Relativity, Gravitation, and Cosmology A Basic Introduction

by Cheng, Ta-Pei

• ISBN13: 9780198529569
• ISBN10: 0198529562
• Format: Hardcover
• Copyright: 2005-02-24
• Publisher: Oxford University Press

Summary

Einstein's general theory of relativity is introduced in this advanced undergraduate and beginning graduate level textbook. Topics include special relativity in the formalism of Minkowski's four-dimensional space-time, the principle of equivalence, Riemannian geometry and tensor analysis, Einstein's field equation and cosmology. The author presents the subject from the very beginning with an emphasis on physical examples and simple applications without the full tensor apparatus. One first learns how to describe curved spacetime. At this mathematically more accessible level, the reader can already study the many interesting phenomena such as gravitational lensing, precession of Mercury's perihelion, black holes, as well as cosmology. The full tensor formulation is presented later, when the Einstein equation is solved for a few symmetric cases. Many modern topics in cosmology are discussed in this book: from inflation and cosmic microwave anisotropy to the "dark energy" that propels as accelerating universe. Mathematical accessibility, together with the various pedagogical devices (e.g., worked-out solutions of chapter-end problems), make it practical for interested readers to use the book to study general relativity, gravitation and cosmology on their own.

Author Biography

Ta-Pei Cheng is Professor of Physics at the University of Missouri - St. Louis, USA.

Table of Contents

Part I RELATIVITY Metric Description of Spacetime
Introduction and overview	3	(11)
Relativity as a coordinate symmetry	5	(3)
From Newtonian relativity to aether	5	(1)
Einsteinian relativity	6	(1)
Coordinate symmetry transformations	7	(1)
New kinematics and dynamics	7	(1)
GR as a gravitational field theory	8	(6)
Einstein's motivations for the general theory	8	(2)
Geometry as gravity	10	(1)
Mathematical language of relativity	11	(1)
GR is the framework for cosmology	12	(1)
Review questions	12	(2)
Special relativity and the flat spacetime	14	(24)
Coordinate symmetries	14	(5)
Rotational symmetry	14	(2)
Newtonian physics and Galilean symmetry	16	(1)
Electrodynamics and Lorentz symmetry	17	(1)
Velocity addition rule amended	18	(1)
The new kinematics of space and time	19	(5)
Relativity of spatial congruity	20	(1)
Relativity of simultaneity---the new kinematics	20	(2)
The invariant space--time interval	22	(2)
Geometric formulation of SR	24	(14)
General coordinates and the metric tensor	24	(4)
Derivation of Lorentz transformation	28	(2)
The spacetime diagram	30	(2)
Time-dilation and length contraction	32	(3)
Review questions	35	(1)
Problems	35	(3)
The principle of equivalence	38	(17)
Newtonian gravitation potential---a review	38	(1)
EP introduced	39	(4)
Inertial mass vs. gravitational mass	40	(1)
EP and its significance	41	(2)
Implications of the strong EP	43	(12)
Gravitational redshift and time dialation	43	(5)
Light ray deflection calculated	48	(3)
Energy considerations of a gravitating light pulse	51	(1)
Einstein's inference of a curved spacetime	52	(1)
Review questions	53	(1)
Problems	53	(2)
Metric description of a curved space	55	(16)
Gaussian coordinates	56	(1)
Metric tensor	57	(6)
Geodesic as the shortest curve	59	(2)
Local Euclidean coordinates	61	(2)
Curvature	63	(8)
Gaussian curvature	63	(1)
Spaces with constant curvature	64	(2)
Curvature measures deviation from Euclidean relations	66	(2)
Review questions	68	(1)
Problems	69	(2)
GR as a geometric theory of gravity - I	71	(16)
Geometry as gravity	71	(4)
EP physics and a warped spacetime	73	(1)
Curved spacetime as gravitational field	74	(1)
Geodesic equation as GR equation of motion	75	(4)
The Newtonian limit	76	(2)
Gravitational redshift revisited	78	(1)
The curvature of spacetime	79	(8)
Tidal force as the curvature of spacetime	80	(3)
The GR field equation described	83	(2)
Review questions	85	(1)
Problems	85	(2)
Spacetime outside a spherical star	87	(28)
Description of Schwarzschild spacetime	87	(5)
Spherically symmetric metric tensor	88	(2)
Schwarzschild geometry	90	(2)
Gravitational lensing	92	(5)
Light ray deflection revisited	93	(1)
The lens equation	93	(4)
Precession of Mercury's perihelion	97	(5)
Black holes	102	(13)
Singularities of the Schwarzschild metric	102	(1)
Time measurements in the Schwarzschild spacetime	102	(3)
Lightcones of the Schwarzschild black hole	105	(3)
Orbit of an object around a black hole	108	(1)
Physical reality of black holes	108	(3)
Review questions	111	(1)
Problems	112	(3)
Part II COSMOLOGY
The homogeneous and isotropic universe	115	(21)
The cosmos observed	116	(9)
Matter distribution on the cosmic distance scale	116	(1)
Cosmological redshift: Hubble's law	116	(4)
Age of the universe	120	(1)
Dark matter and mass density of the universe	121	(4)
The cosmological principle	125	(2)
The Robertson---Walker metric	127	(9)
Proper distance in the RW geometry	129	(1)
Redshift and luminosity distance	130	(3)
Review questions	133	(1)
Problems	134	(2)
The expanding universe and thermal relics	136	(29)
Friedmann equations	137	(5)
The quasi-Newtonian interpretation	139	(3)
Time evolution of model universes	142	(3)
Big bang cosmology	145	(4)
Scale-dependence of radiation's temperature	145	(2)
Different thermal equilibrium stages	147	(2)
Primordial nucleosynthesis	149	(3)
Photon decoupling and the CMB	152	(13)
Universe became transparent to photons	153	(1)
The discovery of CMB radiation	154	(1)
Photons, neutrinos, and the radiation-matter equality time	155	(4)
CMB temperature fluctuation	159	(3)
Review questions	162	(1)
Problems	163	(2)
Inflation and the accelerating universe	165	(32)
The cosmological constant	166	(4)
Vacuum energy as source of gravitational repulsion	167	(1)
The static universe	168	(2)
The inflationary epoch	170	(8)
Initial conditions for the standard big bang model	171	(2)
The inflation scenario	173	(2)
Inflation and the conditions it left behind	175	(3)
CMB anisotropy and evidence for k = 0	178	(5)
Three regions of the angular power spectrum	179	(2)
The primary peak and spatial geometry of the universe	181	(2)
The accelerating universe in the present epoch	183	(6)
Distant supermovae and the 1998 discovery	184	(3)
Transition from deceleration to acceleration	187	(2)
The concordant picture	189	(8)
Review questions	193	(1)
Problems	193	(4)
Part III RELATIVITY Full Tensor Formulation
Tensors in special relativity	197	(18)
General coordinate systems	197	(3)
Four-vectors in Minkowski spacetime	200	(5)
Manifestly covariant formalism for E&M	205	(3)
The electromagnetic field tensor	205	(3)
Electric charge conservation	208	(1)
Energy---momentum tensors	208	(7)
Review questions	213	(1)
Problems	213	(2)
Tensors in general relativity	215	(18)
Derivatives in a curved space	215	(7)
General coordinate transformations	216	(2)
Covariant differentiation	218	(2)
Christoffel symbols and metric tensor	220	(2)
Parallel transport	222	(3)
Component changes under parallel transport	222	(2)
The geodesic as the straightest possible curve	224	(1)
Riemannian curvature tensor	225	(8)
The curvature tensor in an n-dimensional space	226	(2)
Symmetries and contractions of the curvature tensor	228	(2)
Review questions	230	(1)
Problems	231	(2)
GR as a geometric theory of gravity - II	233	(17)
The principle of general covariance	233	(3)
Geodesic equation from SR equation of motion	235	(1)
Einstein field equation	236	(3)
Finding the relativistic gravitational field equation	236	(1)
Newtonian limit of the Einstein equation	237	(2)
The Schwarzschild exterior solution	239	(5)
The Einstein equation for cosmology	244	(6)
Solution for a homogeneous and isotropic 3D space	244	(2)
Friedmann equations	246	(1)
Einstein equation with a cosmological constant term	247	(1)
Review questions	248	(1)
Problems	248	(2)
Linearized theory and gravitational waves	250	(21)
The linearized Einstein theory	251	(3)
The coordinate change called gauge transformation	252	(1)
The wave equation in the Lorentz gauge	253	(1)
Plane waves and the polarization tensor	254	(1)
Gravitational wave detection	255	(4)
Effect of gravitational waves on test particles	255	(2)
Gravitational wave interferometers	257	(2)
Evidence for gravitational wave	259	(12)
Energy flux in linearized gravitational waves	260	(2)
Emission of gravitational radiation	262	(2)
Binary pulsar PSR 1913+16	264	(4)
Review questions	268	(1)
Problems	269	(2)
A Supplementary notes	271	(12)
The twin paradox (Section 2.3.4)	271	(4)
A glimpse of advanced topics in black hole physics (Section 6.4)	275	(4)
False vacuum and hidden symmetry (Section 9.2.2)	279	(1)
The problem of A as quantum vacuum energy (Section 9.4)	280	(3)
B Answer keys to review questions	283	(10)
C Solutions of selected problems	293	(37)
References	330	(3)
Bibliography	333	(2)
Index	335