Classical Fields

by Carmeli, Moshe

• ISBN13: 9789810247874
• ISBN10: 9810247877
• Format: Hardcover
• Copyright: 2002-02-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

Presents gravitation and gauge fields as interrelated topics with a common physical and mathematical foundation, such as gauge theory of gravitation and other fields, giving emphasis to the physicist's point of view.

Table of Contents

The Gravitational Field	1	(20)
Newtonian Gravitation	1	(9)
The Galilean group
Newtonian mechanics
Newton's theory of gravitation
Problems
Basic Properties of the Gravitational Field	10	(3)
Null Experiments	13	(3)
Principle of Equivalence	16	(1)
Principle of General Covariance	17	(3)
Suggested References
The Geometry of Curved Spacetime	20	(64)
Transformation of Coordinates	20	(3)
Contravariant vectors
Invariants
Covariant vectors
Tensors	23	(3)
Definition of a tensor
Tensor algebra
Symmetry of Tensors	26	(7)
Problems
The Metric Tensor	33	(2)
Tensor Densities	35	(10)
Definition of a tensor density
Levi-Civita tensor densities
Problems
The Christoffel Symbols	45	(6)
Transformation laws for Christoffel symbols
Some useful formulas
Geodesic coordinate system
Problems
Covariant Differentiation	51	(10)
Rules for covariant differentiation
Some useful formulas
Problems
Geodesics	61	(6)
Affine parameter
Null geodesics
Problems
The Riemann Curvature Tensor	67	(13)
The Ricci identity
Symmetry of the Riemann curvature tensor
Ricci tensor and scalar; Einstein tensor
The Weyl conformal tensor
Properties of the Weyl conformal tensor
Problems
Differential Identities	80	(4)
The Bianchi identities
The contracted Bianchi identities
Problems
Suggested References
The Einstein Field Equations	84	(71)
The Gravitational Field Equations	84	(4)
Derivation of the gravitational field equations
Properties of the Einstein field equations
The Newtonian Limit of the Einstein Field Equations	88	(5)
Problems
Action Integral for the Gravitational Field	93	(12)
Problems
Equations of Electrodynamics in the Presence of Gravitation	105	(8)
Problems
Lie Derivative	113	(9)
Problems
Structure of the Spacetime	122	(8)
The Killing equation, simple example: the Poincare group
Problems
Stationary and Static Gravitational Fields	130	(5)
Tetrad Formulation of the Einstein Field Equations: The Newman-Penrose Equations	135	(8)
The null tetrad
The spin coefficients
Tetrad components
The Newman-Penrose equations
The optical scalars
The electromagnetic field
Perturbation on Gravitational Background	143	(8)
Decoupled gravitational equations
Decoupled electromagnetic equations
Problems
Coordinate Conditions	151	(1)
Definition of coordinate conditions
deDonder coordinate condition, harmonic coordinate system
Problems
Initial-Value Problem	152	(3)
Problems
Suggested References
Gravitational Fields of Elementary Mass Systems	155	(50)
The Schwarzschild Metric	155	(8)
Problems
The Kruskal Coordinates	163	(5)
The Eddington-Finkelstein form for the spherically symmetric metric
Maximal extension of the Schwarzschild metric
Gravitational Field of a Spherically Symmetric Charged Body	168	(4)
Gravitational Field with Rotational Symmetry	172	(5)
Problems
Field of Particle with Quadrupole Moment	177	(6)
Problems
The Vaidya Radiating Metric	183	(6)
Derivation
The Vaidya metric in null coordinates
Problems
The Tolman Metric	189	(9)
Fluid without pressure
Comoving coordinates
Field equations
Solutions of the field equations
Problems
The Einstein-Rosen Metric	198	(7)
Cylindrical gravitational waves
Periodic solutions
Pulse solutions
Suggested References
Properties of the Gravitational Field	205	(61)
Weak Gravitational Field	205	(8)
Linear approximation
The linearized Einstein equations
Problems
Gravitational Red Shift	213	(2)
Motion in a Centrally Symmetric Gravitational Field	215	(7)
Deflection of Light in a Gravitational Field	222	(5)
Problems
Other Tests of General Relativity Theory	227	(7)
Detection of gravitational waves
Delay of radar pulses in gravitational field
Problems
Gravitational Radiation	234	(13)
The light cone at infinity
The geometry of the manifold m
The general relativistic case
Gravitational waves
Helicity and polarization of gravitational waves
Choice of coordinate system-Bondi coordinates
Problems
The Energy-Momentum Pseudotensor	247	(8)
Conservation laws in the presence of gravitation
Energy-momentum pseudotensor
Four-momentum
Angular momentum
Gravitational radiation from isolated system
The quadrupole radiation formula
Energy loss by two bodies
Problems
Gravitational Bremsstrahlung	255	(11)
Spectral resolution of intensity of dipole and quadrupole
Radiation of low frequencies in collision
Gravitational radiation in nonrelativistic collisions
Solar gravitational radiation
Total gravitational radiation
Comparison with classical sources
Problems
Suggested References
Equations of Motion in General Relativity	266	(99)
The Geodesic Postulate	266	(11)
Motion of a test particle
Test particle in an external gravitational field
Mass particle in gravitational field
Choice of coordinate system
Field equations
Equations of motion
Inclusion of nongravitational field
Problems
Slow-Motion Approximation--The Einstein-Infeld-Hoffmann Equation of Motion	277	(33)
Slow-motion approximation
The double-expansion method
The approximation method
Solution of the first approximation field equations
Solution of the second approximation field equations
Remark
The equations of motion
Remarks
Problems
Motion of Charged Particles in the Presence of Gravitation	310	(6)
The Fokker action principle
Variation of the action
Post-Newtonian Lagrangian	316	(6)
Motion of Spinning Particles	322	(6)
Test particle with structure
The Papapetrou equations of motion
Problems
Motion in the Schwarzschild Field--The Papapetrou-Corinaldesi Equations of Motion	328	(8)
Problems
Motion in the Vaidya Gravitational Field	336	(7)
Geodesic motion in the Vaidya metric
Equations of motion of the spin: supplementary conditions
Derivation of the spin equations
The orbital equations
Problems
Integrals of Motion in Particular Cases	343	(15)
Integrals of Motion in the General Case	358	(7)
Suggested References
Axisymmetric Solutions of the Einstein Field Equations	365	(42)
Stationary, Axisymmetric Metric	365	(3)
Generalization of static metric
General form of the line element
The Papapetrou Metric	368	(5)
Lewis line element
Field equations
The Ernst Potential	373	(4)
Field equations
The Ernst equation
Elementary Solutions of the Ernst Equation	377	(5)
Problems
The Kerr Metric	382	(1)
Derivation
Boyer-Lindguist coordinates
The Tomimatsu-Sato Metric	383	(2)
The NUT-Taub Metric	385	(3)
General solutions
The Demianski-Newman metric
Covariance Group of the Ernst Equation	388	(1)
Nonstationary Kerr Metric	389	(7)
Radiative Kerr metric
Variable-mass Kerr metric
Null tetrad quantities
Energy-momentum tensor and its asymptotic behavior
Perturbation on the Kerr Metric Background	396	(11)
The Teukolsky master equation
Separation of the equations
Boundary conditions
Energy and polarization
Problems
Suggested References
Spinor Formulation of Gravitation and Gauge Fields	407	(74)
Two-Component Spinors	407	(8)
Spinor representation of the group SL(2, C)
Realization of the spinor representation
Two-component spinors
Problems
Spinors in Curved Spacetimes	415	(10)
Correspondence between spinors and tensors
Covariant derivative of a spinor
Useful formula
Problems
The Electromagnetic Field Spinors	425	(3)
Electromagnetic potential spinor
Electromagnetic field spinor
Problems
The Curvature Spinor	428	(6)
Spinorial Ricci identity
Symmetry of the curvature spinor
Relation to the Riemann tensor
Bianchi identities
Problems
The Gravitational Field Spinors	434	(11)
Decomposition of the Riemann tensor
The gravitational spinor
The Ricci spinor
The Weyl spinor
The Bianchi identities
Problems
The SU(2) Gauge Field Theory	445	(5)
Potential and field strength
Local SU(2) transformation
Gauge covariant derivative
Gauge field equations
Conservation of isospin
The Gauge Field Spinors	450	(5)
The Yang-Mills spinor
Energy-momentum spinor
SU(2) spinors
Transformation Rules for the Yang-Mills Spinors	455	(9)
General transformation properties
Transformation under rotations and boosts
Rotations around null vectors
Change of basis for spinors
Problems
The Geometry of Gauge Fields	464	(7)
Spinor formulation
Conformal mapping of gauge fields
Problems
The Euclidean Gauge Field Spinors	471	(10)
Algebra of the matrices sμ
Spinor formulation of the Euclidean gauge fields
Self-dual and anti-self dual fields
Problems
Suggested References
Classification of the Gravitational and Gauge Fields	481	(72)
Classification of the Electromagnetic Field	481	(7)
Invariants of the electromagnetic field
The eigenspinor-eigenvalue equation
Classification
Problems
Classification of the Gravitational Field	488	(21)
Properties of the Wey1 tensor
Classification of the Weyl tensor
The geometry of the invariants of gravitation
The invariants in the presence of an electromagnetic field
Classification by the spinor method
Problems
Classification of Gauge Fields: The Eigenspinor-Eigenvalue Equation	509	(8)
Invariants of the Yang-Mills field
The eigenspinor-eigenvalue equation
Problems
The Matrix Method of classification of SU(2) Gauge Fields	517	(13)
The electromagnetic field
SU(2) gauge fields
Problems
Lorentz Invariant versus Gauge Invariant Methods of Classification	530	(2)
The Matrix Method of Classification-A Four-Way Scheme	532	(21)
Preliminaries
Four-way scheme of classification
Concluding remarks
Problems
Suggested References
Gauge Theory of Gravitation and Other Fields	553	(64)
Differential Geometrical Analysis	553	(5)
Preliminary remarks
Differential geometry-an introduction
Fiber Bundles and Gauge Fields	558	(4)
General relativistic interpretation of differential geometry
Fiber bundles, Abelian gauge fields, Non-Abelian gauge fields
Spinors and spacetime structure
Fiber Bundle Foundations of the SL(2, C) Gauge Theory	562	(7)
Gauge potentials and field strengths
Free field equations
The SL(2, C) Theory of Gravitation	569	(3)
Coupling matter and the gauge fields
The SL(2, C) theory and the Newman-Penrose method
Palatini-Type Variational Principle for the SL(2, C) Gauge Theory of Gravitation	572	(7)
Derivation
Remarks on quantization
The Einstein-Maxwell Equations	579	(11)
Preliminary remarks
The electromagnetic field
Pure gravitational field equations
Combined gravitational and electromagnetic fields
Magnetic Monopoles	590	(3)
Non-Abelian Gauge Fields in the Presence of Gravitation	593	(3)
Null Tetrad Formulation of Yang-Mills Theory	596	(5)
Yang-Mills potentials and fields
Explicit relations between potentials and fields
Yang-Mills field equations
Conserved currents
Energy-momentum tensor and the Einstein equations
Abelian solutions of the Yang-Mills theory
Null Tetrad Formulation of the Yang-Mills Theory in Flat Spacetime	601	(2)
Monopole Solution of Yang-Mills Equations	603	(5)
Problems
Solutions of the Coupled Einstein-Yang-Mills Field Equations	608	(9)
Problems
Suggested References
Appendix A Extended Bodies in General Relativity	617	(16)
Index	633