Gravitation, Part 3

Front Cover
Macmillan, Sep 15, 1973 - Science - 1279 pages
This 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
Bibliography and Index of Names
1221
Subject Index
1255
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