Group Theory and General Relativity : Representations of the Lorentz Group and Their Applications to the Gravitational Field

by Carmeli, Moshe

• ISBN13: 9781860942341
• ISBN10: 1860942342
• Format: Hardcover
• Copyright: 2000-12-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

This is the only book on the subject of group theory and Einstein's theory of gravitation. It contains an extensive discussion on general relativity from the viewpoint of group theory and gauge fields. It also puts together in one volume many scattered, original works, on the use of group theory in general relativity theory.There are twelve chapters in the book. The first six are devoted to rotation and Lorentz groups, and their representations. They include the spinor representation as well as the infinite-dimensional representations. The other six chapters deal with the application of groups -- particularly the Lorentz and the SL(2, C) groups -- to the theory of general relativity. Each chapter is concluded with a set of problems.The topics covered range from the fundamentals of general relativity theory, its formulation as an SL(2, C) gauge theory, to exact solutions of the Einstein gravitational field equations. The important Bondi-Metzner-Sachs group, and its representations, conclude the book The entire book is self-contained in both group theory and general relativity theory, and no prior knowledge of either is assumed.The subject of this book constitutes a relevant link between field theoreticians and general relativity theoreticians, who usually work rather independently of each other. The treatise is highly topical and of real interest to theoretical physicists, general relativists and applied mathematicians. It is invaluable to graduate students and research workers in quantum field theory, general relativity and elementary particle theory.

Table of Contents

Introduction	xiii
The Rotation Group	1	(18)
The Three-Dimensional Pure Rotation Group	1	(3)
The Euler angles
The Group SU2	4	(2)
0The Groups O3 and SU2
Homomorphism of the group SU2 onto the group O3
Invariant Integrals over the Groups O3 and SU2	6	(1)
Invariant integral over the group O3
Invariant integral over the group SU2
Representations of the Groups O3 and SU2	6	(7)
Single- and double-valued representations
Infinitesimal generators
Canonical basis
Matrix Elements of Irreducible Representations	13	(3)
Spinor representation of the group SU2
Matrix elements of the operator D(μu)
Properties of the matrices Dj(μu)
Orthogonality relations
Differential Operators of Infinitesimal Rotations	16	(3)
Representations of O3 in space of functions
The basic infinitesimal operators
Angular Momentum operators
Problems	16	(3)
The Lorentz Group	19	(15)
Infinitesimal Lorentz Matrices	23	(1)
Galilean group
Poincare group
Proper orthochronous Lorentz group
Infinitesimal Lorentz matrices
Commutation relations
Infinitesimal Operators	23	(7)
One-parameter group of operators
Decomposition of a representation of the group SU2 into irreducible representations
Further assumptions
Commutation relations
Representations of the Group L	30	(4)
Canonical basis
Unitarity conditions
Problems	33	(1)
Spinor Representation of the Lorentz Group	34	(20)
The Group SL(2, C) and the Lorentz Group	34	(6)
The Group SL(2, C)
Homomorphism of the group SL(2, C) on the group L
Kernel of homomorphism
Subgroups of the group SL(2, C)
Connection with Lobachevskian motion
Spinor Representation of the Group SL(2, C)	40	(10)
Spinor representation in space of polynominals
Two-component spinors
Spinor representation by means of the group SU2
Matrix elements of the spinor operator D(g)
Infinitesimal Operators of the Spinor Representation	50	(4)
One-parameter subgroups
Infinitesimal operators
Further properties of spinor representations
Problems	52	(2)
Principal Series of Representations of SL(2, C)	54	(24)
Linear Spaces of Representations	54	(9)
The Hilbert space L2(Z)
The Hilbert space L2(SU2)
The Hilbert space L22s(SU2)
Fourier transform on the group SU2
The Hilbert space l22s
Linear spaces of homogeneous functions
Other realizations of the space D(χ)
The Group Operators	63	(4)
Representation of SL(2, C) on D(χ)
Other realizations for D(g; χ)
Conjugate representations
Realization of the representation of the principal series
SU2 Description of the Principal Series	67	(6)
Properties of the principal series
Realization of the principal series by means of the group SU2
Realization of the principal series in the space l22s
The principal series as a representation for the group SU2
Functions on the group SL(2, C)
Comparison with the Infinitesimal Approach	73	(5)
Comparison of the parameters (s, ρ) and (j0, c)
Tangent space to the group SL(2, C)
Lie operators
Laplacian operators
Problems	76	(2)
Complementary Series of Representations of SL(2, C)	78	(20)
Relization of the Complementary Series	78	(4)
Complementary series
Space of representation
Value of the parameter σ
Realization of the complementary series
SU2 Description of the Complementary Series	82	(7)
The Euclidean space H
The Hilbert space H(σ)
Complementary series in the space H(σ)
Canonical basis in the space H
The normalization factor Nj
The spaces h and h(σ)
Realization of the complementary series in the space h(σ)
Comparison with the infinitesimal approach
Operator Formulation	89	(9)
Casimir operators
The z-basis of the group SL(2, C)
Unitary representations
Problems	96	(2)
Complete Series of Representations of SL(2, C)	98	(25)
Realization of the Complete Series	98	(5)
Realization of the complete series in the space L22s(SU2)
The complete series in the space l22s
Equivalence of representations
Relation to the complementary series
Condition of reducibility
Complete Series and Spinors	103	(6)
Relation to spinors
Relation between spinors and φmj
Invariant bilinear functionals
Intertwining operators
Unitary Representations Case	109	(5)
Equivalence of representations
Unitary representations
Invariant Hermitian functionals on D(χ)
Positive definite
Hermitian functionals
Unitary representations on a Hilbert space
Harmonic Analysis on the Group SL(2, C)	114	(9)
Fourier transform on the group SL(2, C)
Properties of Fourier transform on SL(2, C)
Inverse Fourier transform
Plancherel's theorem for SL(2, C)
Decomposition of the regular representation
Problems	121	(2)
Elements of General Relativity Theory	123	(45)
Riemannian Geometry	123	(7)
Transformation of coordinates
Contravariant vectors
Invariants
Covariant vectors
Tensors
Christoffel symbols
Covariant differentiation
Riemann and Ricci tensors
Geodesics
Bianchi identities
Principle of Equivalence	130	(4)
Null experiments
Eotvos experiment
Strong and weak principle of equivalence
Negative mass
Principle of General Covariance	134	(1)
Gravitational Field Equations	135	(4)
Einstein's field equations
Deduction of Einstein's equations from variational principle
Maxwell's equations in curved space
Stationary and static gravitational fields
Solutions of Einstein's Field Equations	139	(12)
Schwarzschild solution
Maximal extension of the Schwarzschild metric
Gravitational field of a point electric charge
Solution with rotational symmetry
Particle with quadrupole moment
Cylindrical gravitational waves
Experimental Tests of General Relativity	151	(7)
Gravitational red shift
Effects on planetary motion
Deflection of light
Gravitational radiation experiment
Radar experiment
Low temperature experiments
Equations of Motion	158	(10)
Geodesic postulate
Equations of motion as a consequence of field equations
Self-action terms
Einstein-Infeld-Hoffmann method
Newtonian equation of motion
Einstein-Infeld-Hoffmann equation
Problems	165	(3)
Spinors in General Relativity	168	(23)
Connection between Spinors and Tensors	168	(2)
Spinors in Riemannian space
Equivalence of spinors and tensors
Covariant derivative of spinors
Maxwell, Weyl and Riemann Spinors	170	(3)
The electromagnetic field
The gravitational field
The Weyl spinor
Ricci's and Einstein's spinors
Classification of Maxwell Spinor	173	(7)
Complex 3-space
Classification
Change of frame
Invariants
Canonical forms
Spinor method
Tensor method
Classification of Weyl Spinor	180	(11)
Complex 5-space
Classification
Change of frame
Invariants
Canonical forms
Spinor method
Tensor method
Problems	189	(2)
SL(2, C) Gauge Theory of the Gravitational Field: the Newman-Penrose Equations	191	(25)
Isotopic Spin and Gauge Fields	191	(6)
Isotopic spin
Conservation of isotopic spin and invariance
Isotopic spin and gauge fields
Isotopic gauge transformation
Field equations
Nonlinearity of the field equations
Internal holonomy of gauge fields
Lorentz Invariance and the Gravitational Field	197	(4)
Homogeneous Lorentz group and the gravitational field
Invariance of the action integral
Generalized Lorentz transformation
Free field case
Poincare invariance and the gravitational field
SL(2, C) Invariance and the Gravitational Field	201	(7)
Spin frame gauge
Potentials and fields
Spin coefficients as potentials
Symmetry of Fab'cd'
Gravitational Field Equations	208	(8)
Identities
Field equations
Gravitational Lagrangian
Conservation laws
Problems	214	(2)
Analysis of the Gravitational Field	216	(20)
Geometrical Interpretation	216	(4)
Geometrical meaning of the spin coefficients
Geometrical meaning of the Weyl spinor components
Goldberg-Sachs theorem
Choice of Coordinate System	220	(6)
Coordinate system
Tetrad
Operators
Free field equations
Maxwell-Einstein's equations
Neutrino equations
Asymptotic Behavior	226	(10)
Asymptotic behavior of Weyl spinor
Sachs peeling off theorem
Comparison with electrodynamics
Problems	233	(3)
Some Exact Solutions of the Gravitational Field Equations	236	(56)
Solutions Containing Hypersurface Orthogonal Geodesic Rays	236	(18)
Divergence, curl and shear
Robinson-Trautman solution
Newman-Tamburino solutions: spherical class
Cylindrical class
Final remarks on the Newman-Tamburino solutions
The NUT-Taub Metric	254	(11)
Tetrad system and coordinate conditions
Field equations
Coordinate and tetrad transformations
Integration of field equations
Summary of calculations
Generalized Schwarzschild metric
The groups of motion
Discussion
Type D Vacuum Metrics	265	(27)
Field equations
Solutions for ρ ≠ 0
Radial integration
Transverse equations
Choice of tetrad and coordinates
Classification of solutions
Generalized Kerr metric
Solution for ρ = 0
Resulting metrics
Discussion
Problems	287	(5)
The Bondi-Metzner-Sachs Group	292	(28)
The Bondi-Metzner-Sachs Group	292	(16)
Definition of the BMS group
The conformal group
The irreducible representation D(j1, j2)
Spin-s spherical harmonics
Spin-s weighted functions on a sphere
Simple example: Maxwell's equation
Further remarks on the spin-s spherical harmonics
Isometries
The Euclidean group
Asymptotic isometries
The Structure of the Bondi-Metzner-Sachs Group	308	(12)
Supertranslation, translation and proper subgroups of the BMS group
Normal subgroups
Lie transformation group, Lie commutator and Lie algebra
Infinitesimal transformations
Representations of the BMS group
Problems	318	(2)
Appendix A: Review of Group Theory	320	(3)
A.1 Group and Subgroup	320	(1)
A.2 Normal Subgroup and Factor Group	321	(1)
A.3 Isomorphism and Homomorphism	322	(1)
Appendix B: Basic Concepts of Representations Theory	323	(2)
B.1 Linear Operators	323	(1)
B.2 Finite-Dimensional Representation of a Group	324	(1)
B.3 Unitary Representations	324	(1)
Appendix C: Infinite-Dimensional Representations	325	(4)
C.1 Banach Space	325	(1)
C.2 Operators in Banach Space	326	(1)
C.3 General Definition of a Representation	326	(1)
C.4 Continuous Representations	327	(1)
C.5 Unitary Representations	327	(2)
Appendix D: Gravitational Field Equations	329	(5)
D.1 Gravitational Field Equations	329	(3)
D.2 Commutation Relations	332	(2)
Appendix E: Transformation Properties of the Newman-Penrose Field Variables	334	(9)
E.1 General Transformation Properties	334	(1)
E.2 Transformations under One-Parameter Subgroups	335	(1)
E.3 Transformation under Null Rotation about lμ	336	(1)
E.4 Boost in lμ -- nμ Plane and Spatial Rotation in mμ -- mμ Plane	337	(1)
E.5 Transformation under Null Rotation about nμ	338	(1)
E.6 Transformation under other Factorization	339	(4)
Bibliography	343	(33)
Index	376