## 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 |

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

3-geometry 4-momentum 4-velocity accelerated observer angular momentum antisymmetric arbitrary basis vectors Bianchi identities boundary calculate Chapter clock components Compute connection coefficients conservation law constant coordinate system covariant derivative curved spacetime defined density DIFFERENTIAL FORMS distance dynamic Einstein Einstein field equation electromagnetic field energy equation of motion equivalence principle Euclidean event Exercise exterior derivative field equation Figure flat spacetime formula function Galilean coordinates gaß geodesic deviation geodesic equation geometric object geometrodynamics gradient gravitational field hypersurface inertial frame initial-value integral linear Lorentz frame manifold mass mass-energy mathematical measured metric Newtonian notation orthonormal parallel transport photon physics rest frame Riemann curvature tensor rotation scalar Show slice slot spacelike spacetime curvature special relativity stress-energy tensor surface symmetry tangent space tangent vector test particles theory of gravity tubes vanish variational principle vector field velocity world line zero αβ αλ βγ μν