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