Gravitation

Charles W. Misner is professor emeritus of physics at the University of Maryland. Kip S. Thorne is the Feynman Professor of Theoretical Physics, Emeritus at the California Institute of Technology. His books include Modern Classical Physics(Princeton), Black Holes and Time Warps, and The Science of Interstellar. John Archibald Wheeler (1911-2008) was professor of physics at Princeton University and later at the University of Texas, Austin. His books include Spacetime Physics and Geons, Black Holes, and Quantum Foam.

"Kip S. Thorne, Co-Winner of the 2017 Nobel Prize in Physics"

• LIST OF BOXESLIST OF FIGURESFOREWORD TO THE 2017 EDITIONPREFACE TO THE 2017 EDITIONPREFACEACKNOWLEDGMENTSPart I SPACETIME PHYSICS1. Geometrodynamics in Brief1. The Parable of the Apple2. Spacetime With and Without Coordinates3. Weightlessness4. Local Lorentz Geometry, With and Without Coordinates5. Time6. Curvature7. Effect of Matter on GeometryPart II PHYSICS IN FLAT SPACETIME2. Foundations of Special Relativity1. Overview2. Geometric Objects3. Vectors4. The Metric Tensor5. Differential Forms6. Gradients and Directional Derivatives7. Coordinate Representation of Geometric Objects8. The Centrifuge and the Photon9. Lorentz Transformations10. Collisions3. The Electromagnetic Field1. The Lorentz Force and the Electromagnetic Field Tensor2. Tensors in All Generality3. Three-Plus-One View Versus Geometric View4. Maxwell’s Equations5. Working with Tensors4. Electromagnetism and Differential Forms1. Exterior Calculus2. Electromagnetic 2-Form and Lorentz Force3. Forms Illuminate Electromagnetism and Electromagnetism Illuminates Forms4. Radiation Fields5. Maxwell’s Equations6. Exterior Derivative and Closed Forms7. Distant Action from Local Law5. Stress-Energy Tensor and Conservation Laws1. Track-1 Overview2. Three-Dimensional Volumes and Definition of the Stress-Energy Tensor3. Components of Stress-Energy Tensor4. Stress-Energy Tensor for a Swarm of Particles5. Stress-Energy Tensor for a Perfect Fluid6. Electromagnetic Stress-Energy7. Symmetry of the Stress-Energy Tensor8. Conservation of 4-Momentum: Integral Formulation9. Conservation of 4-Momentum: Differential Formulation10. Sample Application of ▼ · T = 011. Angular Momentum6. Accelerated Observers1. Accelerated Observers Can Be Analyzed Using Special Relativity2. Hyperbolic Motion3. Constraints on Size of an Accelerated Frame4. The Tetrad Carried by a Uniformly Accelerated Observer5. The Tetrad Fermi-Walker Transported by an Observer with Arbitrary Acceleration6. The Local Coordinate System of an Accelerated Observer7. Incompatibility of Gravity and Special Relativity1. Attempts to Incorporate Gravity into Special Relativity2. Gravitational Redshift Derived from Energy Conservation3. Gravitational Redshift Implies Spacetime Is Curved4. Gravitational Redshift as Evidence for the Principle of Equivalence5. Local Flatness, Global CurvaturePart III THE MATHEMATICS OF CURVED SPACETIME8. Differential Geometry: An Overview1. An Overview of Part III2. Track 1 Versus Track 2: Difference in Outlook and Power3. Three Aspects of Geometry: Pictorial, Abstract, Component4. Tensor Algebra in Curved Spacetime5. Parallel Transport, Covariant Derivative, Connection Coefficients, Geodesics6. Local Lorentz Frames: Mathematical Discussion7. Geodesic Deviation and the Riemann Curvature Tensor9. Differential Topology1. Geometric Objects in Metric-Free, Geodesic-Free Spacetime2. “Vector” and “Directional Derivative” Refined into Tangent Vector3. Bases, Components, and Transformation Laws for Vectors4. 1-Forms5. Tensors6. Commutators and Pictorial Techniques7. Manifolds and Differential Topology10. Affine Geometry: Geodesics, Parallel Transport and Covariant Derivative1. Geodesics and the Equivalence Principle2. Parallel Transport and Covariant Derivative: Pictorial Approach3. Parallel Transport and Covariant Derivative: Abstract Approach4. Parallel Transport and Covariant Derivative: Component Approach5. Geodesic Equation11. Geodesic Deviation and Spacetime Curvature1. Curvature, At Last!2. The Relative Acceleration of Neighboring Geodesics3. Tidal Gravitational Forces and Riemann Curvature Tensor4. Parallel Transport Around a Closed Curve5. Flatness is Equivalent to Zero Riemann Curvature6. Riemann Normal Coordinates12. Newtonian Gravity in the Language of Curved Spacetime1. Newtonian Gravity in Brief2. Stratification of Newtonian Spacetime3. Galilean Coordinate Systems4. Geometric, Coordinate-Free Formulation of Newtonian Gravity5. The Geometric View of Physics: A Critique13. Riemannian Geometry: Metric as Foundation of All1. New Features Imposed on Geometry by Local Validity of Special Relativity2. Metric3. Concord Between Geodesics of Curved Spacetime Geometry and Straight Lines of Local Lorentz Geometry4. Geodesics as World Lines of Extremal Proper Time5. Metric-Induced Properties of Riemann6. The Proper Reference Frame of an Accelerated Observer14. Calculation of Curvature1. Curvature as a Tool for Understanding Physics2. Forming the Einstein Tensor3. More Efficient Computation4. The Geodesic Lagrangian Method5. Curvature 2-Forms6. Computation of Curvature Using Exterior Differential Forms15. Bianchi Identities and the Boundary of a Boundary1. Bianchi Identities in Brief2. Bianchi Identity dR = 0 as a Manifestation of “Boundary of Boundary = 0”3. Moment of Rotation: Key to Contracted Bianchi Identity4. Calculation of the Moment of Rotation5. Conservation of Moment of Rotation Seen from “Boundary of a Boundary is Zero”6. Conservation of Moment of Rotation Expressed in Differential Form7. From Conservation of Moment of Rotation to Einstein’s Geometrodynamics: A PreviewPart IV EINSTEIN’S GEOMETRIC THEORY OF GRAVITY16. Equivalence Principle and Measurement of the “Gravitational Field”1. Overview2. The Laws of Physics in Curved Spacetime3. Factor-Ordering Problems in the Equivalence Principle4. The Rods and Clocks Used to Measure Space and Time Intervals5. The Measurement of the Gravitational Field17. How Mass-Energy Generates Curvature1. Automatic Conservation of the Source as the Central Idea in the Formulation of the Field Equation2. Automatic Conservation of the Source: A Dynamic Necessity3. Cosmological Constant4. The Newtonian Limit5. Axiomatize Einstein’s Theory?6. “No Prior Geometry”: A Feature Distinguishing Einstein’s Theory from Other Theories of Gravity7. A Taste of the History of Einstein’s Equation18. Weak Gravitational Fields1. The Linearized Theory of Gravity2. Gravitational Waves3. Effect of Gravity on Matter4. Nearly Newtonian Gravitational Fields19. Mass and Angular Momentum of a Gravitating System1. External Field of a Weakly Gravitating Source2. Measurement of the Mass and Angular Momentum3. Mass and Angular Momentum of Fully Relativistic Sources4. Mass and Angular Momentum of a Closed Universe20. Conservation Laws for 4-Momentum and Angular Momentum1. Overview2. Gaussian Flux Integrals for 4-Momentum and Angular Momentum3. Volume Integrals for 4-Momentum and Angular Momentum4. Why the Energy of the Gravitational Field Cannot be Localized5. Conservation Laws for Total 4-Momentum and Angular Momentum6. Equation of Motion Derived from the Field Equation21. Variational Principle and Initial-Value Data1. Dynamics Requires Initial-Value Data2. The Hilbert Action Principle and the Palatini Method of Variation3. Matter Lagrangian and Stress-Energy Tensor4. Splitting Spacetime into Space and Time5. Intrinsic and Extrinsic Curvature6. The Hilbert Action Principle and the Arnowitt-Deser-Misner Modification Thereof in the Space-plus-Time Split7. The Arnowitt-Deser-Misner Formulation of the Dynamics of Geometry8. Integrating Forward in Time9. The Initial-Value Problem in the Thin-Sandwich Formulation10. The Time-Symmetric and Time-Antisymmetric Initial-Value Problem11. York’s “Handles” to Specify a 4-Geometry12. Mach’s Principle and the Origin of Inertia13. Junction Conditions22. Thermodynamics, Hydrodynamics, Electrodynamics, Geometric Optics, and Kinetic Theory1. The Why of this Chapter2. Thermodynamics in Curved Spacetime3. Hydrodynamics in Curved Spacetime4. Electrodynamics in Curved Spacetime5. Geometric Optics in Curved Spacetime6. Kinetic Theory in Curved SpacetimePart V RELATIVISTIC STARS23. Spherical Stars1. Prolog2. Coordinates and Metric for a Static, Spherical System3. Physical Interpretation of Schwarzschild coordinates4. Description of the Matter Inside a Star5. Equations of Structure6. External Gravitational Field7. How to Construct a Stellar Model8. The Spacetime Geometry for a Static Star24. Pulsars and Neutron Stars; Quasars and Supermassive Stars1. Overview2. The Endpoint of Stellar Evolution3. Pulsars4. Supermassive Stars and Stellar Instabilities5. Quasars and Explosions In Galactic Nuclei6. Relativistic Star Clusters25. The “Pit in the Potential” as the Central New Feature of Motion in Schwarzschild Geometry1. From Kepler’s Laws to the Effective Potential for Motion in Schwarzschild Geometry2. Symmetries and Conservation Laws3. Conserved Quantities for Motion in Schwarzschild Geometry4. Gravitational Redshift5. Orbits of Particles6. Orbit of a Photon, Neutrino, or Graviton in Schwarzschild Geometry7. Spherical Star Clusters26. Stellar Pulsations1. Motivation2. Setting Up the Problem3. Eulerian versus Lagrangian Perturbations4. Initial-Value Equations5. Dynamic Equation and Boundary Conditions6. Summary of ResultsPart VI THE UNIVERSE27. Idealized Cosmologies1. The Homogeneity and Isotropy of the Universe2. Stress-Energy Content of the Universe—the Fluid Idealization3. Geometric Implications of Homogeneity and Isotropy4. Comoving, Synchronous Coordinate Systems for the Universe5. The Expansion Factor6. Possible 3-Geometries for a Hypersurface of Homogeneity7. Equations of Motion for the Fluid8. The Einstein Field Equations9. Time Parameters and the Hubble Constant10. The Elementary Friedmann Cosmology of a Closed Universe11. Homogeneous Isotropic Model Universes that Violate Einstein’s Conception of Cosmology28. Evolution of the Universe into Its Present State1. The “Standard Model” of the Universe2. Standard Model Modified for Primordial Chaos3. What “Preceded” the Initial Singularity?4. Other Cosmological Theories29. Present State and Future Evolution of the Universe1. Parameters that Determine the Fate of the Universe2. Cosmological Redshift3. The Distance-Redshift Relation: Measurement of the Hubble Constant4. The Magnitude-Redshift Relation: Measurement of the Deceleration Parameter5. Search for “Lens Effect” of the Universe6. Density of the Universe Today7. Summary of Present Knowledge About Cosmological Parameters30. Anisotropic and Inhomogeneous Cosmologies1. Why Is the Universe So Homogeneous and Isotropic?2. The Kasner Model for an Anisotropic Universe3. Adiabatic Cooling of Anisotropy4. Viscous Dissipation of Anisotropy5. Particle Creation in an Anisotropic Universe6. Inhomogeneous Cosmologies7. The Mixmaster Universe8. Horizons and the Isotropy of the Microwave BackgroundPart VII GRAVITATIONAL COLLAPSE AND BLACK HOLES31. Schwarzschild Geometry1. Inevitability of Collapse for Massive Stars2. The Nonsingularity of the Gravitational Radius3. Behavior of Schwarzschild Coordinates at r = 2M4. Several Well-Behaved Coordinate Systems5. Relationship Between Kruskal-Szekeres Coordinates and Schwarzschild Coordinates6. Dynamics of the Schwarzschild Geometry32. Gravitational Collapse1. Relevance of Schwarzschild Geometry2. Birkhoff’s Theorem3. Exterior Geometry of a Collapsing Star4. Collapse of a Star with Uniform Density and Zero Pressure5. Spherically Symmetric Collapse with Internal Pressure Forces6. The Fate of a Man Who Falls into the Singularity at r = 07. Realistic Gravitational Collapse—An Overview33. Black Holes1. Why “Black Hole”?2. The Gravitational and Electromagnetic Fields of a Black Hole3. Mass, Angular Momentum, Charge, and Magnetic Moment4. Symmetries and Frame Dragging5. Equations of Motion for Test Particles6. Principal Null Congruences7. Storage and Removal of Energy from Black Holes8. Reversible and Irreversible Transformations34. Global Techniques, Horizons, and Singularity Theorems1. Global Techniques Versus Local Techniques2. “Infinity” in Asymptotically Flat Spacetime3. Causality and Horizons4. Global Structure of Horizons5. Proof of Second Law of Black-Hole Dynamics6. Singularity Theorems and the “Issue of the Final State”Part VIII GRAVITATIONAL WAVES35. Propagation of Gravitational Waves1. Viewpoints2. Review of “Linearized Theory” in Vacuum3. Plane-Wave Solutions in Linearized Theory4. The Transverse Traceless (TT) Gauge5. Geodesic Deviation in a Linearized Gravitational Wave6. Polarization of a Plane Wave7. The Stress-Energy Carried by a Gravitational Wave8. Gravitational Waves in the Full Theory of General Relativity9. An Exact Plane-Wave Solution10. Physical Properties of the Exact Plane Wave11. Comparison of an Exact Electromagnetic Plane Wave with the Gravitational Plane Wave12. A New Viewpoint on the Exact Plane Wave13. The Shortwave Approximation14. Effect of Background Curvature on Wave Propagation15. Stress-Energy Tensor for Gravitational Waves36. Generation of Gravitational Waves1. The Quadrupole Nature of Gravitational Waves2. Power Radiated in Terms of Internal Power Flow3. Laboratory Generators of Gravitational Waves4. Astrophysical Sources of Gravitational Waves: General Discussion5. Gravitational Collapse, Black Holes, Supernovae, and Pulsars as Sources6. Binary Stars as Sources7. Formulas for Radiation from Nearly Newtonian Slow-Motion Sources8. Radiation Reaction in Slow-Motion Sources9. Foundations for Derivation of Radiation Formulas10. Evaluation of the Radiation Field in the Slow-Motion Approximation11. Derivation of the Radiation-Reaction Potential37. Detection of Gravitational Waves1. Coordinate Systems and Impinging Waves2. Accelerations in Mechanical Detectors3. Types of Mechanical Detectors4. Vibrating, Mechanical Detectors: Introductory Remarks5. Idealized Wave-Dominated Detector, Excited by Steady Flux of Monochromatic Waves6. Idealized, Wave-Dominated Detector, Excited by Arbitrary Flux of Radiation7. General Wave-Dominated Detector, Excited by Arbitrary Flux of Radiation8. Noisy Detectors9. Nonmechanical Detectors10. Looking Toward the FuturePart IX. EXPERIMENTAL TESTS OF GENERAL RELATIVITY38. Testing the Foundations of Relativity1. Testing is Easier in the Solar System than in Remote Space2. Theoretical Frameworks for Analyzing Tests of General Relativity3. Tests of the Principle of the Uniqueness of Free Fall: Eotvos-Dicke Experiment4. Tests for the Existence of a Metric Governing Length and Time Measurements5. Tests of Geodesic Motion: Gravitational Redshift Experiments6. Tests of the Equivalence Principle7. Tests for the Existence of Unknown Long-Range Fields39. Other Theories of Gravity and the Post-Newtonian Approximation1. Other Theories2. Metric Theories of Gravity3. Post-Newtonian Limit and PPN Formalism4. PPN Coordinate System5. Description of the Matter in the Solar System6. Nature of the Post-Newtonian Expansion7. Newtonian Approximation8. PPN Metric Coefficients9. Velocity of PPN Coordinates Relative to “Universal Rest Frame”10. PPN Stress-Energy Tensor11. PPN Equations of Motion12. Relation of PPN Coordinates to Surrounding Universe13. Summary of PPN Formalism40. Solar-System Experiments1. Many Experiments Open to Distinguish General Relativity from Proposed Metric Theories of Gravity2. The Use of Light Rays and Radio Waves to Test Gravity3. “Light” Deflection4. Time-Delay in Radar Propagation5. Perihelion Shift and Periodic Perturbations in Geodesic Orbits6. Three-Body Effects in the Lunar Orbit7. The Dragging of Inertial Frames8. Is the Gravitational Constant Constant?9. Do Planets and the Sun Move on Geodesics?10. Summary of Experimental Tests of General RelativityPart X. FRONTIERS41. Spinors1. Reflections, Rotations, and the Combination of Rotations2. Infinitesimal Rotations3. Lorentz Transformation via Spinor Algebra4. Thomas Precession via Spinor Algebra5. Spinors6. Correspondence Between Vectors and Spinors7. Spinor Algebra8. Spin Space and Its Basis Spinors9. Spinor Viewed as Flagpole Plus Flag Plus Orientation-Entanglement Relation10. Appearance of the Night Sky: An Application of Spinors11. Spinors as a Powerful Tool in Gravitation Theory42. Regge Calculus1. Why the Regge Calculus?2. Regge Calculus in Brief3. Simplexes and Deficit Angles4. Skeleton Form of Filed Equations5. The Choice of Lattice Structure6. The Choice of Edge Lengths7. Past Applications of Regge Calculus8. The Future of Regge Calculus43. Superspace: Arena for the Dynamics of Geometry1. Space, Superpace, and Spacetime Distinguished2. The Dynamics of Geometry Described in the Language of the Superspace of the (3)y’s3. The Einstein-Hamilton-Jacobi Equation4. Fluctuations in Geometry44. Beyond the End of Time1. Gravitational Collapse as the Greatest Crisis in Physics of All Time2. Assessment of the Theory that Predicts Collapse3. Vacuum Fluctuations: Their Prevalence and Final Dominance4. Not Geometry, but Pregeometry, as the Magic Building Material5. Pregeometry as the Calculus of Propositions6. The Black Box: The Reprocessing of the UniverseBibliography and Index of NamesSubject Index