Ph 236 Lee Lindblom General Relativity 2000-01 Tentative Course outline FIRST TERM [tentative] 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. 3+1 split of spacetime into space plus time 2. EM Theory 3. Volume Elements, Integration, Gauss's Theorem, 4. Stress-Energy Tensor; Perfect fluids 5. Conservation laws for charge and for 4-momentum 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 [Week5] 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. Static, spherical, relativistic star a. embedding diagrams 2. Schwarzschild solution outside horizon 3. Slowly rotating star a. dragging of inertial frames 4. Kerr metric outside horizon 5. Spatially Flat Friedman universes 6. Gravitational Plane Wave II. Physics in Curved Spacetime A. Proper Reference Frame of Accelerated, Rotating Observers [Week 7] 1. Local Lorentz frame 2. Decomposition of Riemann into Ricci plus Weyl 3. Construction of observer's proper reference frame; spacetime metric a. Special case: uniformly accelerated frame in flat spacetime 4. Geodesic motion in proper reference frame B. Spacetime Symmetries and Conservation Laws [Week 8, 1st lecture] 1. Killing's equation and spacetime symmetries 2. Conservation laws associated with symmetries: a. conservation laws for geodesic motion b. examples: motion in a plane-wave spacetime, gravitational redshift from a star C. Linearized and Newtonian Approximations to General Relativity [Week 8, 2nd lecture; Week 9, 1st lecture] 1. Philosophy of formalism: gravity as a field theory in flat spacetime 2. Basic equations of linearized theory a. self-inconsistency of the theory b. post-Linear theory 3. Newtonian approximation; its relationship to linearized theory 4. Frame dragging in linearized theory D. Multipole Moments of an Isolated System [Week 9] 1. STF tensors and spherical harmonics 2. Mass moments and current moments for stationary (time-independent) systems with weak internal gravity 3. Weak external field of an arbitrary, isolated body expressed in terms of dynamically evolving multipole moments. 4. Global conservation laws for mass, momentum, and angular momentum of an isolated body. a. Relationship of conservation laws to asymptotic symmetries of spacetime. b. Loss of these conservation laws in generic spacetimes with no symmetries. E. Thermodynamics and Fluid Dynamics for Perfect Fluids in Curved Spacetime [Week 10] 1. Basic equations of thermodynamics and fluid mechanics 2. Fluid conservation law (both local and global) associated with a Killing vector field 3. Bernoulli's equation in a stationary curved spacetime 4. Thought experiment: Injection energy for a star, and Schwarzschild criterion for convection SECOND TERM [Tentative] F. Viscous Fluid Mechanics and Magnetohydrodynamics [Week 11] 1. Decomposition of gradient of fluid 4-velocity into 4-acceleration, expansion, rotation, and shear a. Irreducible tensorial parts of a tensor; relation to theory of group representations 2. Viscous stress-energy tensor; heat-flow contribution to stress-energy tensor; entropy 4-vector 4. Equation of diffusive heat flow 6. Navier-Stokes equation 7. Energy conservation: Law of entropy increase 8. Application: The structure of a thin accretion disk around a black hole 9. Magnetohydrodynamics in general relativity 10. Application: Alfven waves G. Kinetic Theory [Week 12, first part] 1. Distribution function; its Lorentz invariance 2. Liouville's theorem; Vlasov equation - Application: relativistic star clusters 3. Boltzman transport equation - Application: derivation of equation of diffusive heat transport III. Black Holes and Wormholes A. Orbits Around a Schwarzschild Black hole [Week 12, second part] 1. Orbits for particle with finite rest mass 2. Orbits for photons 3. The radial fall of a particle into a black hole 4. Observation of an infalling particle by a distant observer B. Schwarzschild Black Holes and Wormholes [Week 13] 1. Metric of Schwarzschild wormhole in Kruskal coordinates 2. Dynamics of Schwarzschild wormhole 3. Penrose diagram; Causal structure of Schwarzschild wormhole 4. Formation of Schwarzschild black hole by collapse of a spherical star 5. Eddington-Finkelstein coordinates for black hole 6. Rindler approximation near horizon 7. Optical appearance of black hole from various vantage points C. Kerr (Rotating) Black Holes and Wormholes [Week 14] 1. Kerr metric 2. Kerr black hole as seen from outside: - frame dragging - multipole moments 3. Hamilton-Jacobi theory for orbits in curved spacetime - action principles for geodesics 4. Orbits in the Kerr metric 5. Penrose diagram; Causal structure of Kerr wormhole - event horizons - Cauchy horizons 6. Instability of Cauchy horizons -- first pass 7. Collapse of a Star to form a Kerr black hole -- first pass D. Traversable Wormholes and Time Machines [Week 15, first part] 1. Stress-energy to hold them open - violation of averaged null energy condition 2. Formation of time machine by relative motion of mouths 3. Formation of time machine by relative gravitational redshift at mouths 4. Chronology protection conjecture and possible enforcement mechanisms -- first pass IV. Gravitational Waves and Other Perturbations of Curved Spacetime A. Linearized Gravitational Waves in Flat Spacetime [Week 15, 2nd part] 1. Riemann tensor; tidal forces 2. Two polarizations and their gravitatoinal-wave fields 3. Tidal forces 4. TT coordinates 5. Energy carried by waves - first pass 6. Multipolar expansion of waves 7. Slow-motion sources and wave generation 8. Radiation reaction in slow-motion approximation B. Gravitational Wave Detection and Astrophysical Sources [Week 16] 1. Frequency bands and detection methods: overview 2. Resonant-mass detectors 3. Interferometric detectors 4. LIGO, VIRGO, and the International Network 5. Sources in the high-frequency (LIGO/VIRGO) band 6. LISA 7. Sources in the low-frequency (LISA) band 8. Pulsar timing 9. Sources in the very-low-frequency (pulsar-timing) band C. General Relativity as a Nonlinear Field Theory in Flat Spacetime [Week 17] 1. Conceptual framework 2. Gauge conditions 3. Field Equations 4. Landau-Lifshitz pseudotensor 5. Conservation laws for energy, momentum, and angular momentum in asymptotically flat spacetime 6. Tidal coupling between a black hole or other body and the external universe - laws of motion and precession - tidal heating D. Linear & Quadratic Perturbations of Curved Spacetime [Week 18] 1. General equations for perturbations of curved spacetime 2. Short-wave approximation a. gravitational waves b. geometric optics approximation for gravitational waves and electromagnetic 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 F. Pulsations, Stability, and Evolution of Black Holes and Wormholes [Week 19] 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. 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 c. The membrane paradigm 4. Magnetic fields around a spinning black hole a. Deposition on hole by an accretion disk; cleaning of field b. Blandford-Znajek effect; the powering of jets in active galactic nuclei THIRD TERM [tentative] V. Post-Newtonian Approximations and Experimental Tests of General Relativity [Week 20] 1. General concepts of post-Newtonian expansion 2. Einstein Infeld Hoffman (EIH) post-Newtonian equations of motion for compact binary systems and the solar system --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. Tests of general relativity in the solar system 7. Tests of general relativity in binary pulsars VI. Cosmology A. Fully Homogeneous and Isotropic Models of the Universe [Week 21] 1. Friedman solution of Einstein equations: Closed, Open and Flat 2. Standard (non-inflationary) model of evolution of the universe - primordial nucleosynthesis - matter-radiation decoupling - cosmic microwave background 3. Observational studies of large-scale geometry of universe - magnitude-redshift relation - angular diameter-redshift relation 4. DeSitter solution of Einstein equations 5. Inflationary model of evolution of the universe 6. String-motivated cosmologies B. Inhomogeneities in Cosmology [Week 22] 1. First-order perturbations and their evolution 2. Primordial gravitational waves and their parametric amplification in early universe 3. Phase transitions and defects arising therefrom: cosmic strings, domain walls, textures 3. Anisotropies of the cosmic microwave background and their observation 4. Formation of large-scale structure 5. Gravitational lensing VII. Causal Structure of Spacetime, and Global Methods for Analyzing It A. Differential Topology and Causal Structure [Week 23] 1. Basic concepts of topology 2. Futures and pasts 2. Causality conditions 3. Domains of dependence, Cauchy surfaces, global hyperbolicity, chronology horizons C. Applicaton: Black-Hole Event Horizons [Week 24, first part] 1. General structure a. null generators b. caustics 2. Proof of Second Law of Black-Hole Mechanics (horizon area increase) D. Application: Chronology Horizons (time machines) [Week 24, 2nd part] 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 - preview of divergent vacuum polarization 4. Examples E. Application: Singularity theorems [Week 25] 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 VIII. Structures of Singularites [Week 26] A. Some Non-Generic Examples 1. Friedman 2. Schwarzschild 3. Kasner 4. Mixmaster 5. Mass-Inflation B. Generic Singularities; Applications 1. BKL 2. Generalized mass-inflation 3. The big-bang singularity 4. The big-crunch singularity 5. Singularities inside black holes 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 IX. The Dynamics of Spacetime Geometry [Week 27] 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 X. Quantum Field Theory of a Massless Scalar Field in Curved Spacetime: Unruh Effect and Hawking Radiation from a Black Hole [Week 28] A. Quantized Scalar Field in Flat Spacetime 1. Minkowski Viewpoint: a. Modes b. Creation and annihilation operators c. Vacuum state 2. Rindler [accelerated observer] viewpoint a. Modes b. Nature of the Minkowski vacuum c. Unruh radiation C. Black-Hole atmosphere and evaporation 1. Hawking's original derivation of evaporation 2. Modes of scalar field in Schwarzschild spacetime and their quantization 3. Hartle-Hawking state 4. Unruh state: black-hole evaporation