General Relativity, Astrophysics, and Cosmology

by Raychaudhuri, A. K.; Banerji, S.; Banerjee, A.

• ISBN13: 9780387406282
• ISBN10: 038740628X
• Format: Paperback
• Copyright: 2003-12-01
• Publisher: Springer Verlag

Summary

This book is intended for students interested in the applications of general relativity in astrophysics and cosmology but who would like to avoid mathematical complications. This volume thus combines relativity, astrophysics, and cosmology in a single volume. It provides an introduction to the subject that will enable students to consult more detailed treatises as well as the current literature. For prospective researchers in these fields, the book includes an appendix on differential forms, and an extensive, though not exhaustive list of references. The book is divided into three parts. The section on general relativity gives the case for a curved space-time, presents the mathematical background (tensor calculus, Riemannian geometry), discusses the Einstein equation and its solutions (including black holes, Penrose processes, and similar topics), and considers the energy-momentum tensor for various solutions. The section on relativistic astrophysics discusses stellar contraction and collapse, neutron stars and their equations of state, black holes, and accretion onto collapsed objects. The section on cosmology discusses various cosmological models, observational tests, and scenarios for the early universe.

Table of Contents

Preface	v
PART I. The General Theory of Relativity	1	(120)
Introduction	3	(6)
The Case for Nonflat Space--Time	3	(1)
The Principle of Equivalence	4	(2)
Conflict Between the Equivalence Principle and the Pseudo-Euclidean Metric: Gravitational Redshift	6	(2)
A Fifth Force	8	(1)
Tensor Calculus and Riemannian Geometry	9	(26)
Riemannian Geometry and the Metric Tensor	9	(1)
Vectors and Tensors	10	(6)
Invariant Volume and Volume Integral	16	(1)
Affine Connection---Parallel Transport	17	(3)
Covariant Differentiation	20	(3)
The Differential Equation of a Geodesic	23	(2)
The Integrability of Parallel Displacement	25	(3)
The Riemann--Christoffel Tensor	28	(1)
The Bianchi Identity	29	(1)
The Ricci Tensor and the Einstein Tensor	29	(1)
The Weyl Tensor	30	(2)
Geodesic Deviation	32	(3)
Einstein's Field Equations	35	(17)
Einstein's Formulation of the Field Equations	35	(1)
Weak Field Approximation (Static Case)	36	(2)
Gravitational Waves in Weak Field Approximation	38	(2)
Detection of Gravitational Waves	40	(1)
Integration of the Linearized Equations for a Stationary Axially Symmetric Distribution	41	(4)
The Action Principle and the Energy--Momentum Tensors	45	(2)
The Energy--Stress Tensor	47	(2)
The Einstein Equations from the Variational Principle	49	(3)
The Schwarzschild Metric and Crucial Tests	52	(27)
The Schwarzschild Solution	52	(2)
Birkhoff's Theorem	54	(1)
Three Crucial Tests	55	(10)
The PPN Formalism	65	(4)
The Schwarzschild or the Spherically Symmetric Black Hole	69	(2)
Frequency Shift of Spectral Lines of Light Emitted by a Collapsing/Exploding Spherical Body	71	(2)
Fall in Apparent Luminosity of a Collapsing Body	73	(1)
Kruskal--Szekeres Coordinates	74	(2)
Historical Note on the Schwarzschild Black Hole	76	(3)
Electromagnetism in General Relativity	79	(8)
Introduction	79	(1)
The Field of a Charged Particle	80	(2)
Static Electrovac	82	(1)
The Already Unified Field Theory	83	(4)
Axially Symmetric Fields	87	(11)
The Lie Derivative and the Killing Equation	87	(2)
Static and Stationary Metrics	89	(1)
The Axially Symmetric Static Metric	90	(1)
Weyl's Canonical Form	91	(2)
The Case of Two Mass Particles	93	(2)
The Schwarzschild Metric in the Form (6.21)	95	(1)
Stationary Axisymmetric Vacuum Solutions	96	(2)
The Kerr Metric or the Rotating Black Hole	98	(7)
The Kerr Metric in Boyer--Lindquist Coordinates	98	(1)
The Black Hole Property	99	(1)
Locally Nonrotating Observers	100	(1)
The Horizon as a Null Surface	100	(2)
The Kerr--Newmann Metric	102	(1)
The Penrose Process	102	(3)
The Energy--Momentum Pseudotensor of the Gravitational Field and Loss of Energy by Gravitational Radiation	105	(9)
The Pseudo-Energy--Momentum Tensor	105	(2)
Historical Note	107	(1)
Loss of Energy by Gravitational Radiation	108	(3)
The Case of a Binary Star	111	(3)
Analysis of the Observational Data of the Hulse--Taylor Pulsar. Confirmation of the Einstein Quadrupole Radiation Formula	114	(7)
PART II. Relativistic Astrophysics	121	(86)
White Dwarf Stars	123	(15)
Introduction	123	(1)
The Contraction of a Radiating Star in the Absence of Energy Generation	123	(2)
Degeneracy and the Equation of State	125	(3)
Limiting Mass for White Dwarfs	128	(1)
A Simple Argument for the Mass Limit	129	(1)
Critique of Chandrasekhar's Result and Later Works	130	(1)
Historical Note	131	(1)
Observational Data on White Dwarfs	132	(1)
The Cooling and Age of White Dwarfs	132	(6)
Stellar Evolution, Supernovae, and Compact Objects	138	(6)
Introduction	138	(1)
The Evolution of Stars	138	(2)
The Dynamical Collapse	140	(1)
Some Numerical Results	141	(1)
Explosive Processes	141	(2)
Supernova 1987 A	143	(1)
Pulsars	144	(15)
Introduction	144	(1)
Distance from Dispersion Measure	145	(2)
Identification of Pulsars as Neutron Stars	147	(1)
The Energetics of Pulsar Emission	148	(1)
The Magnetic Field at the Pulsar Surface	149	(1)
The Age of Pulsars	150	(1)
Calculation of the Braking Index	150	(1)
The Nonvacuum Model	151	(2)
Observational Determination of Pulsar Masses	153	(1)
Cooling of Neutron Stars---Theory and Observation	153	(2)
The Influence of Superfluidity	155	(1)
The Influence of Pion Condensation	155	(1)
The Influence of Quarks	156	(3)
Spherically Symmetric Star Models	159	(16)
Introduction	159	(1)
The Tolman, Oppenheimer--Volkoff Equation	160	(1)
The Equation of State for Cold Catalyzed Matter	161	(3)
A Model of a Neutron Star and the Mass Limits	164	(3)
The Problems of the Upper Mass Limit of Neutron Stars	167	(4)
The Influence of Rotation, etc., on the Mass Limit	171	(1)
Note on the Stability of Compact Objects	172	(3)
Black Holes	175	(17)
Introduction	175	(1)
The No-Hair Theorem	175	(2)
The Laws of Black Hole Physics	177	(1)
Black Hole Thermodynamics	178	(2)
The Identification of a Black Hole---Cygnus X-1	180	(3)
The Possible Locale of the Occurrence of Black Holes	183	(1)
The Quasi-Steller Objects (Quasars)	184	(1)
Gravitational Lens	185	(7)
Accretion onto Compact Objects	192	(15)
Introduction---Spherically Symmetric Accretion	192	(7)
Disk Accretion	199	(4)
Compact X-Ray Sources	203	(4)
PART III. Cosmology	207	(71)
The Standard Cosmological Model	209	(19)
Introduction to the Friedmann Metric	209	(4)
Elementary Discussion of Standard Cosmology	213	(8)
The Observational Background of Cosmology	221	(5)
Summary	226	(2)
The Singularity Problem	228	(7)
Introduction	228	(1)
The Raychaudhuri Equation	228	(1)
The Meaning of Shear, Vorticity, and Expansion	229	(1)
An Elementary Singularity Theorem	230	(1)
The Godel Universe	231	(1)
General Singularity Theorems	232	(3)
Thermal History of the Universe---Cosmological Nucleosynthesis	235	(8)
The Thermal History	235	(4)
Cosmological Nucleosynthesis	239	(4)
Structure Formation in the Universe	243	(10)
The Problem	243	(1)
The Linear Growth Formula	244	(5)
Finite Perturbation	249	(1)
Structure Formation with Dark Matter	250	(3)
Grand Unified Theory and Spontaneous Symmetry Breaking	253	(11)
Introduction	253	(1)
Gauge Fields	253	(1)
Weak Interaction	254	(1)
Strong Interaction and Grand Unification	255	(5)
Baryon Asymmetry and the Baryon/Photon Ratio	260	(4)
The Inflationary Scenario	264	(11)
Introduction	264	(1)
The Problems in Terms of Entropy	265	(1)
The Vacuum Energy-Stress Tensor and the de Sitter Phase	266	(1)
The Different Models of Inflation	267	(3)
A Critique of the Inflationary Models	270	(1)
Fluctuations in the Inflationary Models	270	(5)
Concluding Remarks	275	(3)
Appendix. Differential Forms	278	(7)
Introductory Ideas and Definitions	278	(2)
Connection 1-Forms and Ricci Rotation Coefficients	280	(1)
Cartan's Equations of Structure	281	(1)
Bianchi Identities and Symmetry Properties of the Riemann--Christoffel Tensor	282	(1)
An Example of the Calculation of the Riemann--Christoffel Tensor	282	(3)
References	285	(4)
Bibliography	289	(4)
Index	293