## Gravitation, Part 3This landmark text offers a rigorous full-year graduate level course on gravitation physics, teaching students to: • Grasp the laws of physics in flat spacetime • Predict orders of magnitude • Calculate using the principal tools of modern geometry • Predict all levels of precision • Understand Einstein's geometric framework for physics • Explore applications, including pulsars and neutron stars, cosmology, the Schwarzschild geometry and gravitational collapse, and gravitational waves • Probe experimental tests of Einstein's theory • Tackle advanced topics such as superspace and quantum geometrodynamics The book offers a unique, alternating two-track pathway through the subject:• In many chapters, material focusing on basic physical ideas is designated as Track 1. These sections together make an appropriate one-term advanced/graduate level course (mathematical prerequisites: vector analysis and simple partial-differential equations). The book is printed to make it easy for readers to identify these sections.• The remaining Track 2 material provides a wealth of advanced topics instructors can draw from to flesh out a two-term course, with Track 1 sections serving as prerequisites. |

### Contents

Geometrodynamics in Brief | 3 |

PHYSICS IN FLAT SPACETIME | 45 |

The Electromagnetic Field | 71 |

Electromagnetism and Differential Forms | 90 |

StressEnergy Tensor and Conservation Laws | 130 |

Accelerated Observers | 163 |

Incompatibility of Gravity and Special Relativity | 177 |

THE MATHEMATICS OF CURVED SPACETIME | 193 |

Search for Lens Effect of the Universe | 795 |

Density of the Universe Today | 796 |

Summary of Present Knowledge About Cosmological Parameters | 797 |

Anisotropic and Inhomogeneous Cosmologies | 800 |

The Kasner Model for an Anisotropic Universe | 801 |

Adiabatic Cooling of Anisotropy | 802 |

Particle Creation in an Anisotropic Universe | 803 |

Inhomogeneous Cosmologies | 804 |

CONTENTS xiii | 225 |

Vector and Directional Derivative Refined into Tangent Vector | 226 |

Bases Components and Transformation Laws for Vectors | 230 |

1Forms | 231 |

Tensors | 233 |

Commutators and Pictorial Techniques | 235 |

Manifolds and Differential Topology | 240 |

Geodesics Parallel Transport and Covariant Derivative | 244 |

Pictorial Approach | 245 |

Abstract Approach | 247 |

Component Approach | 258 |

Geodesic Equation | 262 |

Geodesic Deviation and Spacetime Curvature | 265 |

Tidal Gravitational Forces and Riemann Curvature Tensor | 270 |

Parallel Transport Around a Closed Curve | 277 |

Flatness is Equivalent to Zero Riemann Curvature | 283 |

Riemann Normal Coordinates | 285 |

Newtonian Gravity in the Language of Curved Spacetime | 289 |

Stratification of Newtonian Spacetime | 291 |

Galilean Coordinate Systems | 292 |

Geometric CoordinateFree Formulation of Newtonian Gravity | 298 |

A Critique | 302 |

Metric as Foundation of All | 304 |

Metric | 305 |

Concord Between Geodesics of Curved Spacetime Geometry and Straight Lines of Local Lorentz Geometry | 312 |

Geodesics as World Lines of Extremal Proper Time | 315 |

MetricInduced Properties of Riemann | 324 |

The Proper Reference Frame of an Accelerated Observer | 327 |

Calculation of Curvature | 333 |

Forming the Einstein Tensor | 343 |

More Efficient Computation | 344 |

Curvature 2Forms | 348 |

Computation of Curvature Using Exterior Differential Forms | 354 |

Bianchi Identities and the Boundary of a Boundary | 364 |

Bianchi Identity dR 0 as a Manifestation of Boundary of Boundary0 | 372 |

Key to Contracted Bianchi Identity | 373 |

Calculation of the Moment of Rotation | 375 |

Conservation of Moment of Rotation Seen from Boundary of a Boundary is Zero | 377 |

Conservation of Moment of Rotation Expressed in Differential Form | 378 |

A Preview | 379 |

EINSTEINS GEOMETRIC THEORY OF GRAVITY | 383 |

Equivalence Principle and Measurement of the Gravitational Field | 385 |

FactorOrdering Problems in the Equivalence Principle | 388 |

The Rods and Clocks Used to Measure Space and Time Intervals | 393 |

The Measurement of the Gravitational Field | 399 |

How MassEnergy Generates Curvature | 404 |

A Dynamic Necessity | 408 |

Cosmological Constant | 409 |

The Newtonian Limit | 412 |

Axiomatize Einsteins Theory? | 416 |

A Feature Distinguishing Einsteins Theory from Other Theories of Gravity | 429 |

A Taste of the History of Einsteins Equation | 431 |

Weak Gravitational Fields | 435 |

Gravitational Waves | 442 |

Nearly Newtonian Gravitational Fields | 445 |

Mass and Angular Momentum of a Gravitating System | 448 |

Measurement of the Mass and Angular Momentum | 450 |

Mass and Angular Momentum of Fully Relativistic Sources | 451 |

Mass and Angular Momentum of a Closed Universe | 457 |

Conservation Laws for 4Momentum and Angular Momentum | 460 |

Gaussian Flux Integrals for 4Momentum and Angular Momentum | 461 |

Volume Integrals for 4Momentum and Angular Momentum | 464 |

Why the Energy of the Gravitational Field Cannot be Localized | 466 |

Conservation Laws for Total 4Momentum and Angular Momentum | 468 |

Equation of Motion Derived from the Field Equation | 471 |

Variational Principle and InitialValue Data | 484 |

The Hilbert Action Principle and the Palatini Method of Variation | 491 |

Matter Lagrangian and StressEnergy Tensor | 504 |

Splitting Spacetime into Space and Time | 505 |

Intrinsic and Extrinsic Curvature | 509 |

The Hilbert Action Principle and the ArnowittDeserMisner Modification Thereof in the SpaceplusTime Split | 519 |

The ArnowittDeserMisner Formulation of the Dynamics of Geometry | 520 |

Integrating Forward in Time | 526 |

The InitialValue Problem in the ThinSandwich Formulation | 528 |

The TimeSymmetric and TimeAntisymmetric InitialValue Problem | 535 |

Yorks Handles to Specify a 4Geometry | 539 |

Machs Principle and the Origin of Inertia | 543 |

Junction Conditions | 551 |

CONTENTS XV | 557 |

Hydrodynamics in Curved Spacetime | 562 |

Electrodynamics in Curved Spacetime | 568 |

Geometric Optics in Curved Spacetime | 570 |

Kinetic Theory in Curved Spacetime | 583 |

RELATIVISTIC STARS | 591 |

Spherical Stars | 593 |

Coordinates and Metric for a Static Spherical System | 594 |

Physical Interpretation of Schwarzschild coordinates | 595 |

Description of the Matter Inside a Star | 597 |

Equations of Structure | 600 |

External Gravitational Field | 607 |

How to Construct a Stellar Model | 608 |

The Spacetime Geometry for a Static Star | 612 |

Pulsars and Neutron Stars Quasars and Supermassive Stars | 618 |

The Endpoint of Stellar Evolution | 621 |

Pulsars | 627 |

Supermassive Stars and Stellar Instabilities | 630 |

Quasars and Explosions In Galactic Nuclei | 634 |

The Pit in the Potential as the Central New Feature of Motion in Schwarzschild Geometry | 636 |

Symmetries and Conservation Laws | 650 |

Conserved Quantities for Motion in Schwarzschild Geometry | 655 |

Gravitational Redshift | 659 |

Orbit of a Photon Neutrino or Graviton in Schwarzschild Geometry | 672 |

Spherical Star Clusters | 679 |

Stellar Pulsations | 688 |

Setting Up the Problem | 689 |

Eulerian versus Lagrangian Perturbations | 690 |

InitialValue Equations | 691 |

Dynamic Equation and Boundary Conditions | 693 |

Summary of Results | 694 |

THE UNIVERSE | 701 |

Idealized Cosmologies | 703 |

StressEnergy Content of the Universethe Fluid Idealization | 711 |

Geometric Implications of Homogeneity and Isotropy | 713 |

Comoving Synchronous Coordinate Systems for the Universe | 715 |

The Expansion Factor | 718 |

Possible 3Geometries for a Hypersurface of Homogeneity | 720 |

Equations of Motion for the Fluid | 726 |

The Einstein Field Equations | 728 |

Time Parameters and the Hubble Constant | 730 |

The Elementary Friedmann Cosmology of a Closed Universe | 733 |

Homogeneous Isotropic Model Universes that Violate Einsteins Conception of Cosmology | 742 |

Evolution of the Universe into Its Present State | 763 |

Standard Model Modified for Primordial Chaos | 769 |

Other Cosmological Theories | 770 |

Present State and Future Evolution of the Universe | 771 |

Cosmological Redshift | 772 |

Measurement of the Hubble Constant | 780 |

Measurement of the Deceleration Parameter | 782 |

The Mixmaster Universe | 805 |

Horizons and the Isotropy of the Microwave Background | 815 |

GRAVITATIONAL COLLAPSE AND BLACK HOLES | 817 |

Schwarzschild Geometry | 819 |

The Nonsingularity of the Gravitational Radius | 820 |

Behavior of Schwarzschild Coordinates at r 2M | 823 |

Several WellBehaved Coordinate Systems | 826 |

Relationship Between KruskalSzekeres Coordinates and Schwarzschild Coordinates | 833 |

Dynamics of the Schwarzschild Geometry | 836 |

Gravitational Collapse | 842 |

Birkhoffs Theorem | 843 |

Exterior Geometry of a Collapsing Star | 846 |

Collapse of a Star with Uniform Density and Zero Pressure | 851 |

Spherically Symmetric Collapse with Internal Pressure Forces | 857 |

The Fate of a Man Who Falls into the Singularity at r 0 | 860 |

Realistic Gravitational CollapseAn Overview | 862 |

CONTENTS xvii | 872 |

The Gravitational and Electromagnetic Fields of a Black Hole | 875 |

Mass Angular Momentum Charge and Magnetic Moment | 891 |

Symmetries and Frame Dragging 892 4 | 895 |

Equations of Motion for Test Particles | 897 |

Principal Null Congruences | 901 |

Storage and Removal of Energy from Black Holes | 904 |

Reversible and Irreversible Transformations | 907 |

Global Techniques Horizons and Singularity Theorems | 916 |

Infinity in Asymptotically Flat Spacetime | 917 |

Causality and Horizons | 922 |

Global Structure of Horizons | 924 |

Proof of Second Law of BlackHole Dynamics | 931 |

Singularity Theorems and the Issue of the Final State | 934 |

GRAVITATIONAL WAVES | 941 |

Propagation of Gravitational Waves | 943 |

Review of Linearized Theory in Vacuum | 944 |

PlaneWave Solutions in Linearized Theory | 945 |

The Transverse Traceless TT Gauge | 946 |

Geodesic Deviation in a Linearized Gravitational Wave | 950 |

Polarization of a Plane Wave | 952 |

The StressEnergy Carried by a Gravitational Wave | 955 |

Gravitational Waves in the Full Theory of General Relativity | 956 |

An Exact PlaneWave Solution | 957 |

Physical Properties of the Exact Plane Wave | 960 |

Comparison of an Exact Electromagnetic Plane Wave with the Gravitational Plane Wave | 961 |

A New Viewpoint on the Exact Plane Wave | 962 |

The Shortwave Approximation | 964 |

Effect of Background Curvature on Wave Propagation | 967 |

StressEnergy Tensor for Gravitational Waves | 969 |

Generation of Gravitational Waves | 974 |

Power Radiated in Terms of Internal Power Flow | 978 |

Laboratory Generators of Gravitational Waves | 979 |

General Discussion | 980 |

Gravitational Collapse Black Holes Supernovae and Pulsars as Sources | 981 |

Binary Stars as Sources | 986 |

Formulas for Radiation from Nearly Newtonian SlowMotion Sources | 989 |

Radiation Reaction in SlowMotion Sources | 993 |

Foundations for Derivation of Radiation Formulas | 995 |

Evaluation of the Radiation Field in the SlowMotion Approximation | 996 |

Derivation of the RadiationReaction Potential | 1001 |

Detection of Gravitational Waves | 1004 |

Accelerations in Mechanical Detectors | 1006 |

Types of Mechanical Detectors | 1012 |

Introductory Remarks | 1019 |

Idealized WaveDominated Detector Excited by Steady Flux of Monochromatic Waves | 1022 |

Idealized WaveDominated Detector Excited by Arbitrary Flux of Radiation | 1026 |

General WaveDominated Detector Excited by Arbitrary Flux of Radiation | 1028 |

Noisy Detectors | 1036 |

Nonmechanical Detectors | 1040 |

EXPERIMENTAL TESTS OF GENERAL RELATIVITY | 1045 |

Testing the Foundations of Relativity | 1047 |

Theoretical Frameworks for Analyzing Tests of General Relativity | 1048 |

EötvösDicke Experiment | 1050 |

Tests for the Existence of a Metric Governing Length and Time Measurements | 1054 |

Gravitational Redshift Experiments | 1055 |

Tests of the Equivalence Principle | 1060 |

Tests for the Existence of Unknown LongRange Fields | 1063 |

Other Theories of Gravity and the PostNewtonian Approximation | 1066 |

Metric Theories of Gravity | 1067 |

PostNewtonian Limit and PPN Formalism | 1068 |

PPN Coordinate System | 1073 |

Description of the Matter in the Solar System | 1074 |

Nature of the PostNewtonian Expansion | 1075 |

Newtonian Approximation | 1077 |

PPN Metric Coefficients | 1080 |

Velocity of PPN Coordinates Relative to Universal Rest Frame | 1083 |

PPN StressEnergy Tensor | 1086 |

PPN Equations of Motion | 1087 |

Relation of PPN Coordinates to Surrounding Universe | 1091 |

SolarSystem Experiments | 1096 |

The Use of Light Rays and Radio Waves to Test Gravity | 1099 |

Light Deflection | 1101 |

TimeDelay in Radar Propagation | 1103 |

Perihelion Shift and Periodic Perturbations in Geodesic Orbits | 1110 |

ThreeBody Effects in the Lunar Orbit | 1116 |

The Dragging of Inertial Frames | 1117 |

Is the Gravitational Constant Constant? | 1121 |

Do Planets and the Sun Move on Geodesics? | 1126 |

Summary of Experimental Tests of General Relativity | 1131 |

FRONTIERS | 1133 |

Spinors | 1135 |

Infinitesimal Rotations | 1140 |

CONTENTS xix | 1142 |

Thomas Precession via Spinor Algebra | 1145 |

Spinors | 1148 |

Correspondence Between Vectors and Spinors | 1150 |

Spinor Algebra | 1151 |

Spin Space and Its Basis Spinors | 1156 |

Spinor Viewed as Flagpole Plus Flag Plus OrientationEntanglement Relation | 1157 |

An Application of Spinors | 1160 |

Spinors as a Powerful Tool in Gravitation Theory | 1164 |

Regge Calculus | 1166 |

Simplexes and Deficit Angles | 1167 |

Skeleton Form of Field Equations | 1169 |

The Choice of Lattice Structure | 1173 |

The Choice of Edge Lengths | 1177 |

Past Applications of Regge Calculus | 1178 |

The Future of Regge Calculus | 1179 |

Arena for the Dynamics of Geometry | 1180 |

The Dynamics of Geometry Described in the Language of the Superspace of the 3s | 1184 |

The EinsteinHamiltonJacobi Equation | 1185 |

Fluctuations in Geometry | 1190 |

Beyond the End of Time | 1196 |

Assessment of the Theory that Predicts Collapse | 1198 |

Their Prevalence and Final Dominance | 1202 |

Not Geometry but Pregeometry as the Magic Building Material | 1203 |

Pregeometry as the Calculus of Propositions | 1208 |

The Reprocessing of the Universe | 1209 |

1221 | |

1255 | |

### Other editions - View all

Gravitation, Part 3 Charles W. Misner,Kip S. Thorne,John Archibald Wheeler No preview available - 1973 |

### Common terms and phrases

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