The Isaac Newton institute series of lectures, 2010 -- Princeton University
From two of the world's great physicists―Stephen Hawking and Nobel laureate Roger Penrose―a lively debate about the nature of space and time
Einstein said that the most incomprehensible thing about the universe is that it is comprehensible. But was he right? Can the quantum theory of fields and Einstein's general theory of relativity, the two most accurate and successful theories in all of physics, be united in a single quantum theory of gravity? Can quantum and cosmos ever be combined? On this issue, two of the world's most famous physicists―Stephen Hawking (A Brief History of Time) and Roger Penrose (The Emperor's New Mind and Shadows of the Mind)―disagree. Here they explain their positions in a work based on six lectures with a final debate, all originally presented at the Isaac Newton Institute for Mathematical Sciences at the University of Cambridge.
How could quantum gravity, a theory that could explain the earlier moments of the big bang and the physics of the enigmatic objects known as black holes, be constructed? Why does our patch of the universe look just as Einstein predicted, with no hint of quantum effects in sight? What strange quantum processes can cause black holes to evaporate, and what happens to all the information that they swallow? Why does time go forward, not backward?
In this book, the two opponents touch on all these questions. Penrose, like Einstein, refuses to believe that quantum mechanics is a final theory. Hawking thinks otherwise, and argues that general relativity simply cannot account for how the universe began. Only a quantum theory of gravity, coupled with the no-boundary hypothesis, can ever hope to explain adequately what little we can observe about our universe. Penrose, playing the realist to Hawking's positivist, thinks that the universe is unbounded and will expand forever. The universe can be understood, he argues, in terms of the geometry of light cones, the compression and distortion of spacetime, and by the use of twistor theory. With the final debate, the reader will come to realize how much Hawking and Penrose diverge in their opinions of the ultimate quest to combine quantum mechanics and relativity, and how differently they have tried to comprehend the incomprehensible. - The Nature of Space and Time (Amazon)
- Chapter One: Classical Theory
- Chapter Two: Structure of Spacetime Singularities
- Chapter Three: Quantum Black Holes
- Chapter Four: Quantum Theory and Spacetime
- Chapter Five: Quantum Cosmology
- Chapter Six: The Twistor View of Spacetime
- Chapter Seven: The Debate
- Afterword to the 2010 Edition: The Debate Continues
Lecture Overview:
- Roger Penrose and Stephen Hawking's related but differing viewpoints on nature of space and time
- Emphasis on technical lectures with basic knowledge of general relativity and quantum theory assumed
- Classical work agreement, difference in approach to quantum gravity and quantum theory
- Hawking's positivist perspective versus Roger's Platonism
- General relativity successful, may require modifications on Planck scale but still agrees with observations
- Differences between general relativity and string theory: predictions, mathematical beauty vs. observational tests
- Lack of distinctive predictions from string theory questioned as a scientific theory
- Topics covered: singularities, black holes, global causal structure of spacetime
Background:
- Roger Penrose's contributions to general relativity and discovery of singularity theorem inspired Hawkings work on classical work.
- Differences in perspectives on quantum gravity and quantum theory.
- Classical work agreement but disagreement in approach to quantum theories.
General Relativity:
- Beautiful theory, agrees with every observation made so far.
- May require modifications at Planck scale.
- Supergravity or string theory as possible fundamental theory but lacks testable predictions and observational evidence.
Hawking's Perspective:
- Physical theory is just a mathematical model, meaningless to ask whether it corresponds to reality.
- All that can be asked is for its predictions to agree with observations.
- String theory oversold, no clear distinctive predictions or observational evidence yet.
Singularities and Black Holes:
- Spacetime may have a beginning and end due to gravity's effect on shaping the arena it acts in.
- Gravity leads to regions of the universe that cant be observed, giving rise to concept of gravitational entropy as measure of what we cannot know.
- Critics claim these predictions are only artifacts of semiclassical approximation and will disappear with string theory. However, Hawking believes they are distinctively different from other field theories due to their impact on space-time structure.
Techniques:
- Roger Penrose introduced study of global causal structure of spacetime using I+(p) set, which defines all points reachable by future-directed timelike curves starting from p. Similarly defined for past and future directions. Self-evident definitions.
Causal Structure in Spacetime:
Boundary of Chronological Future:
- Cannot be timelike or spacelike for a set S, except at the set itself
- Boundary is null apart from the set S
- If q is not in closure of S, there's a past-directed null geodesic segment through q lying in the boundary
- More than one such segment may exist and q will be their future endpoint
Example:
- Minkowski space with a horizontal line segment removed
- Boundary has generators with no past endpoint
- Spacetime is incomplete but can be cured by multiplying the metric with a conformal factor
Global Hyperbolicity:
- An open set U is globally hyperbolic if:
- Intersection of future of p and past of q has compact closure (bounded diamond shaped region)
- Strong causality holds on U (no closed or almost closed timelike curves)
Significance:
- Implies a family of Cauchy surfaces for U, allowing prediction and formulation of quantum field theory
- Not clear if sensible quantum field theory can be formed on nonglobally hyperbolic backgrounds
Global Hyperbolicity vs. Singularity Theorems:
- In globally hyperbolic space, there's a timelike or null geodesic of maximum length joining any pair of points that can be joined by a curve
- Significance for singularity theorems: Maximum length geodesic between p and q if they can be joined by a timelike or null curve.
Spacetime Singularities and Conjugate Points
Introduction:
- Spacetime singularities: regions where curvature becomes unboundedly large, objectionable feature is particles with finite beginning/ending history
- Definition of spacetime as maximal manifold on which metric is smooth
- Incomplete geodesics indicate presence of singularities
Conjugate Points:
- Occur when null or timelike geodesies encounter regions of gravitational focusing, pairs exist if extension possible in both directions
- Defined by the existence of incomplete geodesies that cannot be extended to infinite affine parameter values
Energy Conditions:
- Weak Energy Condition (WEC): energy density T nonnegative for any frame, satisfied by reasonable matter
- Strong Energy Condition (SEC): Rablalb nonnegative for every timelike vector la, physically reasonable in classical theory
- Generic Energy Condition (GEC): both strong and weak conditions hold, curvature not specially aligned with geodesic at some point
- Gravity's Effect on Spacetime: attractive force causing neighboring geodesics to curve toward each other, leads to conjugate points via Raychaudhuri equation or Newman-Penrose equation
Singularity Theorems:
- Conditions: energy condition (WEC, SEC, GEC), no closed timelike curves, gravity so strong that nothing can escape
- Types: Hawking-Penrose theorem and others, prove spacetime must be geodesically incomplete if certain combinations of conditions hold.
- Weakening Conditions: stronger versions of other conditions may weaken the required energy condition.
Theorems Predicting Singularities in Spacetime
- Theorems predict singularities under certain energy conditions: generic, strong, and no-escape
- Singularities can occur in future (gravitational collapse) or past (beginning of the universe)
- Two main themes of the lectures: gravity's distinctiveness from other physical fields due to its effect on time and entropy.
Theorems for Singularities in Spacetime:
- Future Cauchy development D+(S): Region of points q where every past-directed timelike curve intersects S
- If future Cauchy development was compact, there would be a Cauchy horizon H+(S) and limiting null geodesic without endpoints in the horizon
- Contradicts generic energy condition if limit geodesic is geodesically complete
- Proves theorem if limit geodesic is not geodesically complete or future Cauchy development is not compact
- If future Cauchy development was compact, there would be a Cauchy horizon H+(S) and limiting null geodesic without endpoints in the horizon
- Past Cauchy development D-(S): Similar to future Cauchy development but for past-directed timelike curves
- If the past Cauchy development is not compact, there exists a timelike curve from S that never leaves the past Cauchy development
Weak Cosmic Censorship:
- Assumes null geodesic generators of I+ (future infinity) are complete in conformal metric
- Observers far from collapse live to old age and not wiped out by singularity
- Assumes past of I+ is globally hyperbolic.
Cosmic Censorship and Black Holes
Weak Cosmic Censorship:
- Assumes spacetime is globally hyperbolic
- Singularities in gravitational collapse are not visible from large distances
- Predicted singularities cannot be seen from "I+" (future null infinity)
- Implies the existence of a region of spacetime not in the past of I+, called a black hole
- Event horizon is the boundary of this region and the past of I+
Properties of Black Holes:
- No light or anything can escape from the black hole to infinity
- Boundary of the black hole is the event horizon
- Generators of the horizon cannot converge within finite distance
Black Hole Formation and Growth:
- When matter falls into a black hole, the event horizon area cannot decrease
- Area will generally increase over time
- If two black holes merge, the final event horizon has greater area than sum of original horizons
Similarities to Thermodynamics:
- Entropy analogy: event horizon area is similar to thermodynamic entropy
- Surface gravity (strength of gravitational field on horizon) analogous to temperature
- Zeroth law of black hole mechanics: surface gravity same everywhere on event horizon
Bekenstein's Proposal and Resolution:
- Bekenstein proposed black holes have entropy proportional to horizon area
- However, this led to inconsistency with thermal radiation absorption by black holes
- Consistency restored when it was discovered black holes emit thermal radiation
- Suggests black holes have intrinsic gravitational entropy related to their nontrivial topology
Singularities in Spacetime
Nature of Singularities:
- Not explicitly implied by singularity theorems
- Assumed that curvature diverges, but not what they imply
Types of Singularities:
- Big bang singularity: Occurs at the beginning of the universe
- Black hole singularity: Inside a black hole, not visible from outside
- Big crunch singularity: Could occur if universe collapses on itself
- Naked singularity: A singularity without an event horizon
Cosmic Censorship Hypothesis:
- Asserts that naked singularities do not occur
- Based on the idea that event horizons surround singularities, preventing them from being seen
Arguments for Cosmic Censorship:
- Oppenheimer-Snyder (OS) model of black hole formation:
- Singularity inside, but surrounded by event horizon
- Not clear if OS model is representative of general gravitational collapse
- Trapped surface arguments:
- Trapped surface implies singularity
- Time-reversed trapped surface implies singularity in the past
- Event horizons have certain properties:
- Boundary of past of future null infinity
- Smooth, generated by null geodesics
- Area of spatial cross sections cannot decrease with time
- Asymptotic future is the Kerr spacetime, a solution to Einstein's equations
Challenges to Cosmic Censorship:
- Attempts to find counterexamples have been made
- Cosmic censorship hypothesis is important for the theory of black holes
Theory of General Relativity and Cosmic Censorship Hypothesis
IFs, PIFs, TIFs, and -TIFs:
- In a spacetime with singularities (naked or covered), we have IFs (indecomposable futures)
- Divided into: PIFs (past-inaccessible futures) and TIFs (time-like inextendible futures)
- TIFs are further subdivided into -TIFs (future-events that can't be reached by a timelike curve) and singular TIFs
Assumptions:
- No two points have the same future or past
- Absence of closed timelike curves (or, more weakly, no two points have the same future or past)
Singularities and Cosmic Censorship Hypothesis:
- Naked singularities: something a timelike curve can both enter and exit from
- Doesn't count as naked if it's part of the big bang (as it always comes out of it)
- Defining naked TIP (or, equivalently, TIF) as contained in a PIP
- Essentially local definition, no requirement for observer to be at infinity
- Strong cosmic censorship hypothesis: exclusion of naked singularities in generic spacetime with matter and reasonable equations of state
- Equivalent to global hyperbolicity or the whole domain of dependence of some Cauchy surface
- Symmetric in time: interchange IPs and IFs for strong cosmic censorship
Thunderbolts:
- A singularity reaching null infinity, destroying the spacetime as it goes
- Doesn't necessarily violate cosmic censorship if we add stronger conditions
The Status of Cosmic Censorship:
- Probably not true in quantum gravity due to exploding black holes
- Classical general relativity: results both for and against cosmic censorship
- Violations of genericity condition or numerical evidence are subject to objections
- Recent indications that some inequalities do not hold if the cosmological constant is positive
Timelike vs Spacelike Singularities:
- Strong cosmic censorship says generic singularities are never timelike
- Two classes: past types (defined by TIFs) and future types (defined by TTPs)
- Naked singularities could unite both possibilities into one
- Typical examples of class (F): singularities in black holes, big crunch; class (P): big bang, white holes (if they exist)
- Disagreement with Stephen Hawking on this matter
Evidence for Different Laws:
- Figure 2.4 shows effects of spacetime curvature on acceleration and volume reduction.
The Weyl Curvature Hypothesis:
- The second law of thermodynamics and observations of early universe suggest that:
- Big Bang singularity was extremely uniform
- It was free of white holes (which violently disobey the second law)
- From these findings, we can argue that:
- The big bang had vanishing Weyl tensor
- Black hole/white hole singularities have diverging Weyl tensor in the generic case
Singularities and Weyl Curvature:
- In standard cosmological models:
- Big Bang has a vanishing Weyl tensor
- Converse: A universe with an initial singularity of a conformally regular type and vanishing Weyl tensor must be an FLRW (Friedmann, Lematre, Robertson, and Walker) universe
- Black hole/white hole singularities have diverging Weyl tensor
Implications:
- The constraint that the early universe was fairly smooth and free of white holes reduces the phase space in the early universe by a factor of at least 10^80 baryons
- A true theory of quantum gravity should replace our present concept of spacetime at a singularity
- It should provide a clear-way of talking about what we call a singularity in classical theory, rather than simply a nonsingular spacetime
Question and Answer:
- Question: Do you think that quantum gravity removes singularities?
- Answer: I don't believe it can be quite like that.
- The big bang could not have resulted from a previously collapsing phase without explaining how that previous phase had such low entropy
- Singularities of collapsing and expanding universes seem to have different geometries, so they would need to be joined together in a way that is unclear
- Answer: I don't believe it can be quite like that.
- A true theory of quantum gravity should provide a clear understanding of what happens at singularities.
Quantum Theory of Black Holes: The New Level of Uncertainty (Chapter Three)
Classical Theory of Black Holes:
- Gravity tends to draw matter together, forming stars and galaxies
- Eventually, some objects will continue contracting due to lack of support
- When mass exceeds certain limit (1.5 solar masses), it collapses into a black hole
- Critical size: event horizon formed, trapping light rays within
- No cosmic censorship conjecture: region of spacetime where information cannot escape called a "black hole"
No-Hair Theorem:
- Large amount of information lost during black hole formation
- Only parameters retained: mass (M) and angular momentum (J) for stationary black holes
- Extended to include electric charge (Q) with Robinson's work
- Information loss doesn't matter in classical theory as it's still accessible, albeit difficult to obtain
- Only parameters retained: mass (M) and angular momentum (J) for stationary black holes
Quantum Theory's Impact:
- Collapsing body sends out limited photons before crossing event horizon
- Outside observer unable to measure state of collapsed body completely
- Black holes radiate and lose mass through quantum theory
- Information might be lost forever as they disappear over time.
Controversial Claims:
- Some physicists reject the idea that information about a quantum system could be lost in a black hole
- Hawking suggests accepting loss of information based on lack of success in showing how it can escape black holes.
The Quantum Theory of Black Holes
Background:
- Discovered by quantum field theory on a black hole formed by collapse
- First diagrams used: Carter diagrams (named after Stephen Hawking's research)
Minkowski Space vs. Collapsing Star: Minkowski space:
- No black hole, no event horizon
- Diagram: triangle standing on one corner (Fig. 3.2) Collapsing star:
- Singularity at the top of the diagram
- Event horizon: diagonal line from top right to center symmetry
- Figure 3.3 shows the difference
Scalar Field in Space-time: Time-independent space:
- Wave equation solution with positive frequencies on I- would remain so on I+
- No particle creation Time-dependent collapse:
- Mixing of positive and negative frequencies leads to particle creation
- Steady rate of particle creation and emission after burst during collapse
- Thermal temperature (T) = 1/2, fixing constant of proportionality in Planck units
Reason for Exact Thermality: Discovery of thermal emission from black holes seemed like a miracle Revealed by joint work with Jim Hartle and Gary Gibbons:
- Singularities caused by bad choice of coordinates
- Euclidean metrics: positive definite, curved but not necessarily singular (Fig. 3.4)
- Identifying imaginary time (t) with some period reveals the significance.
Amplitude between Field Configurations
- Amplitude from configuration 1 at t1 to configuration 2 at t2: matrix element of e^(-iH(t2-t1))
- Path integral representation: all fields between t1 and t2 that agree with 1 and 2 on surfaces
- Choose time separation (t2 - t1) = pure imaginary, equal to iT
- Set initial field 1 = final field 2 and sum over a complete basis of states n
- Left hand: expectation value of e^(-HT), thermodynamic partition function Z at T = -1
- Right hand: path integral over all fields on Euclidean spacetime with period T in imaginary time direction
Path Integral for Partition Function
- If flat spacetime identified with period in imaginary time, gives result for black body radiation
- Schwarzschild solution also periodic in imaginary time with period T
- Path integral over fields on curved backgrounds gives physical quantities as thermal expectation values
Extension to Interactions
- Expand action I in a power series of perturbations g about Euclidean-Schwarzschild metric g0: quadratic terms describe gravitons, cubic and higher terms interacting gravitons
- Path integral over quadratic terms is finite, nonrenormalizable divergences at 2 loops cancel with fermions in supergravity theories
- Dominant contribution to partition function Z: action of background metric g0
- Different from general relativity due to surface term proportional to K and subtracted off flat space action
Calculating Mass and Entropy
- Differentiate log Z with respect to period T to get mass or energy expectation value
- Free energy is mass + T times entropy S, thermodynamic relation between partition function and free energy
- Action of black hole gives intrinsic gravitational entropy of 4M^2, same as laws of thermodynamics
Reason for Gravitational Entropy
- Gravity allows different topologies for spacetime manifold (Euclidean-Schwarzschild solution: S^2 x S^1)
- Two possible topologies: R^2 x S^2 and R^3 x S^1, each with different Euler numbers
- Periodically identified flat space has zero Euler number, while Euclidean-Schwarzschild solution has Euler number 2
- Significance: find a periodic time function whose gradient is nowhere zero on topology of periodically identified flat space to calculate action between surfaces.
Black Holes and Euclidean Space
Mass Term at Infinity:
- Surface term over 1 and 2 cancel, net contribution is from infinity
- Gives half mass times imaginary time interval (2 - 1)
Action for Nonzero Mass:
- Requires nonzero matter fields to create mass
- Volume integral of Lagrangian gives M(2 - 1)
Entropy from Background Field:
- Entropy contributed by background field is zero
Euclidean-Schwarzschild Solution:
- Cannot find a time function with nonzero gradient everywhere
- Best option: choose imaginary time coordinate of Schwarzschild solution
- Action between surfaces at constant vanishes due to no matter fields and zero scalar curvature
- Surface term at infinity is still M(2 - 1)
- Corner term at horizon also equals M(2 - 1)
- Total action for region between 1 and 2 is M(2 - 1)
- When using action with 2 - 1 = 1, entropy would be zero
- Leaving out surface term on horizon reduces action by intrinsic gravitational entropy of black hole
Euler Number and Entropy:
- Topological invariant (Euler number) suggests entropy will remain in more fundamental theory
- Some particle physicists believe entropy will vanish when black hole reaches Planck length due to quantum general relativity breakdown
Thought Experiment with Black Holes:
- Pair creation in strong electric or magnetic fields can be described as tunneling through Euclidean space
- Electric field neutralized by electron-positron pair creation before significant probability of black hole pair creation
- Magnetically charged black holes could potentially be pair created without competition from ordinary particles due to their mass-to-charge ratio
- Ernst found a solution representing two magnetically charged black holes accelerating in magnetic field (figure 3.13)
- Analytic continuation to imaginary time gives picture similar to electron pair creation (figure 3.14)
- Temperature of magnetically charged black hole tends to zero as charge approaches mass in Planck units, so adjust parameters for equal temperatures.
Pair Creation of Black Holes in Magnetic Fields
- Weak magnetic fields and low acceleration allow matching periods between Euclidean and Lorentzian solutions (Figure 3.13)
- Pair creation process: Charged black hole tunnels through Euclidean region, emerging as pair of oppositely charged black holes that accelerate away from each other (Figures 3.14 & 3.15)
- Solutions are not asymptotically flat but can estimate local pair creation rate in magnetic field regions
- Black holes could move far apart, then radiate and lose mass without losing magnetic charge before annihilating via time reverse process (Figure 3.16)
- Delicate calculation required to ensure generalized second law of thermodynamics is not violated due to disappearing black hole horizon area
- Conclusion: Information can be lost in macroscopic black holes and virtual microscopic ones; energy/electric charge conservation still applies, but other information and global charges are not (Figure 3.17)
- Unitarity assumption in quantum theory challenged by potential loss of information through black hole appearance and disappearance
- End of scientific determinism as future cannot be predicted with certainty
Chapter Four: Quantum Theory and Spacetime
Physical Theories of the Twentieth Century:
- Quantum theory (QT): Accurate to about one part in 10^11
- Special relativity (SR): Built on, incorporated into general relativity (GR)
- General relativity (GR): Tested with remarkable accuracy to one part in 10^14
- Quantum field theory (QFT): Incomplete, has divergence problems that may be solved by an ultraviolet cutoff from GR
Problems of the Theories:
- QFT's problem: Divergences require renormalization to make calculations finite
- GR's problem: Predicts spacetime singularities (e.g., black holes)
- Measurement problem in quantum theory, which the speaker believes will be resolved by combining GR and QT
Information Loss in Black Holes:
- Information loss occurs when matter falls into a black hole or evaporates
- This is seen as complementary to the uncertainty principle in QT
- In the presence of a black hole, phase space trajectories converge and volumes shrink
- However, this loss is balanced by spontaneous quantum measurements where information is gained
Measurement Problem:
- The two-slit experiment illustrates the principles of quantum theory: interference, non-reachable points
Quantum Mechanics and Superposition Principle:
- Destructive interference: alternative possibilities can cancel out
- Understood through quantum theory's superposition principle
- Combination of z|A> + w|B> possible, where A and B represent different routes
- Complex numbers are not probabilities but denote superpositions at time t=0
- Unitary evolution (U) preserves superpositions: zA0+ wB0 evolves to zA0+ wB0 after a time t
- Measurement process (R): quantum alternatives magnified into distinguishable classical outcomes
- Probabilities only apply upon measurement
- U and R are different processes: deterministic vs. nondeterministic, local vs. nonlocal, time symmetric vs. time asymmetric
- Measurement problem: the relationship between U and R is unclear
Time Asymmetry in Quantum Mechanics:
- Rule R applies to future probabilities but not past ones
- Second law of thermodynamics relates measurement problem and singularity problem in GR
- Aharonov, Bergmann, and Liebowitz's scheme: time asymmetry comes from boundary conditions
Density Matrix:
- Represents our incomplete knowledge of a system's state
- Contains classical uncertainty as well as quantum probabilities
- Not necessarily orthogonal or normalized to the total dimension of Hilbert space
- Can be written as a weighted probability mixture of states (equation 4.1)
- Example: EPR experiment with two particles flying off in opposite directions and detected here and there
- Density matrix depends on the measurement performed on the moon (equations 4.2 and 4.3)
- Multiple ways to write a given density matrix as a probability mixture of states
- Recent theorem by Hughston, Jozsa, and Wootters shows that for any interpretation of the density matrix here, there exists a measurement there that gives rise precisely to this particular interpretation.
John Bell's FAPP (For All Practical Purposes) Approach to Quantum Mechanics
Process of Reduction of State Vector:
- Standard description: |tot> = w|up here>|?> + z|down here>|?>, where |?> describes environment states outside measurement
- If information is lost in environment, density matrix is best approach
- When information cannot be retrieved, consider system as either |up here> or |down here>, with probabilities w^2 and z^2 respectively
- Still need assumption to explain why density matrix does not provide full solution
Schrdinger's Cat Experiment:
- Superposition of cat states (dead/alive) and environment states (detector firing/not firing)
- Many-worlds view: w|dead cat>|know cat is dead> + z|live cat>|know cat is alive>
- Problem: why do we not perceive macroscopic superpositions?
- Proposed solution: something goes wrong with superpositions of alternative spacetime geometries in General Relativity (GR)
Objective Reduction (OR):
- Decay into one or other alternative, proposed name for this process
- Timescale of reduction depends on Planck length and mass of system
- Instability may occur when enough mass is moved to destabilize superposition
- Gravity provides answer as it affects causality with black holes and information loss
Cosmology: Quantum Cosmology
The Shift in Perception of Cosmology:
- Once considered a pseudoscience with no observational basis
- Now has improved observational data due to technological advancements
- Can no longer be dismissed as lacking an observational basis
Challenges and Limitations of Classical General Relativity:
- Requires assumptions about initial conditions to make predictions
- Raises questions about the laws of physics governing the universe's beginning
- Theorems by Hawking and others showed singularities in the past, beyond the reach of classical general relativity
- Breakdown of classical theory at the beginning of the universe could extend to any point
Path Integral for Quantum Gravity:
- The laws of physics must hold everywhere, including at the beginning of the universe
- Path integral should be taken over nonsingular Euclidean metrics
- Natural choices: 1. Asymptotically Euclidean metrics for scattering calculations; 2. Compact metrics without boundary for cosmology
Asymptotically Euclidean Metrics:
- Appropriate for scattering calculations, where measurements are made at infinity
- Allows small fluctuations to be interpreted as particles
- Path integral over all possible histories (asymptotically Euclidean metrics)
Compact Metrics Without Boundary:
- More natural choice for cosmology, as measurements are made in a finite region of the universe
- Avoids issues with disconnected asymptotically Euclidean metrics that could dominate probabilities
- Hartle and Hawking's "No-Boundary Proposal" (1983): Path integral for quantum gravity over all compact Euclidean metrics without boundary
The No-Boundary Proposal for the Universe
Concepts:
- Boundary condition of the universe: it has no boundary
- Three-dimensional manifold M with induced metric hij
- Wave function of the universe
- Momentum constraint equations
- Wheeler-DeWitt equation
The No-Boundary Proposal:
- Accounts for an isotropic and homogeneous expanding universe with small perturbations
- Agrees with observations of microwave background fluctuations
- Predictions will be tested by extending observations to smaller angular scales
Describing the State of the Universe:
- Probability of containing a three-dimensional manifold with induced metric hij
- Given by a path integral over all metrics gab on M that induce hij on
- Factorized into two wave functions: + and -, representing different parts of M
Wave Function of the Universe:
- Does not depend on time explicitly because there is no preferred time coordinate in a closed universe
- Given by a path integral over fields on a compact manifold M+ whose only boundary is
- Cannot depend on or coordinate choices, implying four functional differential equations: momentum constraints and Wheeler-DeWitt equation
Estimating the Wave Function of the Universe:
- Path integral can be approximated using saddle point solution
- Resulting Euclidean metric satisfies field equations and induces given three-metric hij on boundary
- Higher-order terms can be ignored when the radius of curvature is large compared to the Planck scale
Example: Wave Function for a Three-Sphere with Positive Cosmological Constant:
- Exponential increase in wave function for small radii of
- Oscillating wave function for larger radii, corresponding to an imaginary time Euclidean metric or real-time Lorentzian metric
Interpreting the Wave Function:
- De Sitter space as a closed universe that shrinks and expands exponentially
- The exponential wave function corresponds to an imaginary time Euclidean metric, while the oscillating wave function corresponds to a real-time Lorentzian metric
- The de Sitter universe is created "out of nothing" in the Euclidean regime, without requiring preexisting space
De Sitter Universe: Properties and Implications
Solution to Field Equations:
- For boundary metrics with radius less than critical, real Euclidean metric results
- Action is real
- Wave function exponentially damped compared to round three-sphere of the same volume
- Radius greater than critical: complex conjugate solutions and oscillating wave functions
de Sitter Universe's Thermal Properties:
- Resembles black hole properties
- Singularity removal through coordinate transformation
- Event horizon present, personal for each observer
- Static form reveals Euclidean metric (four-sphere) with regular coordinates
Entropy in de Sitter Space:
- Intrinsic entropy of Euclidean-de Sitter solution: quarter the area of event horizon
- Reflects an observer's lack of knowledge about universe beyond their horizon
Hot Big Bang Model vs. Reality:
- Universe contains matter and is not a pure de Sitter space
- Expanding and hotter in the past
- Microwave background and light elements observations
Challenges with Hot Big Bang Model:
- No predictive power due to singularity issue
- Universe's large-scale homogeneity and isotropy, local irregularities unexplained
- Fine-tuning problem: critical expansion rate required for life development
Inflation Theory:
- Attempts to address the need for a theory of initial conditions
- Universe could start in any state
- Regions suitable for life would inflate and expand homogeneously and isotropically, leaving the universe in its present state.
However, inflation alone cannot explain the present state of the universe completely as it does not provide an explanation for why certain regions inflated while others did not. It only increases the size of the region to a large scale but does not ensure that all parts would have the same initial conditions or expand at exactly the critical rate.
Early Universe Evolution and Scalar Fields
Initial Conditions at Big Bang:
- Arbitrary initial conditions at the big bang can lead to any state now
- Cannot argue that most initial states lead to a universe like ours, as both types have infinite natural measures
No-Boundary Condition:
- Can lead to a predictable universe within quantum theory limitations for gravity with cosmological constant but no matter fields
- However, this model does not describe the universe we live in (full of matter and has zero or very small cosmological constant)
Scalar Fields Model:
- Dropping cosmological constant and including scalar field with potential V()
- Potential V() acts like an effective cosmological constant if gradient is small
- Wave function depends on value of at time t and induced metric hij
- Solvable for small round three-sphere metrics and large values of
- Predicts spontaneous creation of exponentially expanding universe in this model as well as de Sitter case
Evolution of Scalar Field Model:
- Scalar field will roll down potential to minimum at = 0
- If initial value of is larger than Planck value, rate of roll down is slow compared to expansion timescale
- Universe expands almost exponentially by a large factor before oscillations start
- Oscillations lead to conversion of energy into pairs of other particles, which depends on arrow of time assumption
Homogeneity and Isotropy:
- Expanding three-metrics can be written as product of wave function for round three-sphere metric and harmonics wave functions
- Scalar harmonics correspond partly to coordinate freedom and partly to density perturbations
- Tensor harmonics, which correspond to gravitational waves, are simplest to consider and have no gauge degrees of freedom or interaction with matter perturbations
- Use no-boundary condition to solve for initial wave function of coefficients dn of tensor harmonics in perturbed metric
- Ground state wave function is found for a harmonic oscillator at frequency of gravitational waves
- When frequency falls below expansion rate during exponential expansion era, wave function freezes and corresponds to highly excited state rather than ground state when it froze
Inflationary Universe Phenomena
Gravitational Waves vs. Scalar Fluctuations:
- Gravitational waves: upper limit on energy density at time of freezing
- Scalar fluctuations: larger perturbations in microwave background
- One physical scalar degree of freedom (density perturbations)
- Amplitudes multiplied by factor of expansion rate and average change rate of potential slope
- At least ten times bigger than gravitational wave fluctuations
- Within range of approximations used
- Agrees with scale free spectrum of observations
- Explains structure of universe, including formation of galaxies and stars
Entropy and Observer's Perspective:
- Universe has entropy associated with cosmological event horizon
- Can be observed as classical behavior in part of the universe (mixed state)
- Summation over possibilities for unobserved parts
- Decoherence necessary for classical behavior
Arrow of Time:
- Distinction between forward and backward directions in our region of the universe
- Local laws are time symmetric or CPT invariant
- Observed difference comes from boundary conditions at each end of the universe
- Closed universe: collapses again before observer sees all of it
- Proposed solution: Weyl tensor vanishes at one end but not the other (Roger Penrose's proposal)
- Distinguishes two ends of time and explains arrow of time.
Weyl Tensor Hypothesis and Universe's Ends
- Weyl tensor plays an important role in distinguishing the two ends of the universe
- If Weyl tensor had been exactly zero, the early universe would have been homogeneous and isotropic, unable to explain fluctuations and perturbations leading to galaxies and ourselves
Objections to Weyl Tensor Hypothesis
- Not CPT invariant: The hypothesis doesn't respect charge conjugation symmetry, which is a fundamental principle in physics
- Weyl tensor cannot have been exactly zero; Small fluctuations remain unexplained by the hypothesis as it doesn't account for small deviations from perfect homogeneity and isotropy
Rogers Weyl Tensor Condition
- Implies perturbations are not exactly zero but as close to zero as possible, consistent with uncertainty principle
- Deduced from more fundamental principle: No-boundary proposal (NBP)
No-Boundary Proposal
- Perturbations in three-metric field around half Euclidean four-sphere joined to Lorentzian de Sitter solution are in their ground state
- Explains why Weyl tensor is small at one end and large at the other
Two Possible Solutions
- Approximately half of Euclidean four-sphere joined to a small part of Lorentzian region (smooth, ordered universe)
- Same half Euclidean four-sphere joined to expanding-contracting Lorentzian solution (irregular, chaotic universe)
Implications and Conclusion
- Perturbations are heavily damped in the first solution but large without significant dampening in the second one
- Universe was very smooth and small fluctuations at one end; large Weyl tensor at other end explains observed arrow of time
- No-boundary proposal predicts universe like we observe, even though quantum theory doesn't restore complete predictability due to event horizons.
The Twistor View of Spacetime: R. Penrose's Perspective
Classicality of Cats:
- Stephen argued that inaccessible regions do not explain the classical nature of observations in our region
- Density matrix alone does not distinguish between observing a live or dead cat and their superpositions
Weyl Curvature Hypothesis (WCH):
- Penrose agrees with Stephen's position, but argues that WCH is phenomenological rather than explanatory
- Initial singularities have approximately zero Weyl curvature, while final ones have large Weyl curvature
- Small perturbations are acceptable in the quantum regime, as long as they are constrained near to zero
Black Holes and White Holes:
- WCH applies to both initial (big bang) and final (black hole singularities and big crunch) state solutions
- NPB interpretation does not exclude the existence of white holes
- Euclideanization procedure may only apply for very few Minkowski spacetime sections
Twistor Theory:
- Utility of Euclideanization in QFT arises from the need to split field quantities into positive and negative frequency parts
- Twistor theory provides a framework to accomplish this splitting on spacetime itself
- Complex numbers, which underlie spacetime structure, can be represented on the Riemann sphere and mapped onto a plane
Riemann Sphere in Physics
Role of Riemann Sphere:
- Represents all complex numbers
- Wave function of a spin-j particle: linear superposition of up and down
- Point z/w on the sphere corresponds to where positive axis intersects the sphere
- Relates complex amplitudes to spacetime structure
Relation to Spacetime:
- Observer's angular position of stars mapped by Mbius transformation
- Related to Lorentz transformations, preserving Riemann sphere's complex structure
- Twistor Theory: regards light rays as more fundamental than space-time points
- Spacetime considered secondary, twistor space (projective space of light rays) is more fundamental
Twistor Space:
- Five real dimensions initially, but needs to take into account energy and helicity
- Complex projective three-space (six real dimensions), CP3
- Particles divided into left-handed and right-handed pieces based on twistor variables
Incidence Relation: (6.1) defines the relationship between a light ray in spacetime and its representation as a point in twistor space
Two-Spinor Notation:
- Represents momentum pa, angular momentum Mab of massless particle
- Determines twistor variables up to an overall phase multiplier
- Helicity s > 0: right-handed particles
- s < 0: left-handed particles
- s = 0: actual light rays
Quantum Theory of Twistors:
- Wave function f(Za) on twistor space is a holomorphic or antiholomorphic function
- Momentum and angular momentum expressions are independent of operator ordering, so canonically determined.
Twistor Theory for Massless Particles
Helicity and Homogeneity:
- Helicity depends on ordering, so we must take correct definition
- Can decompose wave function into eigenstates of helicity (s)
- Spinless particle with zero helicity has twistor wave function of homogeneity -2
- Left-handed spin-1 particle has helicity and twistor wave function with homogeneity -1
- Right-handed spin-1 particle (helicity ) has twistor wave function with homogeneity -3
- For spin 2, right- and left-handed twistor wave functions have respective homogeneities -6 and +2
- GR is left-right symmetric, but Nature may be left-right asymmetric
- Ashtekar new variables are also left-right asymmetric
Obtaining Spacetime Description:
- Express field amplitude in terms of contour integral over Za or
- Defines spacetime field that satisfies massless particle's field equations
- Holomorphicity constraint of twistor fields encodes all messy equation
Twistor Space and Positive/Negative Frequency:
- Twistor wave function extends to top or bottom half of twistor space for positive or negative frequency, respectively
- Allows for quantum physics in twistor space
Towards Quantum Field Theory (QFT):
- Hodges has developed QFT using twistor diagrams, similar to Feynman diagrams
- Uncovers novel regularization schemes natural to the twistor picture
Extensions to Curved Spaces:
- Conformally flat spacetimes can be described by twistors (conformally invariant)
- Twistor ideas work for conformally nonflat spacetimes, like quasi-local mass and stationary axi-symmetric vacuums
- Nonlinear graviton construction fully addresses curved anti-self-dual Weyl tensor spacetime
Nonlinear Graviton Construction and Twistor Theory
Background:
- Einstein vacuum equations (Ricci-flatness) encoded as condition for holomorphic fibration over CP1
- Challenge: solving full Einstein equations is difficult
- New approach using twistors, which may provide a relation to Einsteins equations
Observations:
- Vacuum Einstein equations are consistency conditions for massless fields with helicity s = 0 (when given by potential)
- In flat spacetime M, space of charges for s = field is exactly twistor space
Program:
- Find space of charges for Ricci-flat spacetime (difficult task)
- Use free holomorphic functions to construct this nonlinear twistor space
- Reconstruct original spacetime manifold from twistor space
Preferred Universe:
- In a k < 0 universe, symmetry group at singularity is Mbius group of holomorphic self-transformations (restricted Lorentz group)
- Preferred for "twistor ideological" reasons
Questions and Answers:
Question: What is the physical significance of helicity state?
- Auxiliary field in twistor theory, not a physical field
- Superpartner of graviton from the perspective of supersymmetry
Question: Where does time-asymmetrical R-process appear in twistor point of view?
- Twistor theory is conservative and doesn't explicitly discuss this yet
- May emerge when fully understanding GR in the twistor framework
Question: Which nonlinear QFT might be most amenable to twistor theory?
- Standard model has been analyzed using twistor diagrams so far
Question: String theory predicts particle spectrum; where does it appear in twistor theory?
- Connection between string theory and twistor theory not fully understood yet
- Understanding GR in the twistor framework may help solve this problem
Question: What is the twistor point of view on continuity/discontinuity?
- Early motivation for twistor theory was theory of spin networks (discrete rules)
- Trend towards holomorphic methods over combinatorial ones, but discrete concepts might be connected to holomorphic ones.
The Debate Between Hawking and Penrose
Hawking's Perspective:
- A positivist
- Believes that quantum theory predicts results of measurements successfully, without needing to correspond to reality
- Does not see the issue with Schrdinger's cat being in a superposition of dead and alive states as it cannot be tested
- Suggests that the decoherence of Schrdinger's cat is due to its size and inability to isolate from environmental forces, rather than quantum gravity
Penrose's Perspective:
- A Platonist
- Worried about the "absurd" nature of Schrdinger's cat being in a superposition
- Believes that quantum theory should correspond to reality
Comparison of Approaches:
- Hawking: Focuses on the predictive power of quantum theory, without demanding it correspond to reality
- Penrose: Wants a quantum theory that corresponds to reality
Misunderstandings:
- Hawking's use of Euclidean methods has been misunderstood by Penrose
- However, Hawking's approach is similar to how Yang-Mills path integrals are handled in physics
Path Integral Approach:
- Path integral can be taken over positive definite or Euclidean metrics for gravity, allowing for more interesting topologies
- Despite potential issues with the convergence of the path integral, this approach has made observable predictions
String Theory vs. Hawking's Program:
- String theory has not yet been able to describe the structure of objects like the Sun or black holes
- In contrast, Hawking's Euclidean quantum gravity program has made testable predictions
Observation and Measurement:
- Penrose believes that measurement through the R process introduces CPT violation into physics
Disagreements Between Hawking and Penrose
Perception of Reality:
- Hawking: believes there is more agreement than disagreement with Penrose
- Penrose: disagrees with Hawking on certain fundamental points, particularly the explanation for how we perceive reality within quantum mechanics (QM)
The Cat Problem:
- The problem lies not in the loss of information but in the equal density matrices before and after the measurement, which cannot be observed
- To explain how we perceive the world, a theory of experience or real physical behavior is required to account for this perception
Superpositions and Stability:
- In Penrose's picture, large-scale superpositions are unstable and must decay spontaneously into one or more stable states (either "live" or "dead")
Stephen Hawking's Views on Quantum Field Theory (QFT) and No-Boundary Proposal (NBP)
Wick Rotation:
- Useful tool in QFT to translate Minkowski space into Euclidean space
- Allows for better definition of certain expressions like path integrals
- Stephens proposal to apply Wick rotation to Lorentzian metrics is different from the standard application in QFT
Phase Space Loss and R-process:
- Stephen disagrees with Roger's claim that R-process is mere magic, not physics
- Believes he has explained why reduction of state should occur at a specific rate (I)
- Also disagrees with Roger about the loss of phase space in black holes
- Argues that quantum gravity theory must be time asymmetric to explain observational differences between past and future singularities
Schrdinger's Cat Experiment:
- Roger worries about ambiguity in density matrix eigenstates when probabilities are equal
- Stephen explains that this ambiguity only occurs when eigenvalues are exactly equal, otherwise a basis is distinguished by being eigenvectors of the density matrix
- Observes either alive or dead cat due to macroscopic differences, not superposition
NBP and Explanation of Weyl Tensor in Early Universe:
- Stephen believes NBP can explain both the smoothness of early universe and irregularity of gravitational collapse
- Explains that the exact saddlepoint metric will be complex, but can be approximated by a Lorentzian region with nearly Euclidean action and a phase-determining Lorentzian part
- All three-geometries are equally probable in gravitational collapse, leading to varied metrics with high Weyl curvature
Black Hole in a Box Experiment:
- Stephen argues that the whole point of black hole thermodynamics was to avoid phase space loss by allowing multiple configurations to form the same black hole.
Stephen Hawking's Perspective:
- Agrees with Roger Penrose that black and white holes are the same for an outside observer in quantum theory
- Believes in gravitational collapse but not collapse of wave function
- Claims that CPT symmetry can be made natural by considering different saddle point metrics for various questions
- Disagrees with Penrose on reality, perceiving macroscopic level as one spacetime
- Sees himself as a positivist and Penrose as a realist
Roger Penrose's Response:
- Addresses cat problem by stating that eigenvalue equality is irrelevant for understanding density matrices
- Argues that QM may need modification for very macroscopic objects
- Believes in the existence of a real world beyond wave function interpretation
- Sees himself as a realist, contrasting with Stephen's positivism
- Compares their debate to Bohr vs. Einstein debate on reality and quantum theory
Gary Horowitz's Input:
- String theory reduces to GR in weak field limit, implying everything GR implies
- Might provide better understanding of singularity behavior
- Still facing challenges but seen as a promising route for understanding quantum gravity
Miscellaneous Topics Discussed:
- Decoherent histories approach and its relationship with Penrose's method
- Entropy in black hole thought experiment not violating second law of thermodynamics due to maximal entropy state
- Possibility of testing quantum measurement mechanism through Leggett-type experiments.
Debate Between Stephen W. Hawking and Roger Penrose
Observational Developments:
- Agreement about the most exciting and important observational development: accelerating expansion of the universe, with evidence for positive cosmological constant or dark energy
- Uncertainty on the nature of this new ingredient (cosmological constant vs. dark energy)
- Support for a flat or slightly positively curved universe, in line with Hartle-Hawking no-boundary proposal and inflationary ideas
Theoretical Developments:
- Inflationary cosmology becoming more part of the standard picture
- Observational support from scale invariance and de Sitter-like early universe structure
- Planck satellite to provide further critical information, particularly on primordial gravitational waves
Roger Penrose's Perspective:
- Skeptical about the extraordinary uniformity of the very early universe and its low entropy
- Introduced Weyl curvature hypothesis (WCH) to explain this, based on conformal geometry and a cyclic universe model (CCC) with no inflation in the very early universe
Stephen Hawking's Perspective:
- Previously proposed that information would be lost during black-hole evaporation, but has since changed his viewpoint
- Now proposes a more complete resolution using ADS-CFT correspondence to regain the lost information
Key Differences:
- Fundamental disagreement on whether standard quantum mechanics rules will remain inviolable in the context of general relativity or if new foundations are needed for quantum gravity
- Strongest disagreement on black-hole information loss paradox and its implications for unitarity.