Ph 236								Kip Thorne
General Relativity						28 Sept. 1998
1998-99

                         Tentative Course outline                         

FIRST TERM 

   I. Foundations of and Formulation of General Relativity

      A. Special relativity [Week 1] 
         1. Principle of Relativity
         2. Flat (Minkowski) spacetime of special relativity 
            a. spacetime diagrams
            b. Lorentz transformations
         3. Deduce how physical rods and clocks behave 
            a. when at rest in lab frame
            b. when accelerated
         4. Quick overview of general relativity
  
      B. Differential Geometry in Flat Spacetime but in Arbitrary Bases 
         [Week 2]

      C. Physics in Flat Spacetime [Week 3]
         1. EM Theory
         2. Volume Elements, Integration, Gauss's Theorem, 
         3. Conservatons Laws
         4. Stress-Energy Tensor 

      D. Differential Geometry in Curved Spacetime [Week4]
         1. Connection Coefficients
	 2. Geodesics
	 3. Local Lorentz Frames
	 4. Curvilinear Coordinates
	 5. Coordinate vs non-coordinate bases, commutators
         6. Geodesic deviation, Riemann curvature tensor

      E. Einstein Field Equations, Equivalence Principle, and
         Principle of Relativity 

      F. Some important exact solutions of the Einstein Field Equations 
         (get them on the table so they can then be used as tools in
         exploring other issues)
	 [Week 6]
         1. Schwarzschild solution outside horizon
         2. Static, spherical, relativistic star
       	    a. embedding diagrams
         3. Kerr metric outside horizon
         4. Spatially Flat Friedman universes
         5. Gravitational Plane Wave


  II. Physics in Curved Spacetime

      A. Proper Reference Frame of Accelerated, Rotating Observers
         [Week 7]
	 1. Construction of reference frame
	 2. Spacetime metric
	 3. Geodesic motion

      B. Conservation Laws, Spacetime Symmetries, and Thought Experiments
         [Week 8]
	 1. Asymptotic structure of spacetime outside an isolated, 
	    gravitating system [asserted, not yet proved]
         2. Conservation of system's mass, energy, angular momentum
	    [asserted; not proved]
         2. Spacetime symmetries, Killing vectors. 
	 3. Geodesic motion: conservation laws associated with symmetries
	 4. Thought experiments for isolated gravitating systems 

      C. The Physics of Matter and Radiation in Curved Spacetime [Week 9]
         1. Thermodynamics
	 2. Fluid mechanics with viscosity and diffusive heat flow
	    -- Bernoulli equation
         3. Geometric optics
         4. Kinetic theory
   
      D. Physics Around Black Holes [Week 10]
         1. Adiabatic, spherical accretion onto a black hole
	 2. Accretion disk around a spinning black hole

SECOND AND THIRD TERMS 

   III. Perturbation Theory, Post-Newtonian Approximation, Gravitational
        Waves, and Experimental Tests of General Relativity

      A. Linearized Perturbations of Flat Spacetime; 
         Gravitational Waves in Flat Spacetime 
	 1. "Linearized Theory"
         2. Newtownian Approximation
         3. Gravitational waves at linear order
         4. Gravitational-wave detectors: LIGO, LISA, and bars 
         5. How LIGO works; analysis of LIGO in different gauges;
            what is gauge invariant and what depends on gauge?

      B. Linear and Quadratic Perturbations of Curved Spacetime;
         Gravitational Waves Propagating Through Curved Spacetime
 	 1. General equations for perturbations of curved spacetime
	 2. Short-wave approximation 
	    a. gravitational waves
	    b. geometric optics approximation for gravitational waves
	    c. propagation of GW's from source to earth
               -- redshifts, ray deflection, gravitational focusing
         3. Quadratic-order perturbations
	    a. Isaacson stress tensor for GW's

      C. Asymptotic Structure of Spacetime Around an Isolated Body;
         Gravitational-wave emission; Radiation reaction 
         1. Concept of an isolated body in asymptotically flat spacetime
         2. STF formalism for spherical harmonics and solutions of 
	    flat-spacetime wave equations
         3. Multipole moments of an isolated body
         3. The general solution of the linearized Einstein equation outside
	    isolated body, using STF formalism
	    a. Gravitational waves in radiation zone
	 4. Conservation Laws for energy, momentum, and angular momentum 
	    of a radiating, isolated source
         5. Matching near-zone field to radiation field
         6. Radiation-reaction potential and radiation reaction
         7. Quadrupole-moment formalism for gravitational-wave generation
         8. Examples: binary stars, pulsating stars.

      D. Tidal Interaction of a Body with External Universe
         1. Buffer zone and local asymptotic rest frame of body
         2. Metric in local asymptotic rest frame
         3. Coupling of body's multipole moments to external curvature
            -- general precession of Earth's spin axis 
            -- tidal heating of Io and of black holes
            -- laws of motion and precession

      E. General Relativity as a Nonlinear Field Theory in Flat Spacetime
         1. Landau-Lifshitz formalism
         2. The curved spacetime paradigm contrasted with the nonlinear
	    field theory paradigm.
	 3. Derivation of Conservation Laws for energy, momentum, and 
	    angular momentum of a radiating, isolated source
         4. The problem of localization of energy in general relativity,
	    and global conservation laws for energy and momentum

      F. Post-Newtonian approximation to General Relativity, Equations of
         Motion, and Experimental Tests
   	 1. General concepts of post-Newtonian expansion
         2. Einstein Infeld Hoffman (EIH) post-Newtonian equations of motion 
            for compact bodies
	    --derivation using nonlinear field theory techniques
            --effects of spin-curvature coupling
            --gravitational-wave emission
         4. Other relativistic theories of gravity
	 5. Parametrized Post-Newtonian (PPN) formalism
         6. Summary of solar-system and binary-pulsar tests of general
	    relativity in the post-Newtonian domain

      G. Perturbations of Black Holes;  Black-hole Physics 
         1. Scalar waves propagating in Schwarzschild geometry
         2. Gravitational perturbations of Schwarzschild geometry
	    a. Regge-Wheeler spherical harmonics
	    b. Gravitational waves at infinity and horizon
	    c. Energy conservation for perturbations
	    d. Proof of stability of Schwarzschild black hole
	    e. No-hair theorem
	    f. Normal modes of Schwarzschild black hole
	    g. Tails of gravitational waves
         3. The Rindler Approximation
         4. Perturbations of Spinning (Kerr) Black Holes
	    a. Evolution of Horizon When Stuff Falls into it
	       --- First law of black hole mechanics
            b. Negative energy particles and perturbations near horizon
	       --- Extraction of rotational energy from a spinning hole


  IV. Causal Structure of Spacetime: Specific Examples Illustrating 
      Horizons, Singularities, Closed Timelike Curves

      A. Schwarzschild Spacetime and its Causal Structure
         1. Eddington-Finkelstein coordinates
	 2. Kruskal coordinates and causal structures in terms of them:
	    a. Event horizons
	    b. Singularities
	    c. Wormhole and its evolution
	    e. Relation to gravitational collapse, black holes, and
	       white holes
	    f. instability of past event horizon
	       -- conversion of white holes into black holes
	 3. Conformal transformations
	 4. Conformal transformation of infinity
         5. Penrose diagram of causal structure

      B. Reissner-Nordstrom Solution and its Causal Structure
         1. The metric and electric field in Schwarzschild-like coordinates
	 2. Transformation to Penrose-type coordinates
	 3. Penrose diagram for Q<M (subcritical)
	    a. Event horizons, Cauchy horizons, singularities
	    b. Evolution of spatial geometry: wormhole -- closed universe
	       -- wormhole
            c. Instability of Cauchy horizon
         4. Penrose diagram for Q=M (critical)
	 5. Penrose diagram for Q>M (supercritical): naked singularity
	 6. Impossibility of pushing a Q<M black hole to
	    supercriticality by throwing in charged particles

      D. Kerr Spacetime and its Causal Structure
         1. Supercritical Kerr: a>M
	    a. Naked ring singularity
	    b. Wormhole connection to negative mass space
	    c. Closed timelike curves
         2. Penrose diagram for subcritical Kerr: a<M
	    -- Event horizons, Cauchy horizons, Chronology horizons, 
	       Chronal and Dischronal regions, Closed timelike curves, 
	       Naked singularities
         3. Impossibility of pushing a subcritical spacetime to
	    supercriticality
         4. Collapse of a spinning star to form a Kerr black hole
	    a. Instability of Cauchy horizon

      E. Misner Spacetimes and its Causal Structures
         1. Misner spacetime constructed from Minkowskii
	 2. Chronal and Dischronal Regions; Chronology horizon
	 3. Covering space
	 4. Instability of horizons 

      F. Wormhole Spacetimes and their Causal Structures
	 1. Construction by removing world tubes from Minkowski
	    spacetime and identifying
         2. Chronal and Dischronal Regions; Chronology horizon
	 3. Stability of chronology horizon

      G. Cosmological Spacetimes and their Causal and Singularity Structures
         1. Friedman Cosmologies
            a. Open, closed, and flat homogeneous hypersurfaces
            b. Dynamical evolution
	    c. Causal structure
	    d. Standard cosmological model for our universe
         2. DeSitter Spacetime
	 3. Inflationary cosmologies
	    a. Causal structure; evolution of horizons
            b. Phase transitions
         4. Evolution of scalar and tensor perturbations
	    a. Parameteric amplification of vacuum fluctuations
	    b. Anisotropy of cosmic microwave background
	    c. Primordial gravitational wave background

  IV. Causal Structure of Spacetime: General Theory and Applications
      
      A. General Theory
         1. Basic concepts of topology
	 2. Futures and pasts
	 2. Causality conditions
	 3. Domains of dependence, Cauchy surfaces, global hyperbolicity,
            Cauchy horizons, chronology horizons

      B. Applicaton: Black-Hole Event Horizons
	 1. General structure 
	    a. null generators
	    b. caustics
	 2. Proof of Second Law of Black-Hole Mechanics (horizon area increase) 

      C. Application: Chronology Horizons (time machines)
	 1. General structure
	    a. null generators
	    b. polarized hypersurfaces
	 2. Compactly generated vs. non-compactly generated chronology horizons
	 3. General structure of compactly generated chronology horizons
	    a. generators
	    b. fountains
	    c. polarized hypersurfaces
         4. Examples 
         5. Chronology protection mechanism: preview 
	    [proof will be sketched later, in quantum field theory section] 

      E. Application: Singularity theorems
	 1. Timelike and null geodesics and their action principles
	 2. The focusing theorem 
	 3. Conjugate points
	 4. Maximum length curves
	 5. Proofs of some simple singularity theorems
	 6. Discussion of the most powerful singularity theorems
	    a. Singularities at the beginning of the universe
	    b. Singularities inside black holes


   V. Structures of Singularites

      A. Some Non-Generic Examples 
	 1. Friedman
	 2. Schwarzschild
	 3. Kasner
	 4. Mixmaster
	 5. Mass-Inflation

      B. Generic Singularities
	 1. BKL
	 2. Generalized mass-inflation

      C. The singularity at the beginning of the universe: 
	 our best understanding

      D. Singularities inside black holes: 
	 our best understanding

      E. Cosmic censorship and naked singularities
	 1. Weak vs strong cosmic censorship
	 2. Examples that seem to violate cosmic censorship
	 3. Current status of the cosmic censorship conjecture


  VI. The Dynamics of Spacetime Geometry

      A. 3+1 split of spacetime and of Einstein's equations
  	 1. Intrinsic and extrinsic curvature of spacelike
	    hypersurfaces
         2. Initial value equations for spacetime geometry 
	 3. Dynamical equations for evolution of spacetime geometry
         4. The Cauchy problem
         5. Junction Conditions and their applications to:
            a. Oppenheimer-Snyder solution for collapsing star 
            b. Collapse of a thin shell; teleological evolution of a
                black-hole horizon 
            c. White holes and their transformation into black holes

      B. Numerical relativity 
	 1. Choices of lapse and shift
	 2. Methods of solving the initial value equations
	 3. Methods of solving the dynamical equations
	 4. Applications:
	    a. Collapse of a spinning star
	    b. Collapse of axisymmetric star clusters
            c. Critical behavior and scaling in gravitational collapse
	    d. Black-hole head-on collisions
	    e. Coalescence of binary black holes

      C. First-Order, Symmetric and Hyperbolic (FOSH) Formulations of the 
         Einstein Equations
	 1. Eardley / Van Putten
	 2. Choquet-Bruhat / York
	 3. Possible advantages and applications of these new formulations
   
   
 VII. Quantum Field Theory of a Massless Scalar Field in Curved Spacetime
      (very elementary approach and a few key results)

      A. Elementary theory of quantum fields in flat spacetime
  	 1. Quantization of simple harmonic oscillator
	 2. Quantization of a violin string
	 3. Quantization of a massless scalar field in flat spacetime
	    a. Modes 
	    b. Vacuum state 
	    c. occupation numbers
	    d. Creation and annihilation operators 

      B. Particle production in inflationary cosmological model
	 1. Review of classical parametric amplification of scalar waves
	 2. Particle production derived
         3. Particle production viewed as parametric amplification of vacuum
	    fluctuations

      C. Black-Hole atmosphere and evaporation
	 1. Hawking's original derivation of black-hole evaporation 
	 2. Acceleration radiation in flat spacetime
	 3. Rindler approximation; black-hole atmosphere
	    a. Hawking-Hartle and Unruh states
	 4. Black-hole thermodynamics

      D. Renormalization of scalar-field stress-energy tensor
	 1. Brief overview of general theory
         2. Stress-energy tensor in Rindler spacetime and its
            renormalization
	 2. Stress-energy tensor in the thermal atmosphere of an 
            evaporating black hole
            a. Evolution of the hole's horizon
         3. Stress-energy tensor near a chronology horizon
	    a. Divergence at polarized hypersurfaces
	    b. Divergence at horizon
	    c. Chronology protection