|
|
xiii | |
|
|
xix | |
Preface |
|
xxi | |
How to use this book |
|
xxvii | |
|
|
1 | (53) |
|
|
1 | (1) |
|
The principles of special relativity |
|
|
1 | (5) |
|
Transformation of coordinates and velocities |
|
|
6 | (7) |
|
|
8 | (2) |
|
Transformation of velocities |
|
|
10 | (1) |
|
Lorentz boost in an arbitrary direction |
|
|
11 | (2) |
|
|
13 | (6) |
|
Four-velocity and acceleration |
|
|
17 | (2) |
|
|
19 | (4) |
|
Tensors as geometrical objects |
|
|
23 | (3) |
|
Volume and surface integrals in four dimensions |
|
|
26 | (3) |
|
|
29 | (6) |
|
The distribution function and its moments |
|
|
35 | (10) |
|
The Lorentz group and Pauli matrices |
|
|
45 | (9) |
|
Scalar and electromagnetic fields in special relativity |
|
|
54 | (53) |
|
|
54 | (1) |
|
|
54 | (1) |
|
|
55 | (11) |
|
Dynamics of a particle interacting with a scalar field |
|
|
55 | (2) |
|
Action and dynamics of the scalar field |
|
|
57 | (3) |
|
Energy-momentum tensor for the scalar field |
|
|
60 | (2) |
|
Free field and the wave solutions |
|
|
62 | (2) |
|
Why does the scalar field lead to an attractive force? |
|
|
64 | (2) |
|
|
66 | (9) |
|
Charged particle in an electromagnetic field |
|
|
67 | (4) |
|
Lorentz transformation of electric and magnetic fields |
|
|
71 | (2) |
|
|
73 | (2) |
|
Motion in the Coulomb field |
|
|
75 | (4) |
|
Motion in a constant electric field |
|
|
79 | (2) |
|
Action principle for the vector field |
|
|
81 | (2) |
|
|
83 | (7) |
|
Energy and momentum of the electromagnetic field |
|
|
90 | (5) |
|
Radiation from an accelerated charge |
|
|
95 | (5) |
|
Larmor formula and radiation reaction |
|
|
100 | (7) |
|
Gravity and spacetime geometry: the inescapable connection |
|
|
107 | (29) |
|
|
107 | (1) |
|
Field theoretic approaches to gravity |
|
|
107 | (1) |
|
Gravity as a scalar field |
|
|
108 | (5) |
|
Second rank tensor theory of gravity |
|
|
113 | (12) |
|
The principle of equivalence and the geometrical description of gravity |
|
|
125 | (11) |
|
Uniformly accelerated observer |
|
|
126 | (2) |
|
Gravity and the flow of time |
|
|
128 | (8) |
|
Metric tensor, geodesics and covariant derivative |
|
|
136 | (53) |
|
|
136 | (1) |
|
Metric tensor and gravity |
|
|
136 | (5) |
|
Tensor algebra in curved spacetime |
|
|
141 | (5) |
|
Volume and surface integrals |
|
|
146 | (3) |
|
|
149 | (13) |
|
Properties of geodesic curves |
|
|
154 | (2) |
|
Affine parameter and null geodesics |
|
|
156 | (6) |
|
|
162 | (8) |
|
Geometrical interpretation of the covariant derivative |
|
|
163 | (4) |
|
Manipulation of covariant derivatives |
|
|
167 | (3) |
|
|
170 | (3) |
|
Lie transport and Killing vectors |
|
|
173 | (8) |
|
|
181 | (8) |
|
|
189 | (50) |
|
|
189 | (1) |
|
Three perspectives on the spacetime curvature |
|
|
189 | (11) |
|
Parallel transport around a closed curve |
|
|
189 | (3) |
|
Non-commutativity of covariant derivatives |
|
|
192 | (4) |
|
Tidal acceleration produced by gravity |
|
|
196 | (4) |
|
Properties of the curvature tensor |
|
|
200 | (8) |
|
|
200 | (3) |
|
|
203 | (1) |
|
Ricci tensor, Weyl tensor and conformal transformations |
|
|
204 | (4) |
|
Physics in curved spacetime |
|
|
208 | (16) |
|
Particles and photons in curved spacetime |
|
|
209 | (1) |
|
Ideal fluid in curved spacetime |
|
|
210 | (7) |
|
Classical field theory in curved spacetime |
|
|
217 | (4) |
|
Geometrical optics in curved spacetime |
|
|
221 | (3) |
|
Geodesic congruence and Raychaudhuri's equation |
|
|
224 | (7) |
|
|
225 | (3) |
|
|
228 | (2) |
|
Integration on null surfaces |
|
|
230 | (1) |
|
Classification of spacetime curvature |
|
|
231 | (8) |
|
Curvature in two dimensions |
|
|
232 | (1) |
|
Curvature in three dimensions |
|
|
233 | (1) |
|
Curvature in four dimensions |
|
|
234 | (5) |
|
Einstein's field equations and gravitational dynamics |
|
|
239 | (54) |
|
|
239 | (1) |
|
Action and gravitational field equations |
|
|
239 | (22) |
|
Properties of the gravitational action |
|
|
242 | (2) |
|
Variation of the gravitational action |
|
|
244 | (3) |
|
A digression on an alternative form of action functional |
|
|
247 | (3) |
|
Variation of the matter action |
|
|
250 | (8) |
|
Gravitational field equations |
|
|
258 | (3) |
|
General properties of gravitational field equations |
|
|
261 | (7) |
|
The weak field limit of gravity |
|
|
268 | (11) |
|
Metric of a stationary source in linearized theory |
|
|
271 | (5) |
|
Metric of a light beam in linearized theory |
|
|
276 | (3) |
|
Gravitational energy-momentum pseudo-tensor |
|
|
279 | (14) |
|
Spherically symmetric geometry |
|
|
293 | (47) |
|
|
293 | (1) |
|
Metric of a spherically symmetric spacetime |
|
|
293 | (20) |
|
Static geometry and Birkoff's theorem |
|
|
296 | (8) |
|
Interior solution to the Schwarzschild metric |
|
|
304 | (7) |
|
Embedding diagrams to visualize geometry |
|
|
311 | (2) |
|
Vaidya metric of a radiating source |
|
|
313 | (1) |
|
Orbits in the Schwarzschild metric |
|
|
314 | (15) |
|
Precession of the perihelion |
|
|
318 | (5) |
|
Deflection of an ultra-relativistic particle |
|
|
323 | (3) |
|
Precession of a gyroscope |
|
|
326 | (3) |
|
Effective potential for orbits in the Schwarzschild metric |
|
|
329 | (5) |
|
Gravitational collapse of a dust sphere |
|
|
334 | (6) |
|
|
340 | (59) |
|
|
340 | (1) |
|
Horizons in spherically symmetric metrics |
|
|
340 | (3) |
|
Kruskal---Szekeres coordinates |
|
|
343 | (15) |
|
Radial infall in different coordinates |
|
|
350 | (6) |
|
General properties of maximal extension |
|
|
356 | (2) |
|
Penrose---Carter diagrams |
|
|
358 | (7) |
|
Rotating black holes and the Kerr metric |
|
|
365 | (16) |
|
Event horizon and infinite redshift surface |
|
|
368 | (4) |
|
|
372 | (2) |
|
Penrose process and the area of the event horizon |
|
|
374 | (4) |
|
Particle orbits in the Kerr metric |
|
|
378 | (3) |
|
Super-radiance in kerr geometry |
|
|
381 | (4) |
|
Horizons as null surfaces |
|
|
385 | (14) |
|
|
399 | (53) |
|
|
399 | (1) |
|
Propagating modes of gravity |
|
|
399 | (3) |
|
Gravitational waves in a flat spacetime background |
|
|
402 | (11) |
|
Effect of the gravitational wave on a system of particles |
|
|
409 | (4) |
|
Propagation of gravitational waves in the curved spacetime |
|
|
413 | (3) |
|
Energy and momentum of the gravitational wave |
|
|
416 | (6) |
|
Generation of gravitational waves |
|
|
422 | (12) |
|
Quadrupole formula for the gravitational radiation |
|
|
427 | (2) |
|
Back reaction due to the emission of gravitational waves |
|
|
429 | (5) |
|
General relativistic effects in binary systems |
|
|
434 | (18) |
|
Gravitational radiation from binary pulsars |
|
|
434 | (4) |
|
Observational aspects of binary pulsars |
|
|
438 | (5) |
|
Gravitational radiation from coalescing binaries |
|
|
443 | (9) |
|
|
452 | (50) |
|
|
452 | (1) |
|
|
452 | (5) |
|
Kinematics of the Friedmann model |
|
|
457 | (9) |
|
The redshifting of the momentum |
|
|
458 | (3) |
|
Distribution functions for particles and photons |
|
|
461 | (1) |
|
|
462 | (4) |
|
Dynamics of the Friedmann model |
|
|
466 | (13) |
|
|
479 | (4) |
|
Brief thermal history of the universe |
|
|
483 | (4) |
|
Decoupling of matter and radiation |
|
|
484 | (3) |
|
|
487 | (6) |
|
Killing vectors and the symmetries of the space |
|
|
493 | (9) |
|
Maximally symmetric spaces |
|
|
494 | (2) |
|
|
496 | (6) |
|
Differential forms and exterior calculus |
|
|
502 | (28) |
|
|
502 | (1) |
|
|
502 | (8) |
|
|
510 | (3) |
|
|
513 | (3) |
|
|
516 | (3) |
|
Spin connection and the curvature 2-forms |
|
|
519 | (11) |
|
Einstein-Hilbert action and curvature 2-forms |
|
|
523 | (3) |
|
Gauge theories in the language of forms |
|
|
526 | (4) |
|
Hamiltonian structure of general relativity |
|
|
530 | (30) |
|
|
530 | (1) |
|
Einstein's equations in (1+3)-form |
|
|
530 | (5) |
|
Gauss---Codazzi equations |
|
|
535 | (5) |
|
Gravitational action in (1+3)-form |
|
|
540 | (12) |
|
The Hamiltonian for general relativity |
|
|
542 | (3) |
|
The surface term and the extrinsic curvature |
|
|
545 | (2) |
|
Variation of the action and canonical momenta |
|
|
547 | (5) |
|
|
552 | (8) |
|
Collapse of a dust sphere and thin-shell |
|
|
554 | (6) |
|
Evolution of cosmological perturbations |
|
|
560 | (31) |
|
|
560 | (1) |
|
Structure formation and linear perturbation theory |
|
|
560 | (1) |
|
Perturbation equations and gauge transformations |
|
|
560 | (12) |
|
Evolution equations for the source |
|
|
560 | (12) |
|
Perturbations in dark matter and radiation |
|
|
572 | (10) |
|
Evolution of modes with λ» dH |
|
|
573 | (1) |
|
Evolution of modes with λ« dH in the radiation dominated phase |
|
|
574 | (3) |
|
Evolution in the matter dominated phase |
|
|
577 | (1) |
|
An alternative description of the matter---radiation system |
|
|
578 | (4) |
|
Transfer function for the matter perturbations |
|
|
582 | (2) |
|
Application: temperature anisotropies of CMBR |
|
|
584 | (7) |
|
|
586 | (5) |
|
Quantum field theory in curved spacetime |
|
|
591 | (52) |
|
|
591 | (1) |
|
Review of some key results in quantum field theory |
|
|
591 | (11) |
|
Bogolyubov transformations and the particle concept |
|
|
596 | (2) |
|
Path integrals and Euclidean time |
|
|
598 | (4) |
|
Exponential redshift and the thermal spectrum |
|
|
602 | (3) |
|
Vacuum state in the presence of horizons |
|
|
605 | (4) |
|
Vacuum functional from a path integral |
|
|
609 | (9) |
|
Hawking radiation from black holes |
|
|
618 | (7) |
|
Quantum field theory in a Friedmann universe |
|
|
625 | (6) |
|
|
625 | (3) |
|
Application: power law expansion |
|
|
628 | (3) |
|
Generation of initial perturbations from inflation |
|
|
631 | (12) |
|
|
632 | (2) |
|
Perturbations in the inflationary models |
|
|
634 | (9) |
|
Gravity in higher and lower dimensions |
|
|
643 | (27) |
|
|
643 | (1) |
|
Gravity in lower dimensions |
|
|
644 | (2) |
|
Gravity and black hole solutions in(1+2) dimensions |
|
|
644 | (2) |
|
Gravity in two dimensions |
|
|
646 | (1) |
|
Gravity in higher dimensions |
|
|
646 | (7) |
|
Black holes in higher dimensions |
|
|
648 | (1) |
|
|
648 | (5) |
|
|
653 | (10) |
|
Surface term and the entropy of the horizon |
|
|
663 | (7) |
|
Gravity as an emergent phenomenon |
|
|
670 | (19) |
|
|
670 | (1) |
|
The notion of an emergent phenomenon |
|
|
671 | (2) |
|
Some intriguing features of gravitational dynamics |
|
|
673 | (6) |
|
Einstein's equations as a thermodynamic identity |
|
|
673 | (3) |
|
Gravitational entropy and the boundary term in the action |
|
|
676 | (1) |
|
Horizon thermodynamics and Lanczos-Lovelock theories |
|
|
677 | (2) |
|
An alternative perspective on gravitational dynamics |
|
|
679 | (10) |
Notes |
|
689 | (6) |
Index |
|
695 | |