A discourse on time, gravity, and the universe that takes the reader through the subtleties of time, the origin of the universe, and physical evolution in Einstein's theory and its extensions. Can time and causality remain fundamental when the classical ideal of spacetime becomes a concept of limited applicability in quantum gravity? A thorough exposition on the canonical framework of Einstein's theory and its extensions reveals the synergy between gravitation and the cosmic clock of our expanding universe that renders time concrete, physical, and comprehensible. In conjunction with a paradigm shift from four-covariance to just spatial diffeomorphism invariance, causal time-ordering of the quantum state of the universe and its evolution in cosmic time become meaningful. The quantum state of the universe is the embodiment of our shared past, present, and future. The advocated framework prompts natural extensions and improvements to Einstein's theory. A salient feature is the addition of a Cotton-York term to the physical Hamiltonian. Besides bringing improved ultraviolet convergence, this radically changes the solution to the initial data problem and the quantum origin of the universe. It lends support to the quantum beginning of the universe as an exact Chern-Simons Hartle-Hawking state that features Euclidean-Lorentzian instanton tunneling. A signature of this state is that it manifests, at the lowest order approximation, scale-invariant two-point correlation function for transverse-traceless quantum metric fluctuations. This initial quantum state also implies, at the level of expectation values, a low-entropy hot smooth Robertson-Walker beginning that is in accord with Penrose's Weyl Curvature Hypothesis. Consequently, the gravitational arrow of time of increasing spatial volume and the thermodynamic second law arrow of time of increasing entropy concur as our universe expands and ages.
Author(s): Chopin Soo, Hoi-Lai Yu
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 277
City: Singapore
Contents
Preface
List of Figures
List of Tables
1. Gravitation and quantum mechanics: A match made in heaven
1.1 Introduction
1.1.1 Newton, the apple, and the moon
1.1.2 Einstein's Nobel prize
1.1.3 Why quantize gravity?
1.1.4 Why geometrodynamics?
1.1.5 Dirac's clarion call to abandon four-covariance
1.1.6 Quantum gravity and the cosmic clock
1.2 Problem(s) of time
1.3 Pauli's theorem on time, and the cosmic clock of ITG
1.3.1 Mandelstam-Tamm time-energy uncertainty relation
1.4 Dirac's derivation of the Schrödinger equation from an extended phase space
1.4.1 Parallels with ITG
1.5 From many-fingered to gauge invariant global cosmic time
1.6 Horava gravity theories: problems and promise
1.6.1 Conict between renormalizability and unitary in four-covariant theories of gravitation
1.6.2 Problems with Horava gravity
1.6.3 ITG formulation of Horava gravity: new vista to address the challenges of classical and quantum gravity
1.7 Gravitation and local gauge invariance
1.7.1 Canonical analysis of Einstein-Hilbert action
1.7.2 Counting the d.o.f. in canonical theory of gravitation
1.7.3 Spatial diffeomorphisms as local gauge symmetries
1.7.4 Changes associated with the Hamiltonian constraint
1.7.5 Local gauge symmetry generators are first-order in canonical momentum
1.7.6 General Relativity possesses only spatial diffeomorphism invariance
1.7.7 Physical understanding of spatial diffeomorphism invariance in gravitation
1.7.8 Geometrodynamics, and gauging the Lorentz group
1.7.9 SL(2, C) Weyl fermions and the Lorentz group as SO(3, C)
1.7.10 Physical Hamiltonian for Einstein's theory and its extensions
1.7.11 Preferred clock and the foliation of classical spacetimes
1.7.12 Consistent Horava completion of Einstein's theory
1.8 ITG and quantum gravity as Schrödinger-Heisenberg quantum mechanics
1.8.1 Circumventing the difficulties of the Wheeler-DeWitt equation
1.8.2 The cosmic clock and Schrödinger-Heisenberg quantum mechanics
1.8.3 Time-dependent Hamiltonian of ITG
1.9 ITG and the origin of the universe
2. General Relativity without paradigm of space-time covariance; and resolution of the problem of time
2.1 Conceptual and technical challenges of the paradigm of four-covariance
2.1.1 Paradigm shift from four-covariance
2.2 Theory of gravity without paradigm of four-covariance
2.3 Geometrodynamics, symplectic potential, and the Hamiltonian constraint
2.3.1 Symplectic potential, and decomposition of variables
2.3.2 Hamiltonian constraint; and its factorization
2.4 Master constraint as a provisional formalism in disentangling the dual roles of the Hamiltonian constraint
2.4.1 First-order in intrinsic time Wheeler-DeWitt equation
2.5 Emergence of classical spacetime
2.5.1 Constructive interference
2.5.2 Recovery of the classical EOM from constructive interference
2.5.3 Emergent classical ADM spacetime; time evolution in Einstein's theory is physical
2.6 Paradigm shift and resolution of the problem of time
2.6.1 Only spatial diffeomorphism invariance is needed and realized
2.6.2 Four-dimensional space-time covariance in GR is a red herring
2.6.3 Classical proper time and its relation to intrinsic time interval
2.7 Modifications of the Hamiltonian, and improvements to the quantum theory
2.7.1 Chern-Simons term, power-counting renormalizability; and positive-definite Hamiltonian
2.8 Gauge-invariant global time, superspace dynamics, and temporal order
2.8.1 Many-fingered time, Hodge decomposition, and invariant global time interval
2.8.2 Emergent classical spacetime metric, and its relation to intrinsic time and the Hamiltonian density of ITG
2.8.3 Diffeomorphism-invariant time-ordered evolution of physical quantum state of the universe
2.9 Further discussions
2.9.1 Observables, two d.o.f., and the Hamilton-Jacobi equation
2.9.2 Dreibein and the Lorentz group
2.9.3 Generalized Baierlein-Sharp-Wheeler action of the theory
2.9.4 York extrinsic time
2.10 ITG and the dynamics of test particles
2.11 Dynamical basis of relativistic time effects in generic ADM spacetimes
2.11.1 A general formula for relativistic time effects
2.11.1.1 Proper time interval
2.11.1.2 Examples of time dilation effects
2.12 Relativistic point particle dynamics, analogy to Einstein's theory, and master constraint formulation
3. Quantum geometrodynamics with intrinsic time development
3.1 Geometrodynamics and the riddle of time
3.1.1 Geometrodynamics reexplored
3.2 General Relativity and the lapse function a priori and a posteriori
3.2.1 A posteriori lapse function determined by constraints and EOM
3.2.2 Reproducing General Relativity with an equivalent ADM Hamiltonian instead of a constraint
3.3 Global intrinsic time and quantum evolution
3.3.1 Expansion of the universe and global intrinsic time
3.3.2 Temporal ordering and quantum evolution
4. Intrinsic time gravity and the Lichnerowicz-York equation (Dedicated to the memory of Niall O Murchadha)
4.1 Introduction
4.2 Einstein's theory and initial data
4.3 Intrinsic time gravity made simple
4.3.1 Simple relativistic particle analogy
4.3.2 Reduced Hamiltonian and the lapse function
4.4 Moving away from General Relativity
4.4.1 Constant mean curvature slice
4.4.2 Solving the Lichnerowicz-York equation
4.5 An exceptional extension of Lichnerowicz-York equation: addition of Cotton-York term
4.6 Intrinsic time versus extrinsic time
4.7 Penrose Weyl Curvature Hypothesis and Lichnerowicz-York equation with Cotton-York term
4.7.1 Asymptotic behavior of the Hamiltonian at early and late intrinsic times
5. New commutation relations for quantum gravity
5.1 Motivation for momentric variables
5.2 New commutation relations
5.2.1 The momentric variable and commutation relations
5.3 Self-adjoint momentric operators and pointwise sl(3, R) algebra
5.3.1 Relation to su(3) algebra
5.3.2 Fundamental commutation relations of momentric and dreibien
5.4 Generator of local Lorentz symmetry
5.5 The kinetic operator, the free theory, and CSCO for momentric variables
5.5.1 SL(3, R) symmetry generated by momentric versus diffeomorphism symmetry generated by momentum constraint
5.5.2 Why is the universe intuitively "metric", and not "conjugately realized"?
5.5.3 Two gravitational degrees of freedom
6. Gravitational waves in Intrinsic Time Geometrodynamics
6.1 Introductory remarks
6.2 Gravitational waves in our spatially closed expanding universe
6.3 Einstein's General Relativity as a special case of a wider class of "non-projectable" Horava gravity theories
6.4 Hamilton's equations for ITG
6.5 Gravitational waves in de Sitter background spacetime
6.5.1 Background solution and linearization
6.5.2 Gravitational wave equation
6.5.3 Higher curvature Cotton-York contribution
6.6 S3 tensor harmonics and physical excitations
6.7 Time development of transverse traceless modes
6.7.1 Einstein's theory and gravitational waves in expanding closed de Sitter universe
6.8 Gravitational waves in the Cotton-York era
6.8.1 Summary on time dependence of gravitational waves in expanding k = +1 de Sitter universe
6.9 Energy of gravitational perturbations, and further remarks
7. Intrinsic time gravity, heat kernel regularization, and emergence of Einstein's theory
7.1 Overview and motivations
7.2 Hamiltonian of ITG
7.3 Momentric, and self-adjoint positive-definite Hamiltonian density
7.4 Introducing interactions with similarity transformations
7.5 Analogy between Yang-Mills magnetic field and the Cotton-York tensor
7.6 Regulating the quantum commutator
7.7 Heat kernel regularization, and emergence of Einstein's theory
7.8 Renormalising the coupling constants
7.8.1 Newtonian limit fixes the renormalised coupling phenomenologically
7.8.2 Positive bare cosmological constant term to counter negative fermionic vacuum energy
7.9 Further remarks
7.9.1 Simple harmonic oscillator analogy
7.9.2 Implication of additional Cotton-York term on the initial state
8. Cosmic time and reduced phase space of General Relativity
8.1 An ever-expanding spatially closed universe needs no, and has no, external clock
8.2 Framing the physical contents of GR: Dirac first class versus second class constrained systems
8.2.1 Reduced phase space of GR
8.2.2 Recap on second class systems and the Dirac bracket
8.2.3 Dirac bracket, reduced phase space; and observables
8.3 Supplementary condition in an ever-expanding universe
8.4 Second class constraints and canonical phase space functional integral of GR
8.4.1 Maskawa-Nakajima theorems, Dirac's criterion for second class, Faddeev-Popov determinants, and reduced variables
8.4.2 Addressing spatial di eomorphisms
8.4.3 Note on the physical auxiliary or "gauge-fixing" conditions
8.4.4 Solution of the diffeomorphism constraint
8.4.5 Reduced symplectic potential
8.5 Physical Hamiltonian, and generalization beyond Einstein's theory
8.5.1 Ultraviolet-complete extension of Einstein's theory and the Yang-Mills analogy
8.6 Comparison of extrinsic, intrinsic, and scalar field time
8.6.1 Extrinsic time with Hamilton implicitly dependent upon solution of Lichnerowicz-York equation, versus intrinsic time with explicit Hamiltonian
8.6.2 Scalar field time and problematic negative kinetic term
8.7 Incorporation of matter and Yang-Mills fields and solution of diffeomorphism constraint
9. Chiral fermions, gauging the Lorentz group without four-dimensional spacetime; and solving all the constraints
9.1 Are fermions fundamental when spacetime is not?
9.1.1 Inclusion of fermionic fields and solution of all the constraints: A brief overview
9.2 Weyl fermions, and gauging the Lorentz group: (3 + 1)-decomposition
9.3 Gauging local Lorentz symmetry without four-dimensional spacetime
9.3.1 SO(3, C) and the unimodular spatial metric
9.3.2 Fermions, spatial diffeomorphisms, and fundamental anti-commutation relations
9.3.3 Diffeomorphism-invariant fermion measure and absence of anomalies
9.3.4 Perturbative and global gauge anomalies
9.3.5 Diffeomorphism-invariant measures of other elds
9.3.6 Total diffeomorphism and Lorentz Gauss Law constraints
9.4 Solution of diffeomorphism and Lorentz Gauss Law constraints
9.4.1 Auxiliary conditions for spatial dreibein
9.5 A general scheme to solve first class gauge constraints exactly, and to promote to a Dirac second class system
9.5.1 Electrodynamics as an example
9.5.2 Relation to Dirac's second class system of constraints
9.5.3 Auxiliary conditions and the shift function
9.6 Solving the total Hamiltonian constraint with fermion, Yang-Mills and scalar fields
9.6.1 Torsion
9.7 Generalized spin structure; GUT, sterile neutrino, and cold dark matter
10. Initial state of the universe
10.1 The dog that did not bark: Weyl curvature and Einstein's theory
10.1.1 Bekenstein-Hawking black hole entropy
10.1.2 Penrose's Weyl Curvature Hypothesis
10.2 Quantum origin of our universe from a conuence of circumstances in ITG
10.2.1 Euclidean-Lorentzian tunneling
10.2.2 A hot beginning
10.3 Gravitational Chern-Simons functional and the Cotton-York tensor
10.3.1 One-to-one correspondence of Cotton-York and transverse-traceless metric excitations
10.3.2 Conformal atness and the Cotton-York tensor
10.3.3 On conformal flatness, Poincaré Conjecture and the three-sphere
10.3.4 Standard three-sphere and k = +1 Robert son-Walker metric
10.3.5 Chern-Simons functional and the Pontryagin invariant
10.4 How did the universe begin?
10.4.1 The Hartle-Hawking no-boundary proposal
10.4.2 Chern-Simons functional and the Hartle-Hawking state
10.4.3 Relative Chern-Simons functional
10.4.4 Chern-Simons state and the superficial degree of divergence of associated Feynman graphs
10.5 Quantum metric uctuations and correlation functions of the Chern-Simons Hartle-Hawking state
Bibliography
