cosmologies and their Diagrams:
Extending Hyperbolic Plane
- Fig. 2.31: Smooth conformal manifold extension to Euclidean plane
- Represents entire sphere S2 as white dot on boundary, single points as black dots
Singularities in Friedmann Cosmologies (=0)
- K>0, K=0, K<0 represented in Fig. 2.34(a),(b),(c) respectively
- White dots: conformal subregions of the entire manifold
- Fig. 2.35 shows cases with positive cosmological constant >0
- Future infinity is spacelike, indicated by final bold boundary line being more horizontal than 45
De Sitter Space-Time and Steady-State Model:
- Closely approaches de Sitter space-time in remote future (Fig. 2.36)
- Sketched as a 2D version with one spatial dimension represented (Fig. 2.36a)
- Strict conformal diagram for steady-state model is half of de Sitter space-time
- Incomplete, only present in past directions
Singularities and Incompleteness:
- Single point singularities represented as white dots on boundary
- Internal dotted lines denote black holes event horizons
- Consistent use of lines (broken for symmetry axis, bold for infinity, wiggly for a singularity, jagged for incompleteness) and spots (black representing single point in 4-space, white tracing out an S2) in strict conformal diagrams
OppenheimerSnyder Collapse to a Black Hole:
- Strict conformal diagram constructed from gluing together parts of Friedmann model (Fig. 2.38a) and EddingtonFinkelstein extension of Schwarzschild solution (Fig. 2.38b,c)
- Fig. 2.39: Strict conformal diagrams for spherically symmetrical vacuum: original Schwarzschild solution, extension to EddingtonFinkelstein collapse metric, and full extension to Kruskal/Synge/Szekeres/Fronsdal form.
Black Holes: Schwarzschild Solution and Hawking Radiation
- Simpler description of black holes: discovered by Arthur Eddington in 1930, rediscovered by David Finkelstein in 1958 (Fig. 2.39(b))
- Maximal extension of Schwarzschild solution (Kruskal-Szekeres extension): given in strict conformal diagram Fig. 2.39(c)
- Hawking radiation: discovered by Stephen Hawking in 1974, black holes have a tiny temperature T inversely proportional to mass
- Example: 10M black hole temperature ~6109 K compared to record low lab temperature of ~109 K
- Larger black holes colder, e.g., ~1.510^-14 K for the center of our galaxy
- Ambient temperature: currently around 2.7 K, expected to get down to temperatures of even the largest black holes through universe's expansion
- Black hole evaporation: loses mass and energy by Einstein's E=mc^2, shrinks away until disappearing with a "pop" (energy of an artillery shell)
Conformal Diagrams
- Schematic conformal diagrams: useful for clarifying ideas, bring infinite regions into finite comprehension, and smooth out space-time singularities
- Two types of horizons in cosmology: event horizon and particle horizon.
Event Horizon
- Dependent on observer's perspective, represents an absolute boundary to observable events (Fig. 2.43)
Particle Horizon
- Arises when past boundary is spacelike instead of singularity, related to strong cosmic censorship issue (touched upon in the next section).
Understanding the Way the Big Bang Was Special
Big Bang:
- Extraordinarily special, but peculiar with regard to gravity
- Entropy was enormously low in comparison to what it could have been
- Entropy was close to maximum in every other respect
Inflation Theory:
- Popular idea that the universe underwent an exponential expansion (cosmic inflation) in early stages
- Expansion aimed to explain the uniformity of the early universe, with practical irregularities being ironed out
Issues with Inflation Theory:
- Does not address the fundamental question: Origin of the extraordinary manifest specialness of the Big Bang
- Dynamics underlying inflation are governed by time-symmetrical dynamical laws
- Includes an "inflaton field" responsible for inflation, which would involve a phase transition and entropy increase
Alternative Perspective:
- Consideration of a collapsing universe helps understand high-entropy initial state
- In a collapsing universe, deviations from FLRW symmetry would become more exaggerated, making inflation irrelevant
- Collapse would lead to a highly complicated, enormously high-entropy singularity, unlike the closely FLRW-form singularity in our actual Big Bang
Collapsing Universe and Singularities
Collapsing Lumpy Universe vs Expanding Universe:
- Collapsing lumpy universe could have had a high-entropy singularity as an initial state
- Time reversal shows expanding universe starts with a high-entropy singularity, more probable than Big Bang
- Black holes in final stages provide image of initial singularity consisting of multiply bifurcating white holes
Singularities:
- White hole is time reverse of black hole, violates Second Law, no light can enter horizon
- Entropy value assigned to a black hole according to Bekenstein-Hawking formula (A/4 or SBH = A/4)
- Largest contributor to universe's entropy comes from large black holes in galactic centres
- Probability of such a special universe occurring by chance is extremely low (1/irrespective of inflation)
Initial Singularity as Instantaneous Event:
- Spacelike initial singularities can be considered as the zero point of cosmic time coordinate
- Time-reverse of Oppenheimer-Snyder collapse has a spacelike initial singularity
- Generic BKL singularities also have this spacelike character due to strong cosmic censorship
Geometrical Criterion for Singularities:
- Important question: What distinguishes smooth singularity of low-entropy Big Bang from general high-entropy type?
Gravitational Degrees of Freedom vs. Electromagnetic Field
- Gravitational degrees of freedom need clear identification
- Comparison with electromagnetic field (EMF) analogy
- EMF described by tensor quantity F, named after Maxwell
- Tensors are crucial in general relativity theory
- Differences between metric tensor g and Maxwell tensor F
- Valence: F has a valence of 2, while g has a symmetry that makes it appear like a single tensor
- Components: F has 6 independent numbers per point (3 electric, 3 magnetic), while g has 10 components
- In EM theory, source for electromagnetic field is the charge-current vector J
- Analogues in gravitational field: curvature tensor R (Riemann or Christoffel) and energy-stress tensor E
- E acts as magnifying lens; C, the Weyl conformal tensor, provides astigmatic distortion of distant star images.
Gravitational Field vs. Electromagnetic Field Observations:
- Direct observation of effects on light rays: gravitational lensing effect
- First clear evidence for general relativity through observations during solar eclipse (1919)
- Stars appear displaced due to Sun's gravitational field
- Distortion observed in star images outside the Sun's limb (elliptical pattern) measures amount of Weyl curvature C intercepted by line of sight.
Understanding Gravitational Lensing and Weyl Curvature Hypothesis
Background:
- Galaxies tend to be elliptical, making it difficult to determine if an individual galaxy's image has been distorted by gravitational lenses
- Statistics can help estimate mass distributions from patterns of ellipticity in background field galaxies
- Application: mapping dark matter distributions (2.60)
Conformal Curvature:
- Describes deviation of null-cone structure from Minkowski space
- C, conformal curvature introduces ellipticity into bundles of light rays
- Infinite Weyl curvature at final singularities (opposite to Big Bang)
Weyl Curvature Hypothesis:
- Proposed condition for initial singularities: C = 0
- Appropriate but mathematically ambiguous due to tensor behavior at singularities
- Tod's proposal: smooth past boundary to space-time as a conformal manifold, constraining C to be finite at Big Bang.
Schematic Diagram:
- Fig. 2.49 represents Paul Tod's proposal for WCH
- Assertion that space-time can be continued smoothly prior to the hypersurface before Big Bang is just a mathematical trick with no physical meaning.
Part 3: Conformal Cyclic Cosmology (CCC)
Connecting with Infinity
- In early universe, extremely high temperatures cause rest-mass of particles to be negligible
- Particles become effectively massless and follow conformal space-time structure
- Relevant physical processes become insensitive to local scale changes
The Structure of CCC
- Conformally invariant theories govern electromagnetic and strong interactions
- Maxwell equations are conformally invariant, allowing for scale changes without affecting equations
- YangMills equations also conformally invariant for strong and weak interactions
- Photons, quarks, gluons, W+, W-, Z particles part of a multiplet with mass linked to Higgs
Earlier Pre-Big Bang Proposals
- In standard theory, rest-mass becomes more irrelevant as temperatures increase
- Conformal geometry appropriate for physical processes near Big Bang, possibly extending back to pre-Big Bang region
Squaring the Second Law
- [To be discussed in 3.4]
CCC and Quantum Gravity
- [To be discussed in 3.5]
Observational Implications
- [To be discussed in 3.6]
Pre-Big-Bang Phase
Propagation of Particles/Fields:
- Photons and other massless particles/fields can propagate smoothly between pre- and post-Big-Bang phases
- Information can be carried forward or backward through these phases
Physical Reality of Pre-Big-Bang Phase:
- Question if we should treat it as physically real
- Suggestions: collapsing phase of the universe that bounces back into an expanding one, violating Second Law
- Difficulties with this proposal: against overall purpose, mathematical difficulties
Explaining the Second Law:
- Aim to find explanation or rationale for the Second Law instead of decreeing special state at bounce moment
Ultimate Future of the Universe (Conformal Diagram):
- Ultimately settles into exponential expansion
- Smooth spacelike future conformal boundary
- Contents consist mainly of photons and gravitons
Challenges in Constructing a Clock:
- Massless particles like photons and gravitons cannot be used to make a clock (discussed in 2.3)
- Other potential dark matter not useful for constructing a clock due to its interaction only through the gravitational field.
Philosophical Change:
- Subtle change of philosophy required for understanding the ultimate picture presented.
Conformal Cyclic Cosmology (CCC)
Thought Experiment:
- Considered the idea that our universe may not be the only one, but rather a part of an extended conformal manifold consisting of a succession of expanding universes
- Massless particles do not experience time as meaningful, making measurements and distance impossible in their absence
- Positive cosmological constant allows for space-time extension on either side of future/past singularities (Big Bang and Big Crunch)
Potential Challenges:
- Identifying the future with the past raises concerns about causal inconsistencies and paradoxes, as information can potentially be passed across these boundaries
- Suggesting a physically real region prior to our "present aeon" (future) and after our "past aeon" (Big Crunch) as an alternative
Key Points:
- Universe composed of a possibly infinite succession of aeons, each expanding into the next seamlessly
- Conformal stretching at Big Bang brings infinite temperature and density to finite values, while conformal squashing at infinity brings zero density and temperature up to finite levels
- Phase space describing physical activity has a conformally invariant volume measure due to rescaling of distance and momentum measures.
The Structure of Conformal Cyclic Cosmology (CCC)
Key Issues:
- Contents of the universe in the very remote future
- Main contributors: photons, gravitons, and potentially other particles
- Philosophical standpoint on the nature of time and geometry in the late stages of the universe's existence
Problems with the Philosophical Standpoint:
- Presence of material within bodies that never fall into black holes (e.g., white dwarf stars, electrons/positrons)
- Inability to explain the absence of massless charged particles (electrons and positrons) in today's universe
- Charge conservation and the possibility of a massless charged particle that would maintain the philosophical standpoint
- Possibility of isolated charged particles, which cannot annihilate each other due to event horizons
- Potential violation of charge conservation or rest-mass conservation as solutions to this problem
Radical Resolution:
- Suggestion of a radical resolution: charge conservation is not one of Nature's stringent requirements, and electric charge might eventually vanish over time.
Alternative Solutions:
- Weakening the philosophical standpoint by arguing that isolated electrons/positrons would not be useful for constructing clocks
- Another resolution might be to suppose that the notion of rest-mass is not an absolute constant, and it could gradually fade away over time.
Lack of Observable Evidence:
- Absence of observational evidence for such violations of conventional ideas (charge conservation or rest-mass)
Implications:
- Theoretical backing for the conventional ideas is less substantial than in the case of charge conservation.
Rest-Mass vs. Absolute Constants in Particle Physics
Rest Mass of Fundamental Particles:
- Rest mass is not an absolute constant as it is in the very early universe and may fade away to zero in the very remote future
- The underlying reason for the particular values of rest masses of individual particle types is completely unknown
- Rest mass is a Casimir operator of the Poincar group, which describes symmetries in Minkowski space
- However, this role becomes less fundamental when there is a positive cosmological constant (0) present
- The ultimate status of rest mass becomes more questionable in relation to cosmology, as it is not exactly a Casimir operator of the de Sitter group
Implications for Cosmological Scaling of Time:
- Rest mass was used to provide a well-defined scale of time for passing from conformal structure to full metric
- If particles' masses decay extremely gradually, it raises a quandary about using particle masses for scaling the metric
- Preserving Einstein's equations with constant requires another proposal for scaling the metric
- The coupling of gravity to its sources is not conformally invariant, which complicates the philosophy of CCC
Conformal Invariance in Electromagnetism vs. Gravity:
- Maxwell's equations are preserved under conformal rescaling, but this does not hold for gravity
- The free gravitational field has a conformal invariance, but the coupling to sources is not conformally invariant
- Conformal invariance is more straightforward in electromagnetism compared to gravity
Conformal Invariance and Cosmic Creation Cycles (CCC)
Consequences of Conformal Invariance:
- Approaching from past: need to use conformal factor that tends to zero smoothly but with non-zero normal derivative
- Illustrated in Fig. 3.5 - Conformal time refers to height in a conformal diagram
- Gravitational field measured by tensor K, propagates according to conformally invariant equation
- Finite values on determine strength and polarization of gravitational radiation (analogous to light)
- Same applies to electromagnetic radiation field (light)
- Finiteness of implies that conformal curvature must also be zero at big bang surface of subsequent aeon
Mathematical Representation:
- Encoding essential information of in a way that does not distinguish it from its reciprocal 1
- Using tensor ****, which remains smooth over crossover 3-surface and is unchanged under replacement with its reciprocal
- Demanding to be a smoothly varying quantity across the crossover
- Mathematical conditions required for this transition are satisfactory and unique, detailed arguments in Appendix B
Massless Fields:
- Only massless fields present in very remote future of earlier aeon (i.e., just prior to )
- Scaling freedom in choice of rescaled metric in region just prior to big bang of subsequent aeon
- Described in terms of self-coupled conformally invariant massless scalar field equation, the -equation
- Phantom field refers to particular choice that gives Einstein's physical metric g
- On opposite side of crossover, negative effective gravitational constant leads to unphysical implications, requiring adoption of alternative interpretation using 1 instead of ****
- Phantom field turns into a real physical field on the big-bang side of crossover, potentially providing initial form of new dark matter
Dark Matter:
- Dominant form of matter, accounting for 70% of ordinary matter (excluding cosmological constant)
- Does not fit neatly into standard model of particle physics, only interacts through gravitational effect
The Phantom Field in Cosmology:
- In late stages of prior aeon, phantom field arises as effective scalar component to gravitational field due to conformal rescalings and has no independent degrees of freedom
- Subsequent aeon: new matter takes over gravitational waves' degrees of freedom, forming dark matter
- Dark matter acquisition of mass in Big Bang is necessary for CCC scheme
- Two "dark" quantities (dark matter and energy) required in CCC
- Observational evidence indicates that both are necessary ingredients
- Phantom field's effect on universe expansion comparable to attraction due to matter
The Value of g:
- Interpreted by quantum field theorists as vacuum energy
- Relativity argues it should be a -tensor proportional to g, but missing factor is puzzling
- Observed value much smaller than expected (around 10^(-120) times larger)
- Coincidence with expansion effects of universe not so unusual considering other physical constants and dimensions.
Dirac's Universal Number N:
- Discovered by Paul Dirac in 1937 as a number that appears in various ratios of fundamental physical constants, particularly those involving gravity
- Age of the universe is approximately N^3 (in terms of Planck time)
- Useful for expressing other physical constants as pure numbers using Planck units
- Original idea was that N would increase with time or G decrease; however, more accurate measurements show it's not constant.
- Dicke and Carter's argument: a creature living in the middle of an ordinary star's active existence would find a universe around N^3 age, explaining apparent coincidence of cosmological constant coming into play now.
Pre-Big-Bang Proposals
Oscillating Universe Model (Friedmann, 1922)
- Contrasted with CCC scheme
- Radius of 3-sphere describing the spatial universe has a cycloid shape
- Succession of expanding and collapsing aeons
- Bounce occurs at space-time singularity, no sensible evolution
- No progressive change representing an increase in entropy
Tolman's Modification (1934)
- Oscillating Friedmann model modified with composite gravitating material
- Entropy increase through internal degree of freedom
- Longer durations and greater maximum radii at each stage
- No contributions to entropy from gravitational clumping
- Better description for universe close to Big Bang
- Radiation-filled analogues of all six Friedmann models introduced
- Strict conformal diagrams do not differ greatly from original Tolman solutions.
Tolman's Contributions and Relevance to CCC
- Representation of material contents as a pressureless fluid (dust) in Friedmann models
- More accurate description near Big Bang with radiation-filled models
- Radius function replaced by semicircle for closed radiation-filled universe
- Analytical continuation doesn't make sense for Tolman's radiation model prior to the big bang.
CCC and Conformal Factors Behavior:
- Reciprocal of k used: k = 1/^2 (with being the Friedmann constant)
- Infinite behavior at singularity becomes problematic for continuation across it
- Comparison between Friedmann's dust and Tolman radiation models:
- Friedmann: square of local time parameter, no sign change (Fig. 3.11a)
- Tolman: varies with local time parameter, sign change required for smoothness (Fig. 3.11b)
- CCC solution presents catastrophic reversal of gravitational constant sign if no switch is made at crossover surface
- Proposed idea of varying dimensionless constants in cyclic models like Friedmann oscillating model or Tolman radiation solution
- The anthropic principle: universe constants could affect life existence, controversial concept.
Wheeler's Proposal and Smolin's Model:
- John A. Wheeler proposed that constants might alter during singular state passage
- Dimensionless constants could change in cyclic models like CCC
- Lee Smolin suggested black holes collapse to form new expanding universes with modified dimensionless constants (Fig. 3.12)
- Collapse-expansion process in Smolin's model unlike CCC, relationship to Second Law unclear.
Smolins Romantic View of the Universe
- New aeons emerge from black-hole singularities
- Cosmological proposals based on string theory and extra dimensions
- Pre-big-bang proposal: Gabriele Veneziano's model
- Strong points in common with CCC
- Roles of conformal rescalings, inflationary period
- Dependent on ideas from string-theory culture
- Proposal by Paul Steinhardt and Neil Turok
- Transition between aeons via collision of D-branes
- Difficult to compare with CCC due to reliance on extra dimensions and string theory concepts
- Pre-big-bang proposal: Gabriele Veneziano's model
- Numerous attempts to use quantum gravity for non-singular quantum evolution
- Simplified lower-dimensional models used
- Implications for 4-dimensional space-time not always clear
- Singularities still present in most proposals
- Most successful proposal: using loop-variable approach by Ashtekar and Bojowald
- Quantum evolution through what would classically be a cosmological singularity
- No serious inroad into fundamental issue of suppressing gravitational degrees of freedom in Big Bang, as required by the Second Law.
Pre-Big-Bang Proposals vs. CCC
- Most proposals lie within the scope of FLRW models and do not address essential matters related to the Second Law
- Classical or quantum bounce may not be sufficient for geometrical matching between one aeon and the next
- Cyclic process consistent with Second Law remains a challenge that must be confronted seriously.
Understanding the Second Law of Thermodynamics: The Entropy Conundrum
- The Second Law of Thermodynamics states that entropy, or disorder, in a closed system always increases over time
- Issue arises from apparent increase in entropy despite similarities between early and late universe states
- Conformal changes do not affect entropy due to phase-space volumes (Boltzmann formula)
- Possible solutions: cosmic inflation, CCC interpretation, or non-zero Weyl tensor C
- Early universe dominated by conformally invariant physics with massless ingredients, including photons and dark matter
- Late universe mostly inhabited by massless particles like photons; entropy lies within these particles
- Major contributors to entropy increase: black holes in galaxies (10^21 entropy per baryon) vs. CMB (10^9 entropy per baryon)
- Current entropy per baryon in the universe is much larger than that of CMB, primarily due to black holes
- Black hole entropy will continue growing in the future, making this number insignificant compared to far-future values.
Black Holes and Entropy
Conundrum:
- How can the entropy appear to have shrunk by an enormous factor?
- Understanding how the entropy will ultimately reappear is crucial
Fate of Black Holes:
- After around 10^100 years, black holes will evaporate through Hawking radiation
- Each black hole presumed to disappear finally with a "pop"
Black Hole Entropy and Temperature:
- Consistent with the Second Law of Thermodynamics
- Explanation based on Bekenstein's argument using quantum-mechanical and general-relativistic principles
- Hawking temperature TBH derived from standard thermodynamic principles
Final State of Black Holes:
- Black hole entropy and temperature suggest a very large entropy
- Ultimate fate is hard to predict, but may involve a quantum gravity regime
- CCC's perspective requires that nothing with rest mass persist forever
Classical vs. Quantum Description:
- Classical description of black holes may be insufficient at extremely small scales
- Singularity in classical space-time is hard to believe, but may be a consequence of classical general relativity
Role of Singularity in Classical Picture
- Examine conformal diagrams (Fig. 3.13) to understand singularity's role
- Two parts: Fig. 3.13(a) and Fig. 2.41
- Assume strong cosmic censorship for qualitative accuracy near pop
- Singularity remains spacelike according to strong cosmic censorship
- Extreme irregularities in space-time geometry close to singularity
- Little hope of adopting standpoint like crossover 3-surfaces of CCC
- Two types of singularities: Big Bang and black hole
- Tame vs. chaotic BKL nature
- Information loss at the point of physical evolution into unknown territory
Possible Alternatives for Preserving Information
- Quantum gravity coming to rescue, allowing a bounce with mirrored space-time geometry: unlikely due to lack of time symmetry fundamental processes
- Information leaking out before pop as quantum entanglements: doubtful as it violates basic quantum principles and information cannot emerge much before the moment of singularity
- Hawking's argument on thermal radiation from a black hole: based on the assumption that information falling into the hole is lost, leading to the conclusion of thermal radiation with temperature equal to Hawking temperature.
Personal Opinion
- Information is likely lost in black holes
- Hawking's revised opinion, more conventional among quantum field theorists, may not be closer to the truth.
Black Hole Information Paradox and Quantum Theory
- Physicists struggle with the idea that information can be destroyed in a black hole (information paradox)
- Reason for this is the principle of unitary evolution, which requires quantum states to evolve deterministically and reversibly, preserving information
- However, Hawking evaporation suggests information loss as black holes shrink
Quantum States and Observations
- Quantum state or wavefunction (represented by ) evolves according to Schrdinger equation during unitary evolution
- To find observable values, a different process called measurement is applied to , resulting in probabilities of possible outcomes
- Measurement collapses the wave function onto one of the possible outcomes
Quantum Mechanics' Curiosities
- Strange hybrid of continuous and deterministic Schrdinger equation and discontinuous probabilistic measurements
- Physicists are not satisfied with this state, some argue for a complete story in the unitary evolution equation
Position on Quantum Theory
- The author sides with Schrdinger, Einstein, Dirac, and takes present-day quantum mechanics as provisional theory
- Believes information loss in black holes is a necessary reality, not just plausible
Black Hole Evaporation and Entropy
- Information loss can be described as a loss of degrees of freedom, shrinking the phase space
- This is a new phenomenon in dynamical evolution where the phase space actually decreases during evolution.
Evolution of Phase Space Following Black-Hole Information Loss
Illustration:
- Fig. 3.14: Evolution in phase space following black-hole evaporation
- Fig. 3.15(a) and (b): Spacetime diagrams of Hawking-evaporating black hole
Discussion:
- In general relativity, there is no unique universal time
- Information loss in black holes occurs gradually over their existence
- Two families of spacelike 3-surfaces: one with sudden disappearance at "pop", another with gradual disappearance
- Indifference to when information loss takes place emphasizes its irrelevance to external (thermodynamic) dynamics
- Degrees of freedom lost have no effect on local physics outside black hole, but reduce overall phase space volume
Comparison with Currency Devaluation:
- Reducing phase space volumes in the universe counts as a large constant subtraction from overall entropy
- Additivity of entropy for independent systems, as discussed in 1.3
Consistency and Viability:
- Reduction in phase-space volume needed for resolving the conundrum posed at the beginning of this section
- Reasonableness of overall entropy increase through black hole formation and evaporation
- Need for more detailed study to calculate effective entropy reduction due to information loss
CCC and Quantum Gravity
- CCC provides a different perspective on issues that have long confronted cosmology
- Questions: nature of singularities in classical general relativity, role of quantum mechanics
Big Bang Singularity and Black Holes
- CCC has something to say about the nature of the Big-Bang singularity
- When propagating physics into the future, it either terminates at a black hole singularity or continues into the next aeon
Information Loss in Black Holes
- CCC requires initial phase-space coarse-graining region to match final one
- Acceptance of huge information loss in black holes allows for the phase space to become thinned down
Cosmological Entropy
- Issue of cosmological entropy from event horizons when >0
- Proposed Bekenstein-Hawking formula for cosmological event horizon entropy (S)
- S depends only on the value of and has nothing to do with details of the universe
- Temperature T derived from entropy S is too small to be relevant
Interpretation of Cosmological Entropy S
- Ultimate entropy of whole universe or just portion within an event horizon?
- Difficulty in maintaining Second Law if entire universe has more entropy than S
- More appropriate interpretation: entropy of material within an event horizon (e.g., future particle horizon)
Possible Entropy of Material within Future Particle Horizon
- By the time o+ is reached, universe should contain about 10 times more material than present particle horizon
- Possible black hole with an entropy around 10^(12) could violate Second Law if attainable in a universe with observed value of
Black Hole Temperature and Entropy: Irreversible Processes
Problem with Accepted Value of T for an Irreducible Ambient Temperature
- Limitations of using observed value of to prevent black hole evaporation
- Black holes remain cooler than ambient temperature, never evaporating away
- Contradiction with the Second Law if past light cone intersects or encounters black hole
Alternative Perspectives on Black Hole Entropy
- Multiple regions of smaller black holes instead of one large one
- Total entropy larger than the Second Law's limit, but not by much
- Initial evidence for caution regarding physical interpretation of S as actual entropy
Challenges with Constancy of
- No discernible degrees of freedom in phase space if considered constant
- Difficulties when assuming as a varying field
- Strange form for energy tensor, unlike other fields
- Weak-energy condition violation
- Physical justification lacking for referring to as actual objective entropy
Cosmological Temperature: Observer Dependent and Subjective
- No information loss at a singularity in the universe's context
- Lack of clear physical arguments to justify cosmological entropy S
- Cosmological temperature has an observer-dependent aspect, unlike black hole Hawking temperature
- Unruh effect: accelerating observer feels temperature, not one in free fall.
Unruh Effect and Cosmological Entropy
Background:
- Observer moving freely in de Sitter background is unaccelerated
- Unruh effect: temperature felt by uniformly accelerating observers in Minkowski space (3.67)
Rindler Observers:
- Uniformly accelerating observers, also called Rindler observers
- Experience tiny Unruh temperature even though moving through a vacuum
- Future horizon 0 associated with temperature and entropy issues (Fig. 3.20)
- Similar to cosmological event horizon in some aspects
Questions Raised:
- Relevance of mathematical procedure based on analytic continuation for exactly symmetrical space-times
- Subjective element: observer's state of acceleration and symmetry
- Non-local considerations related to black hole entropy and temperature (Fig. 3.21)
- Assignment of reality to infinite entropy in Minkowski space
- Vacuum energy issue: quantum field theory vs current understandings
- Calculations for obtaining vacuum energy value: inconsistencies with observations
- Interpretation of cosmological constant as vacuum energy (dimensionally)
- Comparison between observed and theoretically expected values of
Points to Consider:
- Inconsistencies in applying rules of quantum field theory directly
- Preferences for interpreting results after applying methods to eliminate infinities
- Favored interpretation: non-zero value for instead of 0 since observational evidence supports it (2.1)
- Calculation issues may indicate problem with interpretation or calculation techniques.
CCC's Perspective on Singularities and Quantum Gravity:
- CCC's understanding of singularities in classical general relativity is not significantly affected by their physical status, as it does not require T or S to be physically true.
- No black hole reaching large sizes would seriously affect its evolution based on this perspective.
- The introduction of entropy S with a fixed value 3/ plays no role in the dynamics and can be ignored.
- Personal position is to ignore both S and T, as they seem to have no discernible role in the dynamics presented by CCC.
- CCC provides a clear yet unconventional perspective on how quantum gravity would affect classical space-time singularities.
Debate over Quantum Gravity:
- Lack of agreement about what quantum gravity actually is or should be.
- Some argue for maintaining a reasonably classical picture of space-time with tiny quantum corrections until extremely large space-time curvatures arise.
- Others suggest abandoning the smooth continuous space-time picture and replacing it with something radically different, such as quantum foam, topological complications, discrete structure, non-commutative geometry, higher-dimensional geometry, or even space-time fading away.
CCC's Conservative Picture:
- CCC provides a more conservative picture of the Big Bang singularity compared to wild or revolutionary ideas about quantum gravity.
- Smooth space-time without conformal scaling and time evolution can be treated using conventional mathematical procedures.
- Singularities occurring deep within black holes have different structures from the Big Bang singularity, requiring exotic information-destroying physics that may incorporate quantum-gravity ideas differing significantly from today's physics concepts.
Personal View:
- Previous view was that these two types of singular space-time geometries should be treated differently by true quantum gravity due to the Second Law's suppression of gravitational degrees of freedom at the initial end but not at the final one.
- Anticipated that true quantum gravity would require modifications to standard present-day rules of quantum mechanics in accordance with aspirations towards the end of 3.4.
- Unexpectedly found CCC treating the Big Bang as part of an essentially classical evolution governed by deterministic differential equations like those of standard general relativity.
Curvature and Quantum Gravity
- Carlo Rovelli's position: there is Weyl curvature C and Einstein curvature E (equivalent to Ricci curvature)
- Agrees that when radii of curvature approach the Planck scale, quantum gravity dominates, but only for Weyl curvature, not Einstein curvature
- Radii of curvature in Einstein tensor can be arbitrarily small while keeping space-time geometry classical and smooth if Weyl curvature radii remain large on the Planck scale
CCC's Perspective vs. Standard Approaches
- In CCC, the detailed nature of a big bang is determined by what happened in the remote future of the preceding aeon, leading to observational consequences (some discussed in 3.6)
- Classical equations continue the evolution of massless fields present in the very remote future into the next aeon's big bang
- Standard approaches to early universe assume that quantum gravity determines behavior at the Big Bang, such as inflationary cosmology and its explanation of CMB temperature deviations through quantum fluctuations
- However, CCC provides a different perspective on this issue.
Observational Implications
- Evidence for or against CCC's validity: difficult due to extreme organization in Big Bang and large temperatures that may obliterate information
- Overarching spatial geometry of an aeon prior to ours is deterministic and must match our own, either Euclidean (K=0), hyperbolic (K<0), or finite
- Small scale matter distributions can leave signatures on crossover 3-surface, readable in subtle irregularities in the CMB
- Important to understand phenomena causing these signals and their propagation between aeons
- Previous aeon likely behaves like our own, with exponential expansion in remote future due to positive cosmological constant
Evidence Supporting Inflationary Cosmology from CCC
- Correlations observed in temperature variations of the CMB that are inconsistent with standard pre-inflationary cosmologies
- Explanation: Pre-Big Bang theories, like CCC, can address this through an exponential expansion of separation between points in a conformal diagram (Fig. 3.23)
- Initial density fluctuations in the CMB appear scale-invariant over a broad range
- Explanation: Self-similar process of exponential expansion can result in a distribution with certain scale invariance if there is randomness in initial fluctuations distribution.
CCC: Cosmological Model and Its Implications
Background:
- E.R. Harrison and Y.B. Zeldovich's proposal of scale invariant initial fluctuations, later supported by inflationary idea
- Inflation gave rationale for assuming initial fluctuations are scale invariant
- Analysis of CMB observations confirmed close scale invariance over a greater range
CCC: Explanation for Scale Invariance and Correlations Beyond Horizon Size
- Displacing the inflationary phase to a previous aeon
- Effectively self-similar expanding universe, leading to density fluctuations with scale-invariant nature
- Correlations outside Friedmann or Tolman models' horizon size are expected due to events in the previous aeon
Big Question Mark: Fundamental Numerical Constants in Previous Aeon
- Wheeler's suggestion of possible change in fundamental numerical constants (N)
- Two sides: assuming same values or observable effects and changes
- CCC expectations for our own aeon into the future: cosmological constant, exponential expansion until eternity, Hawking radiation deposited in low-energy photons and gravitational radiation.
Potential Detection of Previous Aeon's Effects:
- Possibility of detecting Hawking radiation from previous aeon through CMB irregularities
- If CCC is right, this information could be teased out despite the fact that such effects are normally unobservable in our own aeon.
Unconventional Implications:
- Rest-masses of all particles decay away over time, ultimately becoming massless according to CCC
- No clear prescription for decay rate provided by scheme
- Decaying rest-mass effect might appear as very slow weakening of gravitational constant if all particle types have closely in proportion decay rates.
Experimental Limits:
- Best experimental limit on any decay rate for gravitational constant is less than about 1.6 x 10^(-12) per year
- Time scales and time periods to consider: 10^-10 years vs. 10^10 years
Observational Proposals:
- No clear-cut observational proposal exists for testing the aspect of CCC that demands the ultimate decaying away of rest-mass.
Black Holes Encounters and Gravitational Radiation
Significant Bursts of Gravitational Radiation due to Black Hole Encounters:
- If two black holes pass close enough, each would deflect the other's motion violently
- This could result in a significant burst of gravitational radiation
- The relative motions of the two objects would be appreciably reduced
- In extremely close encounters, the two objects might capture each other in orbits, resulting in tighter and tighter orbits until they merge into one black hole
- This merger would result in an enormous emission of gravitational waves
Gravitational Radiation Propagation:
- The gravitational wave burst would be virtually instantaneous
- In the absence of large distorting effects, the radiation would be contained within a thin spherical shell
- This would have implications for the geometry and matter distribution in the subsequent aeon (our own)
Interpretation of Conformal Metric Scaling:
- The gravitational field can be described by a -tensor K, satisfying a conformally invariant wave equation K = 0
- This allows us to regard K as propagating in the space-time depicted in Fig. 3.26
- As K reaches the crossover surface (), it has a non-zero normal derivative, influencing the geometry of the crossover surface and giving the initial material of the succeeding aeon a "kick" in the direction of the radiation
Observable Effects:
- The gravitational wave burst would give the initial material of the succeeding aeon (presumed primordial dark matter) a significant kick in the direction of the radiation
- This effect would be the same all around the entire circle C, where I is the geometrical intersection of the past light cone (u) and future light cone (+e) at the crossover surface
- Thus, for each black-hole encounter in the previous aeon, there would be a circle in the CMB sky that contributes either positively or negatively to the background average CMB temperature over the sky.
Conformal Cyclic Cosmology (CCC) Test and Analysis:
Background:
- Consulted David Spergel, Princeton University expert in CMB data analysis
- Preliminary analysis conducted by Amir Hajian on WMAP satellite observatory data to find evidence of CCC effect
Methodology:
- Choose succession of alternative radii for analysis
- Calculate average CMB temperature around each circle at these radii
- Produce histograms to check for significant deviation from Gaussian behavior
Initial Findings:
- Spurious effects eliminated by suppressing information from regions close to galactic plane
- Departures from randomness remained, particularly an excess of cold circles between 7 and 15
Proposed Explanation:
- Effects could be due to some spurious ingredients unrelated to CCC
- Crucial issue is whether these departures are specifically related to circular nature of regions being averaged over
Suggested Analysis:
- Repeat analysis with area-preserving twist applied to celestial sphere (elliptical shapes instead of circles)
- Three versions: no twist, small twist, large twist
- CCC should predict greatest effect with no twist, reduction with small twist, and elimination with large twist
Unexpected Result:
- Small amount of celestial twist enhanced the effect in specific radii (8.4 to 12.4)
- Possible explanation: presence of significant distortions due to Weyl curvature complicating analysis
Future Work:
- Break up celestial sky into smaller regions to identify regions with significant Weyl curvature along line of sight between observer and decoupling surface
- Clarify matters in the not-too-distant future for resolution of physical status of conformal cyclic cosmology.
Conformal Rescaling, 2-Spinors, Maxwell, and Einstein Theory
2-Spinor Formalism:
- Simpler to express conformal invariance properties
- Provides a more systematic overview of massless field propagation and Schrdinger equation for their constituent particles
- Employs quantities with abstract spinor indices (A, B, C, ...) for the complex 2-dimensional spin-space
- Tangent space refers to indices in upper position, cotangent spaces to lower position
Maxwell Equations:
- Fab (= F_ba) can be expressed in 2-spinor form as a symmetric 2-index 2-spinor AB (= BA)
- Maxwell field equations with source J^a take the forms:
- F = 4J (when sources are present)
- F = 0 (free Maxwell equations, no sources)
- These equations are conformally invariant in the sense of A6.
Massless Free-Field Equation:
- Represents Schrdinger equation for a massless particle of spin n (>0):
- ** = 0**
- For n=0, the field equation is usually taken to be the d'Alembert operator.
Space-Time Curvature Quantities:
- Rabcd has symmetries:
R[a, b] = R[b, a](cyclic permutation)- Trace of each index separately:
Tr(R[a, b]) = 0
- Relates to commutators of derivatives via the Bianchi identity.
Massless Gravitational Sources:
- When the source tensor Tab is trace-free (T_aa = 0), the Einstein equations are:
R[a, b] = T^(ab)
- Conformal invariance of massless gravitational sources is discussed in A6.
Bianchi Identities:
- General Bianchi identity: [a, R_bc]^de = 0
- When the Ricci curvature is constant (Einstein equations with massless sources), the Bianchi identity simplifies.
Conformal Rescalings:
- Conformal rescaling of quantities:
a'^2 = e^(2) a^2 - Preserves the vanishing of the massless free-field equations and Maxwell equations with sources.
YangMills Fields:
- YangMills fields are conformally invariant, as long as we ignore the introduction of mass through the Higgs field.
Scaling of Zero Rest-Mass Energy Tensors:
- Scaling of energy tensors for massless fields preserves their conservation equations.
Weyl Tensor Conformal Scalings:
- The conformal spinor ABCD encodes the information of the conformal curvature of space-time, which is conformally invariant (up to a factor).