General Relativity and Cosmology with Engineering Applications

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
Part I: The General Theory of Relativity and Some of Its Applications
Chapter 1: The Special Theory of Relativity
1.1 Conflict between Newtonian mechanics and Maxwell’s theory of electromagnetism
1.2 The experiments of Michelson and Morley
1.3 Study Projects
1.4 The notion of proper time, time dilation and length contraction
1.5 The twin paradox
1.6 The equations of mechanics in special relativity
1.7 Mass, velocity, momentum and energy in special relativity, Einstein’s derivation of the energy mass relation E = mc2
1.8 Four vectors and tensors in special relativity and their Lorentz transformation laws
1.9 The general from of the Lorentz group consisting of boosts and rotations
1.10 The Poincare group consisting of Lorentz tranformations with space-time translations
1.11 Irreducible representations of the Poincare group with applications to Wigner’s particle classfication theory
1.12 Lorentz transformations of the electromagnetic field
1.13 Relative velocity in inspecial relativity
1.14 Fluid dynamics in special relativity
1.15 Plasma physics and magnetohydrodynamics in special relativity
1.16 Particle moving in a constant magnetic field in special relativity
Chapter 2: The General Theory of Relativity
2.1 Drawbacks with the special theory of relativity
2.2 The principle of equivalence
2.3 Why gravitational field is not a force ?
2.4 Four vectors and tensors in the general theory of relativity
2.5 Basics of Riemannian geometry
2.6 The energy-momentum tensor of matter in a background curved metric
2.7 Maxwell’s equations in a background curved metric
2.8 The energy-momentum tensor of the electromagnetic field in a background curved metric
2.9 The Einstein field equations of gravitation (i) In the absence of matter and radiation, (ii) In the presence of matter and radiation
2.10 Proof of the consistency of the Einstein field equations with the fluid dynamical equations based on the Bianchi identity for the Einstein tensor
2.11 The weak field limit of Einstein’s field equations is Newton’s inverse square law of gravitation
2.12 The post-Newtonian equations of celestial mechanics, gravitation and hydrodynamics
Chapter 3: Engineering Applications of General Relativity
3.1 Applications of general relativity to global positioning systems
3.2 General relativistic corrections to the Klein-Gordon wave propagation
3.3 Calculating the effect of general relativity on the motion of a plasma with applications to estimation of the metric from the radiation field produced by the plasma in motion
3.4 Problems with hints
3.5 Quantum theory of fields
3.6 Energy-momentum tensor of matter with viscous and thermal corrections
3.7 Energy-momentum tensor of the electromagnetic field in a background curved space-time
3.8 Relativistic Fermi fluid in a gravitational field
3.9 The post-Newtonian approximation
3.10 Energy-Momentum tensor of matter with viscous and thermal corrections
3.11 Energy-momentum tensor of the electromagnetic field in a background curved spacetime
3.12 Relativistic Fermi fluid in a gravitational field. The Dirac equation in a gravitational field has the form
3.13 The post-Newtonian approximation
3.14 The BCS theory of superconductivity
3.15 Quantum scattering theory in the presence of a gravitational field
3.16 Maxwell’s equations in the Schwarzchild space-time
3.17 Some more problems in general relativity
3.18 Neural networks for learning the expansion of our universe
3.19 Quantum stochastic differential equations in general relativity
Chapter 4: Some Basic Problems in Electromagnetics Related to General Relativity (gtr)
4.1 Em waves and quantum communication
4.2 Cavity resonator antennas with current source in a gravitational field
4.3 Cq coding theorem
4.4 Restricted quantum gravity in one spatial dimension and one time dimension
4.5 Quantum theory of fields
4.6 Energy-momentum tensor of matter with viscous and thermal corrections
4.7 Energy-momentum tensor of the electromagnetic field in a background curved spacetime
4.8 Relativistic Fermi fluid in a gravitational field
4.9 The post-Newtonian approximation
4.10 The BCS theory of superconductivity
4.11 Quantum scattering theory in the presence of a gravitational field
4.12 Maxwell’s equations in the Schwarzchild spacetime
4.13 Some more problems in general relativity
Chapter 5: Basic Problems in Algebra, Geometry and Differential Equations
5.1 Algebra, Triangle geometry, Integration and basic probability
5.2 Mechanics
5.3 Brownian motion simulation
5.4 Geometric series
5.5 Surface area
5.6 Hamiltonian mechanics from Lagrangians
5.7 Rate of a chemical reaction
5.8 Linearization of the Navier-Stokes Fluid equations with gravitational self interaction
5.9 Wave equations in mechanics
5.10 Surface of revolution
5.11 1-D Schrodinger equation
5.12 Lagrange’s triangle in mechanics
5.13 Number theory
5.14 Blurring of 3-D objects in random motion
5.15 Commutators of products of matrices
5.16 Path of a light ray in an medium having inhomogeneous refractive index
5.17 Re-ection matrices
5.18 Rotation matrices
5.19 Jacobian formula for multiple integrals
5.20 Existence of only five regular polyhedra in nature
5.21 Definition of the derivative and its properties
5.22 Pattern recognition using group representations
5.23 Using characters of group representations to estimate the group transformation element
5.24 Explicit formulas for the induced representation for semidirect products of finite groups
5.25 Applications of the Extended Kalman filter and the Recursive Least Squares Algorithm to System Identification Problems using Neural Networks
5.26 Application of neural networks to the gravitational metric estimation problem
5.27 Problems in quantum scattering theory
5.28 Compact operators
5.29 Estimating the metric parameters from geodesic measurements
5.30 Perturbations to the band structure of semiconductors
5.31 Scattering into cones for Schrodinger Hamiltonians
5.32 Study projects involving conventional field theory in curved background metrics
5.33 Intuitive explanation of an invariance principle in scattering theory
5.34 Scattering theory for the Dirac Hamiltonian in curved space-time
5.35 Derivation of the approximate Schrodinger Hamiltonian for a particle in curved spacetime with corrections upto fourth order in the space derivatives
5.36 Quantum scattering theory in the presence of time dependent Hamiltonians arising in general relativity
5.37 Band structure of a semiconductor altered by a massive gravitational field
5.38 Design of quantum gates using quantum physical systems in a gravitational field
5.39 Quantum phase estimation
5.40 Noisy Schrodinger equations, pure and mixed states
5.41 Constructions using ruler and compass
5.42 Application of the Jordan canonical form for matrices in general relativity
5.43 Application of the Jordan canonical form in solving fluid dynamical equations when the velocity field is a small perturbation of a constant velocity field
5.44 The Jordan canonical form
5.45 Some topics in scattering theory in L2(Rn)
5.46 MATLAB problems on applications of linear algebra to signal processing
5.47 Applications of the RLS lattice algorithms to general relativity
5.48 Knill-Laflamme theorem on quantum coding theory, a different proof
5.49 Ashtekar’s quantization of gravity
5.50 Example of an error correcting quantum code from quantum mechanics
5.51 An application of the Jordan canonical form to noisy quantum theory
5.52 An algorithm for computing the Jordan canonical form
5.53 Rotating blackhole analysis using the tetrad formalism
5.54 Maxwell’s equations in the rotating blackhole metric
5.55 Some notions on operators in an infinite/finite dimensional Hilbert space
5.56 Some versions of the quantum Boltzmann equation
Part II: Quantum Mechanics
1 The De-Broglie Duality of particle and wave properties of matter
2 Bohr’s correspondence principle
3 Bohr-Sommerfeld’s quantization rules
4 The principle of superposition of wave functions and its application to the Young double slit diffraction experiment
5 Schrodinger’s wave mechanics and Heisenberg’s matrix mechanics
6 Dirac’s replacement of the Poisson bracket by the quantum Lie bracket
7 Duality between the Schrodinger and Heisenberg mechanics based on Dirac’s idea
8 Quantum dynamics in Dirac’s interaction picture
9 The Pauli equation: Incorporating spin in the Schrodinger wave equation in the presence of a magnetic field
10 The Zeeman effect
11a The spectrum of the Hydrogen atom
11b The spectrum of particle in a 3 − D box
11c The spectrum of a quantum harmonic oscillator
12 Time independent perturbation theory
13 Time dependent perturbation theory
14 The full Dyson series for the evolution operator of a quantum system in the presence of a time varying potential
15 The transition probabilities in the presence of a stochastically time varying potential
16 Basics of quantum gates and their realization using perturbed quantum systems
17 Bounded and unbounded linear operators in a Hilbert space
18 The spectral theorem for compact normal and bounded and unbounded self-adjoint operators in a Hilbert space
19 The general theory of Events, states and observables in the quantum theory
20 The evolution of the density operator in the absence of noise
21 The Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation for noisy quantum systems
22 Distinguishable and indistinguishable particles
23 The relationship between spin and statistics
24(a) Tensor products of Hilbert spaces
24(b) Symmetric and antisymmetric tensor products of Hilbert spaces, the Fock spaces
24(c) Coherent/exponential vectors in the Fock spaces
25 Creation, Conservation and Annihilation Operators in the Boson Fock Space
26 The general theory of quantum stochastic processes in the sense of Hudson and Parthasarathy
27 The quantum Ito formula of Hudson and Parthasarathy
28 The general theory of quantum stochastic differential equations
29 The Hudson-Parthasarathy noisy Schrodinger equation and the derivation of the GKSL equation from its partial trace
30 The Feynman path integral for solving the Schrodinger equation
31 Comparison between the Hamiltonian (Schrodinger-Heisenberg) and Lagrangian (path integral) approaches to quantum mechanics
32 The quantum theory of fields
33 Dirac’s wave equation in a gravitational field
34 Canonical quantization of the gravitational field
35 The scattering matrix for the interaction between photons, electrons, positrons and gravitons
36 Atom interacting with a Laser
37 The classical and quantum Boltzmann equations
38 Bands in a semiconductor
39 The Hartree-Fock apporoximate method for computing the wave functions of a many electron atom
40 The Born-Oppenheimer approximate method for computing the wave functions of electrons and nuclei in a lattice
41 The performance of quantum gates in the presence of classical and quantum noise
42 Design of quantum gates by applying a time varying electromagnetic field on atoms and oscillators
43 Solution of Dirac’s equation in the Coulomb potential
44 Dirac’s equation in general radial potentials
45 The Schrodinger equation in an electromangetic field described as a quantum stochastic process
46 Dirac’s equation in an electromagnetic field described as a quantum stochastic process
47 General Scattering theory, the Moller and wave operators, the scattering matrix, the Lippman-Schwinger equation for the scattering matrix, Born scattering
48 Design of quantum gates using time dependent scattering theory
49 Evans-Hudson flows and its application to the quantization of the fluid dynamical equations in noise
50 Classical non-linear filtering
51 Derivation of the extended Kalman filter (EKF) as an approximation to the Kushner filter
52 Belavkin’s theory of non-demolition measurements and quantum filtering in coherent states based on the Hudson- Parthasarathy Boson Fock space theory of quantum noise, The quantum Kallianpur-Striebel formula
53 Classical control of a stochastic dynamical system by error feedback based on a state observer derived from the EKF
54 Quantum control using error feedback based on Belavkin quantum filters for the quantum state observer
55 Lyapunov’s stability theory with application to classical and quantum dynamical systems
56 Imprimitivity systems as a description of covariant observables under a group action
57 Schwinger’s analysis of the interaction between the electron and a quantum electromagnetic field
58 Quantum Control
59 Quantum error correcting codes
60 Quantum hypothesis testing
61 The Sudarshan-Lindblad equation for observables in an open quantum system
62 The Yang-Mills field and its quantization using path integrals
63 A general remark on path integral computations for gauge invariant actions
64 Calculation of the normalized spherical harmonics
65 Volterra systems in quantum mechanics
66a RLS lattice algorithms for quantum observable estimation
66b Quantum scattering theory, the wave operators and the scattering matrix
67 Quantum systems driven by Stroock-Varadhan martingales
Appendix
References
Index